Vector Valued Nevanlinna Theory (Research Notes Inmathematics Series)

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Research Not s in Mathematics

H JW Ziegler

Vector valued Nevanlinna Theory

Pitman Advanced Publishing Program BOSTON· LONDON · MELBOURNE

73

H JW Ziegler University of Siegen

Tvector valued Nevanlinna Theory

Pitman Advanced Publishing Program BOSTON· LONDON· MELBOURNE

PIlMAN BOOKS LIMITED 128 Long Acre, London WCZE 9AN PIlMAN PUBLISHING INC 1020 Plain Street, Marshfield, Massachusetts 02050

Associated Companies Pitman Publishing Pty Ltd, Melbourne Pitman Publishing New Zealand Ltd, Wellington Copp Clark Pitman, Toronto First published 1982

© H J W Ziegler 1982 AMS Subject Classifications: (main) 30, 3OC, 30D (subsidiary) 31,32,53 Library of Congress Cataloging in Publication Data Ziegler, H. J. W. (Hans J. W.) Vector valued Nevanlinna theory. (Research notes in mathematics; 73) Bibliography: p. Includes index. 1. Functions, Meromorphic. 2. Value distribution theory. 3. Nevanlinna theory. I. Title. II. Series. QA331.Z53 1982 515.9'82 82-13202 ISBN 0-273-68530-1 British Library Cataloguing in Publication Data Ziegler, H. J. W. Vector valued Nevanlinna theory.-(Research notes in mathematics; v. 73) 1. Functional analysis 2. Vector-valued measures I. Title II. Series 515.7 QA320 ISBN 0-273-08530-1 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording and/or otherwise without the prior written permission of the publishers. The paperback edition of this book may not be lent, resold, hired out or otherwise disposed of by way of trade in any form of binding or cover other than that in which it is published, witbout the prior consent of tbe publishers. Reproduced and printed by photolithography in Great Britain by Biddies Ltd, Guildford

To RENATE

Contents

Preface

1 Extension of the First Main Theorem of Nevanlinna Theory and Interpretation by Hermitian Geometry

§ 1

Generalization of the Formula of Poisson-Jensen-Nevanlinna

1

§ 2

Interpretation by Hermitian Geometry

9

§ 3

The Generalized First Main Theorem

16

§ 4

The Example of Rational Vector Functions

27

2 Some Quantities arising in the Vector Valued Theory and their Relation

34

to N evanlinna Theory § 5

Properties of

V(r,a)

§ 6

Properties of

T(r ,f)

§ 7

The Connection of

§ 8

T(r,f.) ,m(r,f.) and J J The Order of Growth

34 39

T(r,f) ,m(r,f)

and

N(r,f)

with

N(r,f.) J

45

51

3 Generalization of the Ahlfors-Shimizu Characteristic and its Connection

58

with Hermitian Geometry

o

T(r,f)

58

§ 9

The Generalized Ahlfors-Shimizu Characteristic

§ 10

The Generalized Riemann Sphere

§ll

The Spherical Normal Form of the Generalized First Main Theorem 73

§12

The Mean Value Representation of

68

o

T(r,f)

75

82

4 Additional Results of the Elementary Theory

§13

The Genus of a Meromorphic Vector Function

§14

Some Relations between

M,m; N,n ; V,v

and

82 A

89 vii

5 Extension of the Second Main Theorem of Nevanlinna Theory

110

§15

The Generalized Second Main Theorem

110

§16

The Generalized Deficiency Relation

129

§17

Further Results about Deficiencies

141

Appendix: Rudiments of Complex Manifolds and Hermitian Geometry

168

Bibliography

189

Table of Symbols

197

Index

199

viii

Preface

= f (z) be a meromorphic function in the Gaussian complex plane C. 1 Nevanlinna theory or the theory of value distribution gives answers to the

Let

w

1

question of how densely the solutions of the equation (z E C ,01 1 E CU{oo})

are distributed over

C; it also studies the mean approximation of the func-

tion

f 1 (z)

to the value

a1

along large concentric circles around the ori-

gin

z =0

a problem which turns out to be equivalent to the former.

Nevanlinna theory originates from a general formula of F. and R. Ncvanlinna [45], by which they were developing a general method for the investigation of meromorphic functions. This formula includes both the Poisson formula and the Jensen formula as special cases, and in its most important form it expresses the logarithm of the modulus of an arbitrary meromorphic function by the boundary values of the function along a concentric circle around the origin and the zeros and poles of the function inside this circle. Nevanlinna theory was created at the moment when Rolf Nevanlinna gave the formula an ingenious interpretation, This happened about 1924. The most general result of Nevanlinna theory can be summarized by saying that the distribution of the solutions to the equation

f 1 (z)

= a1

is extremely uniform for almost all values of

a1 ;

there can only exist a small minority of values which the function takes relatively rarely. The investigation of these exceptional values constitutes the main task of value distribution theory in the sense of Nevanlinna. The earlier value distribution theory before Nevanlinna can be traced back to the year 1876, when K.Weierstrass [57] showed that in the vicinity of an isolated essential singularity a meromorphic function given value

a1

f 1 (z)

approaches every

arbitrarily closely. In 1879 E.Picard [50] even proved the

surprising fact that a meromorphic function takes in the vicinity of an isolated essential singularity every finite or infinite value

a1

with 2 exceptions at

the most. Points which are not taken are now called Picard exceptional values of the function. The results which were found after by the mathematicians E. Laguerre, H.Poincare, J.Hadamard, E.Borel and others revealed that in spite ix

of the possible existence of Picard exceptional values the distribution of zeros or, more generally, the distribution of a -points of an entire function is controlled, at least in some sen se, by the growth behaviour of the maximum modulus function max !f 1 (z) I Izl= r which has the function of a transcendental analogue of the degree of a polynomial. This approach of early value distrib ution theory breaks down, however, if

f 1 (z)

is meromorphic, since then

has a pole on the circle

M(r,t 1 )

becomes infinite if

f 1 (z)

Izl = r . An attempt by E.Borel [3J himself of in-

cluding meromorphic functions in this framework was not very successful. In Nevanlinna theory the role of valued function

logM(r,f 1 )

is taken by an, increasing real

T(r, f 1) , the "Nevanlinna characteristic function" which is

associated to the given meromorphic function

f 1 (z) . A great deal of work

had been done in establishing the relationship between distribution of values and growth when Rolf Nevanlinna created his epoque making theory. This theory, which applies to entire functions, as well as to meromorphic functions, even improved tremendously the earlier value distribution theory of entire functions. There have been many attempts to extend the Nevanlinna theory in several directions. Besides older investigations of E.Borel, A.Bloch, R.Nevanlinna and H.Cartan the most important of these, known as the theory of holomorphic or meromorphic curves, was initiated by H. and J.Weyl [43] in 1938; the most difficult problem of this extension, the proof of the defect relation for holomorphic curves, was solved by L.Ahlfors [2] ; recently a very modern treatment of this theory was given by H. Wu [46 J • In its most simple form this theory investigates the distribution of the zeros of linear combinations A f (z) + •.• + A f (z) 00 nn

of finitely many integral functions stan t multipliers

A = (A

0'

... ,A) n

w. = f.(z) for different systems of conJ ] or, in other words, this theory analy zes

the position of a non-degenerate meromorphic curve

C -

pn

relative to

the hyperplanes

A w + ... + A w = 0 in complex projective space pn. o 0 n n The theory of holomorphic curves by Weyl-Ahlfors was further extended in a very general way to a higher dimensional theory first by W. Stoll [56] and then in a different direction, stressing Hermitian differential geometric aspects, by

x

H.I.Levine [43], S.S.Chern nO], R.Bott and S.S.Chern [7] and other authors. In 1972, introducing once more fascinating new ideas, the Ahlfors-Weyl theory was extended in a different direction, more regarding to algebraic ge ometry, by J . Carlson and P . Griffiths [IS] to equidimensional holomorphic mappings Cm_ V ,where V is a projective algebraic variety and where

m

m

you are interested in how the image meets the divisors on

V

m

. This theory

was further generalized in the same direction by P.Griffiths and J.King [30] to the study of holomorphic mappings A-V

f

where

A

is an algebraic,

braic subvariety

Z

C

V

V

a projective algebraic variety. Given an alge-

the 2 basic questions which are treated in this

setting are in analogy to Nevanlinna theory: (A) can you find an upper bound on the size of

CI(Z)

in terms of

Z

and the "growth" of the mapping f ;

(B) can you find a lower bound on the size of

f-I(Z)

,again in terms of

Z

and the growth of the mapping. The most important special case of this problem is when A::: C m and V::: pn ,the complex projective space. Then f may be given by n

meromorphic functions Z ::: (z

The subvarieties Pc/wI"" ,w n ) the equations

Z

l'

.. '

z

'm

) E Cm

will be the zero sets of collections of polynomials

and so the questions amount to globally studying solutions to

Concerning the extensions of Nevanlinna's theory for holomorphic mappings between Riemann surfaces, we refer to L.Sario and K . Noshiro [30] and to the more Hermitian differential geometrical versions of S.S.Chern [11] and H.Wu [62] . We disregard here the several extensions of Nevanlinna theory to certain classes of non-holomorphic functions, note however the extension of E.F .Beckenbach and G.A . Hutchison [4] to triples of conjugate real harmonic functions, as it rests on a similar underlying idea as our developments. In this Research Note I am presenting an extensi9n of the formalism of NeyanIinna theory to systems of

... ,f (z) n

n

~

1

meromorphic functions

in a way, which is fundamentally different from the theory of

holomorphic or meromorphic curves of Weyl-Ahlfors and its higher dimensional

xi

generalizations. As in Nevanlinna theory again the starting point is a formula, this time a generalization of the formula of Poisson-Jensen-Nevanlinna, which I discovered in 1964, when I was trying to extend the Nevanlinna formalism to the simultaneous solutions of systems of n

equations

=

f (z) n

=

a

n

z E C

a

n

E

C

w. = f.(z) ,j=l,···,n are n:c I meromorphic functions. J J We note that already G.P61ya [52] and R.Nevanlinna [47] have studied func-

where

tions with respect to values they assume at the same points; contrary to the present study, however, these values were taken to be one and the same complex number

a 1 • and they investigated the condition under which necessarily

fl(z) == f 2 (z) I succeeded in extending formally both the main theorems of Nevanlinna theory. together with the Nevanlinna deficiency relation. Although the above system of equations has only solutions for points set which is rather thin for

a = (a l ,··· ,an) E f(C) a n > I " these results seemed to be quite inter-

esting. However, one difficult main problem was stilI to solve; the problem of finding the true geometric meaning of the extended quantities. a problem which was proposed to me by Helmut Grunsky and by Rolf Nevanlinna in 1967/68. It took me several years to find its solution. and was finally achieved partly in my doctoral thesis and partly in my Habilitationsschrift, on which the present Research Note is based. The main difficulty was the appearence of a new term to the generalized Nevanlinna value distribution quantities and

T(r,£)

V(r,a)

in addition

m(r,a), N(r,a)

in both the extended First and Second Main Theorems. I then

tried to compare my result with the totally different theory of Weyl-Ahlfors and its extensions and with the more recent developments of complex manifolds and Hermitian geometry. So I gradually found out that the notions of the m dimensional complex projective space

pm

,its Fubini-Study Kahler metric and the

complex differential geometry of Hermitian line bundles, which are now a central fact in the theory of holomorphic or meromorphic curves and in generalized multidimensional Nevanlinna-Weyl-Ahlfors theories, are also the key for the

xii

proper understanding of the geometric meaning of the main new term and even for the interpretation of my tic function"

o

II

V(r,a),

generalized Ahlfors-Shimizu characteris-

T(r, f) . A fundamental role is played by the curvature form,

whose cohomology class represents the characteristic Chern class of the hyperplane section bundle over complex projective space. While the theory of holomorphic curves by Weyl-Ahlfors and its generaliza-

tion give results for the problem of how often the image of the mapping meets a set of hyperplanes or subvarieties in the image space, the theory of the present Research Note does not render results in this direction when

n

~

2 . In

contrast to this we study the growth of the projection of the curve f(z) - a

C -

Cn

into

pn- 1

and the connection of this growth with the distribution in

C

of the solutions

to the system of equations f( z)

whilst the point counterpart for

a

a

varies over

Cn

,a problem which has no effective

n = 1 . The theory reduces to Nevanlinna theory if

and stays in close contact with the original Nevanlinna formalism if

n = 1 n

~

1 .

The readers of this book must have hardly any prerequisites from Nevanlinna theory, but only a good lesson in function theory. An advantage. however - even a necessity in some places - would be a little familiarity with the sources R.Nevanlinna [46], [27] or [28]. In Chapter 1, §2 , Chapter 3, §9 and in Chapter 5, §15 some background knowledge of fundamental ideas from Hermitian geometry and algebraic geometry might aid the deeper understanding of certain formulas. The interested reader will find some hints on these matters in the Appendix and further details in the literature of the Bibliography. With pleasure I thank W.Helmrath and Professor R.Schark for correcting part of the manuscript, but above all I want to thank my wife for helping me with the literature and having all the patience during the time of writing. Last but not least, I would like to thank Pitman Publishing for their excellent cooperation.

Siegen, West Germany

Hans J. W. Ziegler

April 1982 xiii

1 Extension of the first main theorem of N evanlinna Theory and interpretation by Hermitian geometry §1

Generalization of the formula of Poisson-Jensen-Nevanlinna

We denote by

en

the coordinates

the usual w

n

dimensional complex Euclidean space with

=

the Hermitian scalar product

+v w

n

n

and the distance

1

Ilv - wll

=

+

2

Let

= be

n

~

•••

J

W

=

n

f (z) n

1

complex valued functions of the complex variable meromorphic and not all constant in the Gaussian plane e 1

z ==

,which are e, or in a

finite disc

{ Iz I

=

0<

Thus in

<

R}

R;;;

C

+00

e

°

(we put

e +00

<

R ==

<

e),

+

00

a vector

valued meromorphic function f( z)

=

is given, which does not reduce to the constant zero vector For such a function the notions liZ e ro", lip

!.J"

are defined as in the scalar case

function Zo

I e" and "m u 1 tip 1 i c i-

of only one meromorphic

f1 (z) t. More explicitly, in the punctured vicinity of each point

eR the vector function Laurent series E

£( z)

t

n = 1

0

0 = (0,. ... ,0) .

w

=

£( z)

can be developed into a

( 1.1)

See, e.g., Dieudonne' [6]'p.236

1

where the coefficients are vectors (0, ... ,0)

of course, if around

dC

denotes a sufficiently small positively oriented circle

r

, then the usual formula f( 1;) dZ;

2;'- J

=

ck

dC

(--)k+l l;;-zO r

is valid. In order to fascilitate the task of describing the vector valued Nevandroff one-point compactification of Laurent expansion, then £( z)

of

II

Zo

m u 1 tip 1 i cit y"

the ideal elemen t. of the Alexan-

"00" t

linna formalism, we will denote by

C n tt

. Now, if "p~"

will be called a -k

in the above 0

i n t" of

at least one of

; in such a point

o

00 -p

or an "

f. (z) has a pole of this multiplicity in J the ordinary sense of function theory, so that in Zo itself f( z) is not dethe meromorphic component functions

fined. If

kO

"zero"of

0

>

fez)

in the Laurent expansion, then of "multiplicity" kO f 1 (z), . . . , fn (z)

component functions multiplicity. Finally, if is holomorphic in accumulation in

Zo

k O ;;;

0

Zo

is called a

; in such a point

Zo

all

vanish, each with at least this

in the expansion (1.1), then

w

==

f( z)

Obviously the set of zeros or poles has no point of

CR

After these preliminaries, we will now prove an ex tension to meromorp hie vector functions of the important formula of Poisson-Jensen-NevanIinna. Let G

C

dG

CR

be a domain with closure

G

C

CR

and assume that its boundary

consists of finitely many closed analytic Jordan curves. Now denote by

zl"",zn

all zeros and poles of

other. Assume that

f(z)

Green's function of

G

Construct

m +1

fez)

in

G

, which are different to each

has no zeros or poles on with pole

Zo

aG

. Let

g(z,zO)

be

,and suppose that zo;tzq(q=l, ... ,m).

pairwise disjoint closed

t The two real infinities will be denoted by

z - centered discs q

+00

and

, respectively.

ttThe fact that in general C n u {oo} cannot be given the structure of a complex manifold, presents no difficulty in this description. 2

q with radius

= O,l, ... ,m

. Putting m-

U dt: q=O q

G -

=

=

U(z)

log [[f(z)[[

Green's formula

j( ~ u

g -au) ds

-

G

g L', U )

(u L', g

i

an

an

dx

1\

dy

t:

is valid, where the derivatives under the integral of the left hand side are with respect to the inner normal of the boundary L',g(z,zo)

=

(z E Gt:)

0

and

g(z,zO)

a G t: =

of

GC

(z E aG)

0

•

Because of this can

be written m

ag(z,zO)

i

u

ds

I:

+

an

q=O

i (u

a g(z, zo) on

g(z,zo) -au) ds an

a dt: +q

aG -+

i

g(",O) 'U(,) dx

(1. 2)

dy

A

Gt: Since the circle

lim t:-+ 0

is harmonic in

g(z,zo) + log [z-zO[ a dt:

o

i (u

G

,the integral over

satisfies dg an

g -au) ds an

a dt: ~-

0

lim t:-~O

i (adt: 0 -{-

,nog [z-zo[ U

+

an

10g[z-zO[ -au) ds an

Here the integral over the second term on the right side tends to zero as 3 d~

The counter-clockwise traversed circle tation

z

= zo + se -i
as

has the parametric represen-

increases from

1

=

and

s-+O.

0 to

21T

• From

ds

3n we obtain

21T dlog Iz-zol

f-

f U('O,,,-i.)

=

ds

U

3n

o

ads +

(1. 3)

d.

0

so that 3g lim s-+ 0

au)

- g -

3n

=

ds

(1. 4)

an

In the punctured vicinity of

z

(q=l, ... ,m)

q

we have the develop-

ment

11f( z) II where

A q

Iz-z

q

A I q

V (z) q

q

"

0, +00

is a positive or negative integer of absolute value equal to the

multiplicity of

q Now, for

,and where q x + iy q q q = 1, ... ,m ,

lim s-+o

( log V (z) -ag q an

around

V (z) q

z

f

z

V (z) q

is

(llogV g

q

(Z»)

with respect to

ds

o

,

x,y

(1. 5)

an

ads q +

since the integrand is continuous in

z

q

. We conclude that for

q=l, ... ,m

lim £:"+0

1(0::

_ g aU)dS an

a dE +q

=

lim E+O

I

ag

(A q log Iz-z q

gA

1

an

3loglz-z I q )dS q an

(1.6)

a dE +q

Since the integral of the first expression on the right hand side tends to zero, we get, repeating the argument which led to (1.4), for (1.6) the limit - 2n A g (z ,zo) q q Using this we obtain from (1. 2) and (1. 4), letting

log Ilf( zO)

Ii

2;

I

ag(z,zO) ds

log Ilf( z) II an

aG m

L q=l 1 2n

A g(z ,zo) q q

I

g(z,zo) lllogllf(z)11 dx I\dy. (1. 7)

G

( 1. 7) holds even if nite; here, if

z

Zo E {z1, .. ·,zm} is a pole of

q

f( z)

, since then both sides become infi, we understand

Also, (1. 7) remains valid if there are zeros or poles of

II£( z q ) II £( z) on

= +00 aG

this can be easily seen, modifying the proof by indenting the boundary suitably at the singular points and by performing a limit process, taking into account the mild logarithmic nature of the singularities. Substituting and

z

for

Zo

and

r;.

for

z

z. (00 ) respectively the zeros and poles of J ted according to its multiplicity, we have for any

,and denoting by f( z)

in

z E G

G

z.( 0) J

,each coun-

the basic formula

log

IIHz)11

I

l211

(lg(r;,z)

log

II£( z;) II

ds

an

3G

L

g(Zj(O),z)

+

z.(O) E G

J

If in particular we choose

r

{I z I

:::

<

r}

(0 < r < R) ,

then

is

z

2

r

g( t;, z)

C

G

Green's function with pole in

( 1. 8)

(t;=o+iT)

G

- t;z

(1. 9)

log r( t; - z)

In order to compute

~ an

,which is needed in formula (1.8). we put

r(t; - z)

=

A(CZ)

r

2

t;

-

:::

a + iT

t;z

Then we can write log A where

h

- (g + ih)

:::

is a function conjugate harmonic to

traversed circle

aC r

• the derivative of

h

g

. Now, on the positively

in the direction of the tangent

is ah

dT

do h

h

+

a ds

as and the derivative of

ag

g

T

in the direction of the inner normal is

dT =

ds

do +

an so that by the Cauchy-Riemann equations

flg

dh

()n

dS

Now, for

£;

dg

1 ds

dg

on the circle,

.-

dn

-

as

(~ as

so that 1 alog II ----

i -d h) ds

+

dS

1 ( dlog II

a

0

dS

+

dlog II dT ---) ds (3 'f ds

+

dlog A i - - - dT) ds dE; ds

do ds

0

1 ( d log II

do ds

d £;

dlog A

1

dE;

as Writing r e i¢

r,

t e

z

i8

we obtain ag -

zE;

E;

ds

=

an

--d¢ E; - z

+ r

2

re i ¢ re i¢ - t e iO

d¢

+

re i ¢ i¢ t i 8 re - e

d¢

+

re

i¢

(re

-i¢

d¢ - zE;

t -i8 i¢ e e t -i8 i¢ r- e e

re

-te

-i6

te -i¢

-i8 -te

-i8

d¢

t e -i8 ( re i¢ - t e i6)

)

=

d¢

d¢

+

c

re -i¢ - t e -i8 1 2

re i¢ - t e i8 1 2

1

1

r 2_t 2

=

2 2 r -2rtcos(¢-s)+t

d¢

which is the Poisson kernel. Introducing it into (1.8) gives the fundarr

Theorem 1.1 (Generalized Poisson-Jensen-Nevanlinna Formula): £( z)

Let

be a meromorphic vector function in i8 , which does not reduce to the constant zero vector. Then for z = te (0

<

r

R)

<

the following formula is valid: 21T

j

1 21T

log IIf(z) II

r 2_t 2 log II£(re i ¢ )11

2 2 r - 2rtcos (o-¢ ) +t

d¢

0 2-r -z.(O) z J log r(z-z.(O» z. (0) E C J r J

L 1 21T

£( z)

z. (0)

and

)

z. (00) )

L

+

)

log

r(z-z.(oo» J

z. (00) EC r J

2 r -I;z

I log C

Here

2-r -z.(co) z

A log 11£(1;) II

da

1\

dT

r( z-I;)

(I; =a+iT)

r

denote respectively the zeros and poles of

,coun ted with multiplicities. Note that in the case

n = 1

of a scalar meromorphic function the

last integral vanishes, since then in the integrand the expression

=

(1. 10)

log If 1 (I;) I

Re log f 1 (I; )

log 11£(011

is harmonic up to isolated points, so

that the ordinary formula of Poisson-Jensen-Nevanlinna t is obtained. Since the Green's function (1. 9) is positive, the sums over the zeros and poles in (1.10) are non-negative quantities. The same applies to the two dimensional integral, observing that Alod£( I;) II

t

~

log

0 tt up to the poles of

lif( 1;) II

is subharmonic, i. e.

f

See, e.g., Nevanlinna[28],p.164

ttSee §5 or look at the explicit form of this expression in §2 8

Interpretation by Hermitian Geometry

§2

In this section it will be shown in which way ideas from the Hermitian Geometry

+

of complex manifolds' can be used to give an interesting interpretation of the import an t term 1lIogllf(.;)11

(2.1)

do /\ dT

which arises in the extended formula (1.10) of Poisson-Jensen-Nevanlinna. For this interpretation we shall need the notion of com pIe x t i v e

spa c e

where

0

. , w)

pn-1

is the point

en - 0 : = en - {OJ

. To define it, we take (0, ... ,0)

pro j e c,

w = (wI'"

, and identify those points

en - 0

of

which differ from each other by a factor. The resultn-1 ing quotient space is P . The numbers (w1' ... ,w n ) are called the hom 0 g e n e 0 u s c 0 0 r din ate s of the point n

'V W

. .In t h ey d etermlne

pn-1

' ; 1' fVIce versa

U.1 , defined respectively by

have the

n -1

•

IS

given, then the numbers w can b e covere d b y n open

P n-1

are defined up to a common factor only. sets

'V

W

w.;r 0 ,

1:;; i

1

a f fin e or i n hom

1 0 cal

:;; n

g e n e

0

In 0

u s

U.

1

we

coo r d i-

nat e s 1 :;; k

= these map

e n-1

b ijective1y onto

U.

1

'V

( .1;;

W

1

1

, ... ,.1;; 1

i-I

,.1;;

k ;r i

:;; n

by i+I

1

, ...

The transition of these local coordinates in

,.r; 1

u.

1

n

)

n u.J

is given by

k k jl;;

=

il;;

1 :;; k

:;; n

k ;r j

.I;;j 1

which are holomorphic functions. In particular,

pO

is a point, and

pI

can be identified with the Rie-

"f The main reference for this theory is ehern [5] .

S2

mann sphere

We denote by

dad

d

a

+

the usual operators on a complex manifold and note that pn -1

dC

and

=

dd c

i(a -

())

2i d a

can be endowed with a Kahler metric, the F ubi n i - Stu d Y 'V

met ric

F

of constant holomorphic sectional curvature 4 ; this metric can

be described as follows: On

-2 F =

(Ek w k w k )

en - 0

we consider the covariant 2-tensor

~ ( Lkw k w k )( l.kdwk 0 dw k ) -

():k wkdw k )

0 ( Lk wkdw k ) ~

and its associated 2-form

(Ekwkdw k )

=

A

[l.kWkdWkJ]

!lw11 4 i

2" ()

(2.2)

= w is closed, since it can be written

= F

and

w

both vanish for

n = 1

. By assigning to a point

w

of en-O

the point it defines in the quotient space, we get a natural projection

Let

*

denote the usual puB-back map under 1T • Then for 'V n-1 Fubini-Study metric F on P and its exterior 2-form 1T

tively defined by

10

1T

n > 1

the

are respec-

* (\; F

1T

*

and

F

:::

1T

AO

W

:::

W

'V

'\,

is closed, so

(,0

i

w

is a Kahler metric and

F d

2"

a log IIwl1 2

is the pull-back to

of the Kahler form

Pn - 1

on

w has an important additional geometric meaning: For the projection

the in verse image of each point is homeomorphic to

e *

::: e 1 - 0

relationship is an example of the fundamental notion of a hoI lin e sal

0

m

0

This r phi c

bun dIe , and this particular example is called the u n i v e r n-l bun dIe J over P (if n > 1). We can describe it more ex-1

plicitly as follows. In

w )

1T

1T

,instead of using the coordinates

(U i )

(wI"'"

,we can use the coordinates

n

1 :;; k ;;; n

:::

This exhibits

IT

-1

(U.)

clearly as the product

1

U.

is the fiber coordinate relative to tively the fiber coordinates

w . . r,

:::

W.

1

k

1

U. x e*

1

W.

1

here

1

. Relative to U. -1 1 w. in 1T (U. 1 J

and

W.

and

;z!

and

nU.)J

w. E e* 1

U. respecJ are related by

w.

i

_J

:::

.1)

J J

1

This shows that the change of fiber coordinates is simply obtained by the multiplication with a non-zero holomorphic function. The universal bundle at a point on

J

en

~

E

pn -1

defines an

J

is characterized by the property that the fiber

is the line

Her mit ian

{A w }A c

en

. The Euclidean norm

met ric

ant h e

fib e r s

of

by setting n

2 I Iw.1J

:::

1

If

w

is any non-zero section of

then the cur vat u r e

J

,i.e. a local lifting

or e her n

U c p n - 1 -7 en-O

for m of the bundle

J

with

respect to this metric is given by

11

= The bundle tion

J*

dual to

bundle

H

1

=

- 2; d d C log Ilwll

7T

w

J

is the importan t h Y per pIa n e sec n-1 over P . The negative of the latter form, i.e.

the form

1

+ -

7T

=

W

is then the curvature form or Chern form of H 1 the Kahler form of the Fubini-Study metric We see that up to the factor n-1 7T of P given in homogeneous coordinates is equal to the curvature form of the Hermitian bundle

H

.!.7T w

. The curvature form

is a real valued closed

differential form of type (1,1), and the cohomology class to which it belongs in the sense of de Rham 's theorem is the c h a r act e r i s tic c I ass

of the bundle

of the second cohomology group

or

C her n

. It represents the positive generator

H

H2(pn-1,Z)

_

Z

and is Poincare' dual

to the fundamental homology cycle of a hyperplane. We can now understand the meaning of the term

as follows. In view of the projection

the given

7T

meromorphic vector function f(z)

defines a holomorphic map or hoi

0

m

0

r phi c

cur v e

'V

f (z) 'V

in complex projective space. A priori f 1 , ... ,fn point

and the common zeros of CR

Zo E

f

is defined only up to the poles of f 1 , ... ,f n

each component function

fj

. However at any such has a local representation of

the form f. (z) J where

p. ]

p

12

o,

= is a well-defined integer. We may set

=

max { - p. } J

00

and the map

1-1---1 ( (z-zO)Pf1(z), . . . , (z-zO)Pfn(z) )

Z

'"

extends

f

Zo . So the map

over

'" z) £(

=

(fl(z), ... ,fn(z»

is well-defined everywhere in zeros and poles of

•

o

f

CR

'"

_ _ _ pn-l

C

R

. If

(2.3)

• CR

denotes

minus the

, then we have the diagram

.--:-- c

1::-CR

'"

n (

0

'pn-l

inc = inclusion .

'\, f

This diagram commutes and we say that, apart from its zeros and poles, f'"

a lifting of 1

to log

27T =

4 'IT

=

. Outside the exceptional set we compute for

11£(1;)11 11 log

d

i

-

1f

i

f* w is the If we set

II £( t,;) II

dl;

1\

tdl;

1\

d~

~ ~ log II£(O I d~ t as

d

a log 11£(011

d

a log I £( F,) 112

denotes the pull-back map under

f

0

n > 1

~

w)

v

is

dT

f* ( 1

2'IT f*

1\

a2 log 1[£(011 d s a~

'IT

-1fi

where

da

f

'IT

.l'IT f* W

=

. By Wirtinger's theorem,

I u m e e l e men t of the curve

f,

pulled back to

CR'

v(r,O)

j

1

=

-

1T

C v(r,D)

then

2i1

C

v

IIf(UI!

do /\ dT (2.4)

I ume

0

(the volume is 1Tv(r,O»

of the holomorphic curve

r

is due to the fact that if tive subspace of pn-l then

The factor

/:, log

cr

r

is the normali zed

of the restriction to

j

1

f * u)

is anyone-dimensional projec-

1[

1T

In the N evanlinna case is a point, and

v(r, 0)

n = 1

f

reduces to a constant map since

pO

vanishes identically in this case.

In concluding this section we note for later reference the following explicit formula, which is obtained by the computation (2.2) or by the direct computation given below:

1 21T !1 log

IlfUJl1

1 f* 1T =

=

i 2'/T

i 2IT

do /\ dT

W

dE.; /\ d'"[

2

1

IifIi4

L

If/k - fkfjl2

d~

(2.5)

/\ d'"[

j < k

outside the exceptional points, where

f(

~)

vanishes or has a pole. Here the

last line comes from

2

L j < k

If/k - fkfjl2

=

L j,k

U/k - fkfj)(I{k -

rk~)

=

=

fl [Crk'

f.fl'rkc J ~ J

~

+

JJ

<[',f'> - - + 2 [ - <£',f> ]

It is easy to see that (2. 5) can be con tin uously extended in to the exceptional

points. From (2.5) we see in particular that the integrand in (2.4) is

~O

The direct computation of (2.5) runs as follows. Using the relations

= for holomorphic d dO log

=

A, we obtain

1

I[ f( t;) [[

2 =

1

d

ao

log

2

+

Re

d

3T log

1

[I f( t;) II'

d

2 3T log

<£,f>

1 +

2

<£,f>

-i(-

+

<£,f'»

= 2 1m

= <

d2 d0 2 d2

(h2

f, £> 2 [Re ] 2

log

[[f(OI[

= 1m

log [I f( 0

2 + 1m

II

2 [1m ] 2

2 [1m ]2 This gives

A log

Ilf(t;) II

2

=

=

§3

- l\2 2 ---------2

The generalized first main theorem

z =0

Putting

in (1. 10), p. 8 we obtain the identity 21f

log

Ilf(O)11

=

2'.

I

r

logll I(reil ) II dj

'"'"' log

L

z.(O) E C

o

J

z.(O)

J

r

r +

'"'"' log

L

z.(oo) E C

J

z.(oo)

J r

which reduces to the formula of Jensen if n = 1 For a = (a 1 , ... ,an) E C n we denote by vector function

f(z) - a

the meromorphic

f(z) - a At this point we need one more definition: if

is a zero of

where

an II a -p

f( z) - a

'$

0

• then we will call

0

i n t

f(z) - a II

of

f( z)

of the same "m u 1 tip I i cit y" as the zero zo . It is clear that for each a E C n U {oo} the set of a - points of f has no points of accumulation in CR

(we already remarked on p. 2 that the set of zeros or poles of

point of accumulation in If

z =0

f

has no

CR ) .

• both sides of (3. 1) become infinite. We wish to avoid this. To this end observe that for each a E Cn a Laurent series

is a zero or pole of

f

£( z) - a

(3.2)

=:

Z =: 0 (c (a) ;r (0,.' .. ,0» is valid in a punctured vicinity of . Here q c (a) does not depend on the choice of a if q < 0 clearly, if z q is not a pole of fez) and if f(O);r a , then q = 0 , c (a) q f( 0) - a in (3.2). If in (3.1) the function is replaced ,by fez)

a)

=:

0

,we get

211 logllc (a)11 q

o

-

q log r

in j

+

lodf(rei .) -

aII d.

o

L lOg~

L

+

z. (a)

o
2;

j

log

I~ I

J

r

log

0< Iz.(oo)l
'log

11'«) - a II

do

A

z. (

00 )

J

(3.3)

d,

er a E en u {oo}

where f

denote the

a - points of

, counted with multiplicities and ordered by non-decreasing moduli. We now need some more notation: Let n(r,oo,f)

n(r,oo)

=:

denote the number of poles of

nCr ,f) f( z)

Iz I ~ r

in

,counted with multiplici-

ties. Then we can write r

r

""'"

L

log

o
J

= z.(oo)

J

and upon integration by parts,

j

log

o

~

d(n(t,£) - n(O,f»

L

log

r

r (n(t,£)-n( 0,0 )logf r -

r z. ( J

00 )

0< 1z. (00 ) 1< r J

t=O

°

r

j

=

Further, if for any

a E en

n(t,O - n(O,f) ------dt

(3.4)

t

{<X>}

n(r,a)

=

n(r,a,O

°u

r

(n(t, £)-n(O, f) )dlog f

denotes the number of

a - points of

f( z)

in

1z I

~ r

,counted with

multiplicities, then we obtain in the same way that r

j

r

"'"

~

log

O
z.(a)

n(t,a) - n(O,a) ------dt

J V(r,a)

In (3.3) the expression

° defined by

.l.IIOg 1£ I Lilog ilf(U - a II da " dT

V(r,a)

t;

211

er can be written in a different way. In order to see this, we decompose

jI 21T

r

V(r .a)

"

,',

(3.5)

t

log:: 'log lI{(tei $)

dt

° °

21T

r

,~Iogr j o

~

dt

j

0

t Mog

II{(tei ;)

a

II

t d;

~ aII

d;

(3.6)

r

-

2~ I

2n

dt

o

I

t log t 'Iodf(te i ,) - a lid,

0

and obtain by differentiation r

dV(r,a)

211

r dtI

=

_1_ 21Tr j

dr

o 1 2nr

v(r,a)

2' ,

A

dT

d<j>

(~=o+h)

r

by putting

f 'log 11£(0 - a II do C

II

lilogllf(S) -alldo

v(r,a)

Introducing the function

- a

0

I C

t l\log !lfCtei<j»

A

(3.7)

d,

r

we obtain the formula r

I

VCr ,a)

v(t,a) - - dt

(3.8)

t

o Here, in view of the interpretation of v(r,a)

is just the normalized

v(r,D)

in §2, p.l4, the function

vol u m e (the volume is

TIv(r,a»

in

C

r

of the holomorphic curve

clearly,

v(r,a)

_ _ _ pn-1 ~

C

(f - a) '\,

if

R

== D

if

n = 1

n;; 2

. Because of this geometric meaning the

important function V(r ,a)

V(r,a,f)

will henceforward be called the

II

vol u m e

fun c t ion II associated with

the meromorphic vector function v(r,a)

volume function II Now the

11

c

0

. Of importance will also be the "little

v(r,a,£)

=

u n tin g

of Nevanlinna theory

f( z)

fun c t ion II

of finite or infinite a - points

will be generalized to vector valued meromorphic func-

tions by the foIlowin g definition: r

-

N(r,a)

N(r,a,f)

n(O,a)logr

j

+

r N(r,£)

-

N(r,co)

=

n( O,£)log r

+

n(t,a) - n(O,a) dt t

°

j

net,£) - nCO,£) dt t

° Applying this notation and (3.4),(3.5),(3.6), we now rewrite (3. 3) ,observq = n(O,O) - nCO,£)

ing that in (3.3) the number

of the number of zeros and the number of poles of

is equal to the difference fez)

at the origin. The

result is the following identity, which because of its importance we formulate as a theorem. Theorem

3. 1: Let

vector function on

o<

r < R

=

w

=

f(z)

C R (0 < R

;$

(fl(z), ... ,fn (z»

+ (0)

• Then for

be a meromorphic a E C n , f( z) "" a ,

the following identity holds

21T

2~ flog here

° c (a) q

not a pole for

Ilf(rei ,)

~ a d, N II

+

V(r,a)

(r.f)

+

N(r,a)

+ logllc (a)11 ;

q

(3.9) is given by the development (3.2); in particular, if f( z)

and if

f( 0) '" a

,then simply

c (a) q

z =0

is

= f( 0) - a

Note that this identity gives a differential geometric interpretation of the integral mean in the left hand side of (3.9). Remark 1 : In the special case

n = 1

and

a =

°

formula (3.9) boils

down to a formula of Nevanlinna theory, for which H. and J. Weyl use the name i'

"condenser formula" , refering to the electric potential of a circular condenser. Remark 2 : From formula (3.9) follows immediately that the mean value 2IT

,', f 'ogllf(rei ;)

,(r ,a)

-

a II d4

o is a continuous function of in (3.9) are continuous in

r

o<

for

r < R , since all other expressions

r

de r

Remark 3 : Up to the r - values such that on of

f

,this mean value is differentiable with respect to

lie poles or a-points rand

2IT

r:r

i, flog IIf(re

i ;) -

a II d; + n(r,f)

c

v(r,a)

+

n(r,a)

o Remark 4: Assume that integral function such that

f(z) = (f 1 (z), ... ,f n (z» is a vector valued fez) $. aE en . Then it follows from (3.9) and

from remarks 2,3 that the mean value ing and convex function of Remark 5: If

log r

\l(r,a) for

fez) = (f1(z), ... ,f n (z»

function without common zeros of

is a continuous, non-decreas-

0 < r < R is a vector valued integral

f 1 (z), ... ,fn (z)

,then up to an additive

constant the integral mean 2IT

].J(r, 0)

,~ flog lIf(re

=

arising in (3.9) for

a

=

° °

i ;)

lid;

equals the characteristic order function of H. and

f 1 (z), f 2 ( z), are arbitrary meromorphic but not all identically vanishing, then

J. Weyl for the corresponding meromorphic curve in

... ,f (z)

n

pn-1 i'i' . If

t Weyl [43] ,p. 73

TTWeyl [43] ,po 81

21

in view of (2.4), (2.5) and (3.8) the characteristic order function of H. and J. Weyl t equals

V(r,O)

.

With the usual abbreviation +

=

logx

(x ~ 0)

max (log x , 0)

we have the decomposition log x

=

+

logx -

+ 1 logx

so that we can substitute in (3.9) log

Ilf - a\I

1 + log - - Ilf - a II

=

We introduce the following generalized quantities of Nevanlinna theory:

21T

m(r,oo,£)

mer,£)

2~ J16gli

=

f(re i .) Ii d.

o 21T

m(r,a)

m(r,a,£)

=

.1..

J l~g _ _1_ _ _

21T

IIHrei
d
(a;t;oo)

,

o T(r,£)

mer,£)

For finite or infinite

1 inn a p r

0

a

xi mit y

m{r,a)

g e n era 1 i zed

t ion

II

•

is the

fun c t ion

vergence of the vector function It

+ N(r,£)

f

II

n ;

g e n era 1 i zed

it measures the asymptotic con-

to the point

N e van 1 inn a

a

+

log Xl + log x 2 + log 2 we obtain

t Weyl [43],p .142

22

T(r,f)

is the

c h a r act e r i s tic

Further, from the general inequality +

N e v a n-

fun c-

and so that

We now introduce the latter ingredients into identity (3.9). The result is the following theorem. which generalizes the first main theorem of Nevanlinna theory to vector functions: Theorem 3.2 (Generalized First Main Theorem): Let

be a meromorphic vector function in f( z) 0/=

CR

. Then for

f(z):: (£1(z), ... ,f n (z» 0 < r < R , a E Cn ,

a

T(r,O

V(r,a)

+ N(r,a)

m(r,a)

+

+ log ilcq(a) II

+

dr,a)

(3.10) Here

dr,a) f(O)

If

+0

is a 'function such that

a

otherwise,

and if c (a) q

z

=0

Idr,a)

~ h~gllall + log

2 , £:(r;O)=O

f(z)

c (a) :: f(O)-a; q

1

is not a pole of

, then

.

is obtained from (3.2) .

In the scalar case

n:: 1

the volume function

V(r,a)

vanishes iden-

tic ally • and theorem 3.2 reduces to the first main theorem of Nevanlinna the ory . In the general case

n;;; 1

we see from

dV(r,a)

o

v(r,a) dlogr that r -)- 0

V(r,a)

is an increasing convex function of

log r

, which vanishes as

. Likewise from

dN (r, a) n(r,a)

;;;

0

dlogr up to isolated values of

r

, it follows that similar to Nevanlinna theory the

counting function of a - points and convex function of a

log r

is the number of solutions in

f( z)

:: a

N(r,a) for

o< Izi ~ r

is an increasing, piecewise linear r

< R

; here

n (r, a)

of the system of

n

for finite equations

, multiple roots counted according to their multiplicities. Obviously,

23

n(r,a)

is positive for some

where

00

denotes

CR

a

Cn

E

,punctured in the poles of

CR

n

a E f(C r

if, and only if,

. We note

f

fez) = (f1(z), ... ,f n (z» be a rtt?n-constant meromorphic vector function defined in C R ,where n > 1 . Then for fixed r (0 < r

Lemma 3.3: Let

< R)

the counting functions

outside a set of

vanish for dimensional Lebesgue measure zero in C n

2n

nCr,a)

and

N(r,a)

f( ~ C R )

For the proof it suffices to show that

is a null set if

n > 1 f(K)

this will hold if we can prove that for any compact null set. Since contained in the set in

K 00

CR

Cn

because measure Hence

n > 1 <

f(K)

is compact, it can be covered by a .finite number of squares

. Since f

m(K) + 1

Q

m(Q)O(

Now we can cover

K

£

2

II:; I)

, the measure of

of side· c

is

)

by squares with side

,so it follows that

m(f(K»

£

and total

(m(K) + l)O(

<

Eh

is a null set.

log r

m(r,a)

is in general neither increasing nor con-

. From (3.9) it is easy to see that the sum log r

tends to a finite limit as In the Nevanlinna case a

=

O(

+

maps a square

OC c 2n-2 ·s 2)

The proximity function vex in

= fez)

fez + 1:;)

on which

=

is a

in a certain sense"

counting function

-+

n = 1

-

00

for large values of

the proximity function

N(r,a)

side of (3.10) for "most a

m(r,a)

is nearly equal to

dent that the volume function II

T(r,i)

T(r,f)

and for "most

is bounded and the . If

n

~

2

it is evi-

VCr,a)

will be the main term in the right hand in a certain sense, so that for II most II a E Cn

the value distribution quantities

N(r,a)

and

m(r,a)

important role than the volume quantity

V(r,a)

be nearly equal to

is large. For .n

T(r,f)

m(r,a) + N(r,a)

if

T(r,f)

will playa much less

,which for these ~

1

a

will

the generali-

zed first main theorem expresses an invariable property: Independently of how the point a E Cn is chosen, the sum of the terms V(r,a), N(r,a) and m(r,a)

has the same "characteristic value"

up to terms, which remain bounded as tion of

VCr ,a)

by putting

VCr,oo)

r

-+

-

T(r,£)::: mer,£)

+

N(r,f)

R

. If we complete the defini-

0

,then this invariable prop-

erty is true also for

o

a =

co

•

The generalized characteristic function

T(r,f)

.

T(r,f), in a slightly modified form

,possesses a geometric interpretation, which will be discussed in

chapter 3 . We conclude this section by formulating the generalized first main theorem for the special case f( z)

n =2

of only two component functions:

=

In this case we can write up to isolated points lI£(z) - all

2

=

=

+

Thus up to isolated points

Lllog II£(O - a II

=

=

Now in Nevanlinna theory of scalar valued meromorphic functions

A(Z)

the characteristic function in the spherical normal form of Shimizu-Ahlfors is r

0

T(r,A)

=

~ I~t I 0

11;1~t

1;\'(01 2 do

1\

d,

(~

=

(J

(3.11)

(1+ IA(I;)1 2 )2 +h)

where

25

represents the spherical surface element of the image of ping

C

under the map-

r

J..(z)

Comparing with formulas (3.7). (3.8) we see that in the special case of only two component functions the volume function

V(r. a)

n

=2

is equal to the

ordinary Shimizu-Ahlfors characteristic function of the quotient f 1(z) - a 1

A(Z)

f 2 (z) - a 2

so that we can write

o

V(r,a)

f -a

T( r, _1_1) f

=

·2- a 2

in (3.9) and (3.10) . In particular, the generalized first main theorem can be formulated in the following form: Corollary 3.4 (Generalized First Main Theorem for f

z< z) )

a E

be a meromorphic vector function in

C2

f 2( z )

T(r,f)

CR

n

= 2):

Let

. Then for

f(z) = (fl(z).

o
< R

,

a2

$

o f I -a 1 T(r, -f) + N(r,a) + m(r,a) + log Ilck(a)11 + dr.a) 2- a 2

=

( 3.12) where

and

have the properties described in theorem 3.2 .

£

Here, using the fact that the Shimizu-AhHors characteristic is equal to the Nevanlinna characteristic up to a bounded term, the expression

can be replaced by f 1-a 1 T(r, -f) 2- a 2

26

+

0(1)

The example of rational vector functions

§4

Highly instructive is already the simple example of rational vector functions. In this section we examine only an application of the generalized first main theorem to this function class; applications to these functions of the later theory will be given in Chapter 5 Let

be a rational function. where

each other and of degrees ory it is shown

PI (z)

PI

and

Q 1 (z)

and ql

are polynomials prime to

• respectively. In Nevanlinna the-

that the characteristic function of

f1

has the form ( 4.1)

here the number

is called the degree of

f1

. Writing

if

we see that the degree of max(Pl. Q l)

f 1 (z)

can be given the following interpretation:

is nothing but the number of poles of

f 1 (z)

in the finite plane

plus the number of poles at infinity. It is exactly in this interpretation that formula (4.1) will generalize to systems of rational functions. £(z) = (£1(z). ...• f n (z)) is given. if all component functions are rational functions A"rational

vector

function"

P. (z)

f. (z) J

=

_J_ _

= 1, ..... ,n

( 4.2)

Q.(z)

J

A non rational meromorphic vector function in

s c end e n til. We will assume that each

j = 1, ... ,n

the plynomials

f( z)

P. J

and

C

will be called " t ran

-

is non-constant and that for are prime to each other

27

and have degrees

Pj

and

qj

,respectively.

As in scalar valued function theory we will say that the rational vector function of

if

f( z)

has a

ff

m u 1 tip 1 i cit y

It

f(.!)

pol e

f1

or an

It

00

poi n t

-

enu{oo}

f1

at

i n fin i t Y f1

denoted by

ff

z = 0 ; more generally, if

has a, pole of this multiplicity at

z

It

,then we will say that

fez)

has an

II

a - poi n t

o f

a E

m u 1 t i-

--~----------------------

plicity

at

infinity"

if

has an

For a given rational vector function a E en u {oo}

for any

The number

f

the integer

0 ~

and satisfies

*( 00 )

a - point of this multiplicity at *(a)

*(a) <

+

z = O.

is thus defined

00

can be computed very easily by the formula max{ 0 , max(p.-q.)} j J J

=

( 4.3)

We also see that mer,£)

*( 00) • log r

=

+

( 4.4)

00)

Example 1 (n=3)

f( z)

I

=

PI - qI P3 - q3

z

2 z

= o-

2

::

1 - 4

::

::

max {O

= m(r,f)

3logr

If we define the number

n(+oo,oo)

then

n( +00,00)

=

lim

3

z + I -4z

-2

)

P2 - q2

=

3 - 0

::

3

::

3

-3 max ( - 2, 3, - 3) } +

00) n ( +00 , 00 )

::

n ( +00 , £)

by putting

n(r,oo)

r-+ +00

is the number of poles of

f( z)

in the finite plane

e

in the latter example e.g., there is only one pole of multiplicity 4(at the origin)

28

= 4 . Using this notation, it is clear that

n(+oo,oo)

so that

N(r,£)

n(+oo,oo) .logr

+

0(1)

( 4.5)

By addition of (4.4) and (4.5) we obtain the characteristic function

T (r, £)

{ *(00) + n(+oo,oo) }logr

+

n ( +00 , 00) = 4

For instance, in example 1 we have 710gr

+

(4.6)

0(1)

T(r,f)

=

0(1)

In general, for any n(+oo,a)

a E en u {oo}

we put

n(r,a) lim r-+ +00

and we denote by

*

n(a)

:

n( +00 ,a)

:::

+ *(a)

(4.7)

the sum of the number of a - points of

f

in the finite plane

e

and at in-

finity, counted by multiplicities. the characteristic function (4.6) of

With this notation applied to

a rational vector function can be written in the easily memorizable form

*

n(oo) .logr

T(r ,£)

+

(4.8)

0(1)

Note that this is the exact analogue of the interpretation we have given above to formula (4.1) of scalar Nevanlinna theory. In view of (4.8) it makes sense now to call

d

=

the " d e g r e e

* nero) It

T(r ,f)

=

lim r-+ +00

log r

of the rational vector function.

Remark 1. A deeper geometric meaning of the degree

*

n( 00)

will be seen

in Chapter 3. Remark 2. If the meromorphic vector function

fez)

in

e

is non-ration-

al, then T(r,£) lim r-+ +00

log r

this follows from the corresponding result of scalar Nevanlinna theory

and from

results of Chapter 2. If

a

is finite, then the number

*(a)

can be computed according to the

formula maxi 0, - max(p.(a) - q.) } j J J

=

(4.9)

q.

where

is as above, and p.(a) denotes the degree of the numerJ J ator of the j-th component of the rational vector function Pl(z) - a 1Q l(z)

(

fez) - a

P n (z) - a n Qn (z) )

( 4.10)

Q (z)

Q 1 (z)

n

Using this, it is clear that m(r,a)

*(a) ·log r

+

( 4.11)

0(1)

which is analogous to (4.4) for finite

a

.

Example 2 (n=2) :

fez)

fez) - a *(a)

1 + z + 2z 3 3

.( 5 + z

=

z

5 z

:=

a=(l,2)

z

1 ; z ) z

max { 0, - max ( 0-1, 1- 3)}

:=

m(r,a)

logr

:=

max {O , - ( - I)}

+ 0(1)

Now, for an arbitrary meromorphic vector function the expressions

v(r,a~

= 1

and

n(r,a)

f

such that

f( z) $ a

are non-decreasing functions of

1',

so that the limits v( +00 ,a)

=

n( +00 ,a)

=

( 4.12)

v(r,a) lim 1'++00

lim

( 4.13)

n(r,a)

1'++00 exist in the sense that they are either finite numbers G 0 or are +00 . This n holds for any a E C U {oo} ,where in (4.12) we complete the definition of v(r,a)

by putting

an integer. We prove

30

v(r,oo) == 0

. If

n(+oo,a)

is finite, then it must be

Proposition 4.1:

Let

fez)

be a meromorphic vector function,

fez) $. a

.

Then with the notations (4.12) and (4.13) V(r,a) lim r++ oo

:::

v( +00 ,a)

a E en

u

{oo}

( 4.14)

:::

n ( +00 , a)

a E en

u {oo}

( 4.15)

logr N (r, a)

lim r++ oo

log r

Proof. In order to prove (4.14), we write r

r VCr ,a)

v(t,a) --dt t

::

J

+

const.

;;i

v(r,a) logr

+ const.

1 so that V(r,a)

v( +00 , a)

lim r++ oo

( 4.16)

log r

t > 1 , k > 1

Further, for

tk

j

k Vet ,a)

v(t,a) dt

v(t,a) (k-1) log t

t

t so that V(r,a)

=

lim r++ oo

Letting

log r

k + +00

lim

(1 -

~)

v( +00 ,a)

t++oo we obtain

V(r,a) lim r++ oo

log r

Together with (4.16) this proves (4.14). Equation (4.15) is proved in the same manner. We now return to the case of a rational vector function

f

. From the gen-

31

eralized first main theorem T(r,f)

V(r,a)

=

+ N(r,a)

+ m(r,a)

+ 0(1)

Proposition 4.1 and formula (4.11) we have on the one hand as T(r,f) v( +00 ,a) + n( +00 ,a) + *(a)

----3>-

=

r -+ +00

* v(+oo,a) +n(a)

Iogr On the other hand we have from formula (4.8) T(r,f)

*

n( 00)

r -+ +00

as

log r We summarize this in If

Proposition 4.2: v( +00, a)

n*

is a rational vector function,

f( z) 0/= a

,then

is a non-negative integer, which can be computed by the formula

v( +00 ,a) with

f( z)

*

n(oo)

=

-

*

a E en u roo}

n(a)

( 4.17)

defined in (4. 7) .

Here, in the case of scalar Nevanlinna theory of only one non-constant rational component function, the left hand side of (4.17) collapses to we have

*

n(a)

*

= n( 00)

for all

a

0 , and

. Thus in the vector valued case Propo-

sition 4.2 generalizes the fact that a non-constant rational function assumes every value including infinity equally often on the Riemann sphere, namely as

*

often as is given by the degree

n( 00 )

. In fact, equation (4.17) can be inter-

a E en u {oo}

pretated by saying that all points

are equidistributed in the

sense that the sum v(+oo,a)

*

* + n(a)

is independent of

n( 00)

a

Example 3 (n=2) : In example 2 we have for

* n(a)

=

*(a) + n(+oo,a)

1 + 0

=

=

a=(l,2) 1

further

* n(oo)

*( (0) + n ( +00

,00 )

=

0 + 3

= 3

so that by Proposition 4.2 v( +00 ,a)

3-1

=

=2.

In concluding we give one last example for the notions of this section. Example 4 (n=2):

f( z)

=

,

( (Z-l)( z-2)

* (0) n(

* (0) ·log r n(

T(r,f)

=

m(r,f)

= =

n( +<x> ,0) v(+oo,O)

=

=

=

degree of 5logr

=

+ 0(1)

,

5logr + 0(1)

* n(O)

1

f

=

2

3

m(r,O)

= =

log r

+

0(1)

N(r,O)

=

logr

+

0(1)

V(r,O)

=

N(r,f)

*(0)

5 - 2

=

5

+ 0(1)

0(1)

1

z2 2) (z-1)(z-3)

+ 0(1)

310gr

The first main theorem applied to V(r,O) + N(r,O) + m(r,O)

a =

°

is

= 310gr + logr + logr + 0(1) = T(r,f)

2 Some quantities arising in the vector valued theory and their relation to N evanlinna Theory §5

Properties of

V(r,a)

We already noticed in Chapter 1, §3 that r v(t,a)

=

VCr ,a)

dt

r

t

J

0

log r

is a non-negative, non-decreasing and convex function of r

which vanishes as

-+

0

; here

2~ J

v(t,a)

'log

v(t,a)

11£(,)

~

a

I

do /\ dT (E;=o+h)

I ~ I ~t Some further growth properties of

V(r,a)

(and

v(r,a) )

meromorphic vector function in for fixed

r

only if

f( z)

where

a

A(z)

+

. Then the following properties hold:

CR

the functions

(0 < r < R)

ish for some

are summarized in is a non-constant

Proposition 5.1 : Assume that

0)

(0 < r < R),

is given by

f

v(r,a)

and

van-

V(r,a)

is of the form (5.1)

A(z)b

is a con -

is a complex scalar valued function and

stant vector; (ii)

if

fez)

tically for all (iii)

if

is of form (5.1), then

r

r

f(z)

-+

0

isnotofform(5.1),then

34

V(r,a)

vanish iden-

v(r,a) r

and

V(r,a)

are

(0 < r < R) , which tend to

.

In particular, for given must occur:

and

(0 < r < R)

strictly increasing and positive functions of zero as

v(r,a)

a E Cn

,one of the following three possibilities

either lim

v(r,a)

=:

o

or

+00

lim

<

r-+R

<

v(r,a)

+00

r-+R

or

v(r,a)

=0

for all

r

V

In view of the definitions of

,

0 < r < R

and

v

what remains to be proved in

Proposition 5.1 results from. Lemma 5.2:

Let

be as in Proposition 5. 1. Then for fixed

f

the

11l0g 11f( z) - a I either vanishes only in isolated points of ~ C R or vanishes identically there. It vanishes identically if, and only if,

function else

it

fez)

is of the form f( z)

+

a

(f. (z)

Jo where

a E Cn

jo

En, ... ,n}

(5.2)

a. ) b Jo

is a suitable index and

is a constant

vector. Proof. In the case

n =: 1

nothing has to be proved, since

all

is harmonic up to isolated points. Now consider the case

Hz)

has the form (5.2), then clearly

lIlogllf(z) - all

direction it suffices to consider the case ity we can assume that

there is a largest index

a=:O jo E

identically. We can assume that

. Since

n f( z)

n, ... ,n} jo

~

~

=0

n > 1

. If

. In the other

2 , and without loss of generaldoes not vanish identically,

, for which

f. JO 2 , because the case

does not vanish jo =: 1

is trivial.

Now put gj

=

-1 I. I. J JO

=: 1, ... ,jO-1

(5.3)

Then gj

=:

f. f: Jo J

-2

I.e ) I. J JO JO

(5.4)

and (5.5)

From (5.5), (5.4), (5.3) and (2.5) we find that for

=

lllog Ilfll

2 IIfr4

L

If/

k - fkfjl2

j
=

2

(5.6) +

Assume now that the set of points, where (5.6) vanishes, is not isolated in CR

. Then for each

not isolated in

CR

j

(l;;;.j:;; jo -1)

. Thus

gj

the set, where

gj( z)

vanishes identically for

1;;; j

vanishes, is ~

jo -1

,so

b. ,for which g. (z) == b. (l:;; j ;;; jO-1) J J J By (5.3) it follows that f.(z) = f.(z)b. , (j=l, ... ,jO-l) , which can be J JO J f(z) = f.(z) b written denotes the n -component vector , where b b JO = (b 1 , ... ,b jo _ 1 ,I,O, ... ,0) . Thus we have obtained (5.2) in this case, which that there must exist constants

completes the proof of Lemma 5. 2. The motivation for the following is the result of (scalar) Nevanlinna theory that the counting function t a 1 "/C f 1 (0)

N(r,a 1 )

is a subharmonic function of

If a meromorphic vector function w = f( z) n is given and if a E C ,then the functions

are subharmonic in

00

CR

= (f 1 (z) , ... , fn (z»

for

,

f( z) $.

°

log [f.(z) - a.1 ' j = 1, ... ,n J J -00 • The following gen-

or identically eqnal to

eral properties of subharmonic function; in the plane can be applied : the product of a subharmonic function and a positive constant is again

0)

a subharmonic function; Oi)

the sum of finitely many functions with subharmonic logarithrll is again

a function with subharmonic logarithm. We conclude that for each fixed log Ilf(z) - a II

tLehto [41J 36

a E Cn

the function

is subharmonic in

00 C R

this result could also have been directly obtained

since by (2.5) {dog

11£( z)

- a II

0

c;

which, by a property of subharmonic functions, characterizes

lod£( z) - a II

as being subharmonic. Now a function

Gc

domain

Cn

u(a)

of

is called

a E Cn

•

u(a)

-00;;;;

p I u r i sub h arm

semi-continuous and if the restriction of

u(a)

< 0

,defined in a

+00

,if it is upper

n i c

to any complex analytic plane

is a subharmonic function in each component of the intersection of

G

with

the plane. The latter means explicitly that for any complex numbers

a

n

the function

and u(al,; + b)

is a subharmonic function of the complex scalar variable

I,;

in every compo-

nent of the open set Using this definition and once more the above properties of sub harmonic functions, we conclude that for each fixed u(a) of

z

E

<'0

CR

the function

log II£(z) - a II

(a 1 , ...• a n ) is plurisubharmonic in Consider now formula (3.9) of Theorem 3.1 : a:::

Cn

21T

2~ flog IIHrei;)-a lid; , N(r.£1

::: V(r.a) + N(r,a) + log

lie q (a)11 (5.7)

o We will apply the following properties of plurisubharmonic functions:

(iii)

let

summable in

lu(a, t) I

<

W(a)

u(a, t) t s(a)

be a family of plurisubharmonic functions, which is

,where s

v

v-

is a finite positive measure. Assume that

being locally summable. Then the integral

J u(a,t)

dv(t)

is also pI urisubharmonic;

37

(iv)

the limit of a decreasing sequence of plurisubharmonic functions is i-

dentically

or is plurisubharmonic.

-00

r

From (iii) we see that for each fixed is finite and

'" a

(0;;; r

R)

<

such that

Hz)

de, the integral mean

on

r

2TI

a lid;

21.!IOollr
"(r .a)

o a E en

is a plurisubharmonic function of sum

)l(r,a)

+ N(r,£)

Now let there be points are poles for

f( z)

of

z

r

fez)

r

and is such that for each

de. By ( 5. 7) the sum

)l(r,a)

r

r

de, for which

on

. Then we can choose a sequence

nously decreases to on

. The same then holds for the

and thus is decreasing if

r

+

rk

k

N(r,f)

=:

a

, or which

' which monoto-

, f (z)

is finite and "'a

is an increasing function

decreases. According to (iv) we con -

elude that )l(r,a)

+

N(r,f)

=:

is again plurisubharmonic in exist points

z

Proposition 5.3

en

, even if

is such that on

r

f( z)

, which are poles or for which

=:

a

de

r

there

. We have obtained

is a non-constant meromor-

If

phic vector function in l1(r,a)

lim {11 (rk,a) + N(rk,f)} k++oo

+ N(r,f)

eR

=

then for each V(r,a)

r

+ N(r,a)

(0;;; r < R) + log

lie q (a) II

the sum (5.8)

is a plurisubharmonic function of One sees immediately that 11 (r,a) + N(r,f)

lim Iiall-r+oo

=

1

log Iiall

so that (5.8) is a plurisubharmonic function of minimal growth in the sense of t Lelong tLelong [42)

38

§6

Properties of

Proposition 6.1:

T ( r, f)

T(r,f)

is an increasing function of

Proof. We represent i8 ting z = re

C

in

p

r

for

(r < p < R)

0 < r < R

.

by (1.10), put-

=

A (z) p

L

B (z) P

z. ("') E C

J

log

2

--

z. ( "') z J p(z - z.(oo» J

p

-

P

Using the usual decomposition of

log

as difference of non-negative terms

and omitting non-positive terms in the right hand side of (1.10) we obtain from this '0 log IIf(re l ) II

+

B

p

(re is )

and since here both terms in the right hand side are not negative, + B

p

(re i8 )

so that

2TI

mer,£)

2~ f Ap (re") de

2TI 1 fBp(rei8)dS 2TI

+

o

o

The first term on the right hand side is the mean value of the harmonic function

A

over the circle

p

aC r

; hence its value is

compute the second term we observe that

B (z)

A (0) p

vanishes for

r

=

m(p,f)

. To

Izl=r,so

that we can write

2TI

J.. 2TI

f o

2TI B p (re i6 ) de

- 1 f (B (re ie ) - B (re ie » de

2TI

p

r

o

Since the difference under the integral of the right hand side is harmonic in

39

C

r

,this integral equals B

P

(0) - B

N(p,f) - N(r,f)

(0)

r

so that 21T

,', f

N ( p ,f) - N (r, f)

=

B p (re") de

( 6.1)

o It follows that

mer ,f)

m(p,f)

or, by definition of

T(r,f)

T(r,f)

+ N(p,f)

-

N(r,f)

T(p,f)

This proves (6.1). Remark. It is easy to see from the definition that r ->- 0

nite limit as

.

T (r ,f) Proof. Let Izl < p

o<

is a con vex function of

r < p < R

log r

for

o<

r

< R

G :r <

. Apply (1. 8) to the circular domain

The function

1 21T

C( z)

j

is positive in

G

---ds

z. (co) J

,and for

f C(tei8 ) d8 o

To this inequality we add the equality obtained from (6.1): 21T N(t,f)

N(p,f)

-

2~ f Bp(tei ') d; o

The result is the inequality

g(zt"'),z)

r< Iz.(oo)l

and harmonic up to the poles

.-l 21T

+

an

21T met,£)

L

ag(l;,z) + logllf(r,)II

aG

40

tends to a fi-

T(r,f)

r

< t < p,

I 211

T(t,f)

;;;

1 211

D(tei $) d$

(6.2)

o where

D

is given by C(z)

D( z)

The function p

D(z)

-

B (z) p

+ N(p,f)

is harmonic in each point of the circular ring

r < Izl <

and thus satisfies there Laplace '5 equation =

t.D

After multiplication by

t2

this can be written

o

=

+

(dlogt)

o

=

+

+

2

hence 211

j o

d 2 D(te i $ ) (dlogt)

o

=

d$

2

From this we conclude that the mean value in the right hand side of (6.2) is a linear function of for

t =r

logt

. On the circular boundary

boundary values D(z)

. We now compute this mean value for

= l6gllf(z)11

l6gllf(z)11

. Since

+ N(p,f)

• so that

Iz I = p

B (z)

t = p

the function

and

C (z)

has

vanishes there, we have there

p

211

~I 2'11"

D(pei
T(p,f)

=

o Izl = r

On the boundary finition of

D ( z)

o

= l6gllf(z)11

C(z)

, so that by de-

,

211

211

~I 211

we have

D(rei $) d$

=

,', fIrigllf(re

i;

o

JIId; .

2~

211

j

.

;

0

41

Using (6.1) we obtain from this

211

~ 211

j

D(re i ¢) d¢

=

+ N(r,f)

m(r,f)

T(r,£)

=

o These considerations show that for the boundary points of the interval

r:S t :::;

(6.2) being linear in function of

=r a n d

t

=P

logt, it follows altogether that

T(t,f)

is a convex

This proves Proposition 6.2 .

log t

Remark. If

t

equality holds in (6.2). The right hand side of

p

f( z)

is an integral vector valued function, then Proposition

6. I and Proposition 6.2 imply that T(r,f)

mer,£)

is increasing and convex in

logr

for

r > 0

•

For a non-constant meromorphic vector function generalized characteristic

T (r, £)

f

C

in the plane

increases unboundedly with

r

, the

,for oth-

erwise by the generalized first main theorem the sum m(r,a)

+ V(r,a)

+ N(r,a)

a E en

would stay bounded for every tends to infinity for

r

-7-

+ V(r,a)

N(r,f(O»

. We conclude:

+00

If for a meromorphic vector function

m(r,a)

,which is impossible since

f

in the plane the sum

+ N(r,a)

rests finite for a given point

a

as

r

-7-

+00

,then

f

reduces to a con-

stant. Vice versa, for a constant vector function + N(r,a)

+ V(r,a)

f( z)

== f( 0)

is constant and bounded for every

, the sum a

up to

m(r,a) a=f( 0) ,

for which this sum becomes infinity. In the remaining part of this section we persue a few considerations, which may be useful in connection with computations of For

w

= (wI"'" w ) n

E

en

we denote by

= Then we obtain n llwll

J.7

T(r,£) , mer,£) , or m(r,a).

2 s

Ilw II

s

the expression

10giiII w i'I s

<

log I!wl!

16gllw lis

<

16gllwll

s

0

16gilwl!

-

5

~

+

+ log lIwll

1 zlogn

+ l6gllwll

(6.3)

s

s

1 zlogn

~

l6g!lw lis

This shows that up to a bounded term, logiiw II s

1 zlogn

16gllw II

can be replaced by

; the latter is often easier to compute than the former. We can define

a modified proximity function

m (r, f) s

by putting

211

2~ J 16gli

m (r, £)

s

'lli,

f (re1

d,

o and a modified characteristic function T (r,£) s

m (r,£) s

T (r, £) s

by putting

+ N(r,£)

Then we have

o :;;

=

T(r,f) - T (r,£) s

~

mer,£) - m (r,£) s

1 zlogn

or mer,£) and m (r,£) , reT (r,f) s s . So in most con sidera tions T (r, £) spectively, differ at most by Z log n s can be used instead of T (r, f) , and m (r,£) instead of mer,£) . Furs ther, from (6.3) we obtain

which shows that

1 - zlog n

T(r,£)

and

1

1 + log

1

1 log - [Iwll

:S

Ilwll s

:;;

log Ilwll s

from which the following two inequalities are obtained + 1 log-[Iwll s

;;;

1

Z log n

+

1

+ log--

Ilwll

+ 1 log-Ilwll

If we now define a modified proximity function

by putting

I

m (r,a) s

+ 1 log-Ilwll s

for finite

21f

m (r,a) s

1

1 0 log + 21f

43

then we conclude the inequalities

1 Z 10 g n

m (r,a) s

m(r, a)

m ( r , a)

+

m (r,a) s

so that

o

m (1' ,a) s

-

1 zlog n

mer ,a) m(r,a)

Thus in most applications

can be replaced by

m (1' ,a). s

Finally we prove If

Proposition 6. 3: formation

A

f l' ... ,fn

are subjected to a linear homogeneous trans-

with constant complex coefficients and non-vanishing determi-

nant, then the following inequalities hold with

K

+

1m (r,Af) s

-

m (r,f) I s

;£

log (nK)

IT (r,Af) s

-

T (r,f) I s

;£

log (nK)

Im(r ,AO

- m(r,f) I

;£

a suitable constant:

+

1 zlogn

+

+

log (nK)

a linear homogeneous transformation Proof. We now perform on the f. ---J with constant complex coefficients and non-vanishing determinant. Let n

L

=

(Af(z». J

k=l

be this substitution and n f.( z)

L

=

J

k a j (Af(z»k

k=l

its inverse. Put K

=

k k max{ IA.I , la.l} .J, k J J

Then we conclude IIAf( z) II s

44

;£

nK Ilf( z) II

s

11f( z) II s

nK

IIAf( z) II s

so that, using the inequality

l6g (nK)

+ 16gllf(z)11

s

or 116g11Af( z) II

s

-

16gllf( z) II

+

log (nK)

I

s

From this we see that 1m (r, Af) s

-

m (r, f) I s

-

T (r,f) I s

+

log (nK)

The inequality IT (r,Af) s

+

log (nK)

follows from this and the definition of

T

s

. For the original proximity func-

tion we have Im(r ,Af) - m(r, f) I

Im(r,Af)-m (r,Af)+m (r,Af)-m (r,f) I s s s

1 + Im(r,AO-m (r,Af) I + 1m (r,AO-m (r,f) I ~ -2 logn + log (nK) s s s This shows Proposition 6.3 .

§7

The Connection of and

m(r,f.)

--J

T (r , f) ,

w::

f( z)

m(r,f.) _ J N(r,f j )

and

N (r , f)

with

T(r f.), J I

N(r,f.)

J

For the componen t functions tion

m (r , f)

w.

)

::

f. ( z) )

of the meromorphic vector func-

we have the usual notions of Nevanlinna theory m(r,co,fj )

_ N(r,co,f.) J

t

n(r,f.) == n(r,co,£.) J ) T(r,f.) J

== m(r,f.) J

+ N(r,f.) )

tNevanlinna [27]

45

m(r,a.) J

_ m(r,a.,£.) J J

N(r,a.) J

-

n(r,a.) J

N(r,a.,f.) J J

_ n(r,a.,£.) J J

etc.

Note that some of these quantities may not be finite or well-defined, if reduces to a suitable constant.

f.( z)

J

We now establish connections of these quantities with the generalized quantities

T(r,£),

To begin

m(r,f),

with let in

For each

m(r,a),

f 1 (z)

N(r,£)

and

N(r,a).

be a (scalar) meromorphic function in

0

.~

we can define an integer

CR • by put-

ting

Zo

==

0 if

==

the multiplicity of the pole if pole of f 1 ( z)

is not a pole of

f 1 (z) is a

The term in the following sum being positive only at finitely many places in 1

z1

r

:;;

n(r,£.) J

If now

1T

fez)

1T

z

(0

we can write

(f.)

1, ... ,n

z J

= (fl(z), ... ,fn(z»

we define for each

Again

n(r,£.) J

,and by definition of

==

z

in

is a meromorphic vector function in

CR

max j=l, ... ,n

an integer 1T

1T

1T

z

(t)

;;:

0

by putting (7.2)

(t.)

(f) is positive only at isolated points in z the definition of n (r ,0 ,it is clear that

L

z

z J

1T

nCr ,f)

(7.1)

(f)

Izi :;; r

and, regarding

(7.3)

Izi :;;r By (7.2) and (7.3) nCr ,f)

L

Izl;;;r

max j==l, ..• ,n

1T

z

(t.)

J

(7.4)

The sum on the right of (7.4) is

L

max j=l •... ,n

~

and

(f.)

1l

J

Z

izi$r

n

L L j=l izi~r

$

1f

z

(f.)

J

From this, (7.1) and (7.2) we obtain the inequalities n

n(r,f.) J

n(r,f)

j=l, ... ,n

which connect the "little" counting function of poles of

f

with the corre -

sponding little counting functions of the component functions. For a in C n we can now treat similarly the a - points of let

a1 E C

and let

f1(z)

f

. Again,

be a (scalar) meromorphic function in

which now is assumed to be not identically equal to in

(7.5)

a1

CR

. Then we put for

'

Zo

CR

:: 0

if

the multiplicity of the zero of f1(zO):: a 1 ' f1(z) =F a 1

if

Note that we do not define

if

f.(z) 1: a.

n(r ,a.) J

L

::

J

J

we have

(7.6)

v (f.-a.) z J J

izi$r f( z)

If now

(f1(z), ... ,fn(z»

morphic function, then we have for n(r ,a)

$

::

min j::1, ... ,n f. =$ a.

J

J

a

is a non-constant vector valued meroE Cn v (f.-a.)

z

L izi ;;;r

J

J

min n(r, a.) . -1 , ... ,n J Jf. =$ a. J J

(7.7)

47

We can summarize these simple facts in Proposition 7. 1: function in

CR

Let

be a meromorphic vector

o ;:;

. Then for

r < R n

n(r,f.) J

nCr,£)

Ln(r,f k )

(7.8)

, j=l, ... ,n

k=l If

f

o ;;;

is non-constant, then for n(r,a)

a.

$-

R

<

and

nCr ,a.) J

min f.

r

(7.9)

J J j=l, ... ,n

In order to establish similar relations between the counting functions z =0

of poles. let first

be no pole for

f( z)

N

, so that simply

r

net,£)

J- - d t

N(r,£)

t

o Then

z =0

f.

is also no pole for each

the definition of

N (r , f.) J

J

we can write

(j=l, ... ,n)

, and by (7.8) and

n

N(r,£.) J

N(r,£)

LN(r,fk) , j=l, ... ,n

.

(7.10)

k=l

z

Secondly, let

=0

be a pole for rO

r

N(r,£)

=

j

net,£) --dt t

+

j

f

. Then we can write for

net,£) - nCO,£) - - - - - - dt

+ n(O.f)Iog rO

t

o and the same holds if

f

is replaced by

hand inequality of (7.8) we conclude that

48

f. (j=1, ... ,n) J

. Using the right

r n N(r,£)

L j=l

75

J

n(t,£.) ~dt t

00)

+

rO n

L

=

00)

+

N(r,fj )

(7.11)

j=l For the other direction we first note that =

n(t,f) - n(t,f.)

J

"

(11

L

Iz! ~t

"

11

L

!zIH

z

(£) -

11 Z

Z

(f.)

J

n(O,£) - n(O,f.)

(f.» J

J

so that, using the left inequality of (7.8), rO

r

N(r,f)

~

J

n(t,£.)· J dt

+

J

n(t,£.) - n(O,f.) ] ] dt

t

+

n(O,f)log rO

t

o For ~

N(r,f) for

n(O,£)logr o

we have

, so that

N(r,f.) ]

.0 < r < 1

we can in general only write

N(r,£)

N(r,£.) )

+ (n(O,£) - n(O,f.»logr o

)

=

N(r,£.) J

+ 0(1) .

We summarize: Proposition 7.2 : Under the conditions of Proposition 7. 1 the following holds: 0)

If

z = 0

is not a pole of

f

,then n

N(r,f.) J

(ii)

If

z =0

~

N(r,£)

L

N(r ,fk )

,

(j=I, ... ,n) .

(7.12)

k=l is a pole of

f . then

49

n

N(r,£.) J

+ 0(1)

L

N(r,£)

+ 0(1)

N(r,f k )

(7.13)

k=l

on the left hand side can be omitted if

O( 1)

where the

Comparing (7.12) with (7.13) we see that (7.13) holds in all cases. In view of (7.9) we now consider the counting functions finite

First, if

a

z = 0

n(O,a) = 0

then also

is no

f. ]

a. - point of J

for

]

j

=

for

j=l, ... ,n

, and we have

r N(r,a.)

N (r, a)

r

(

n(t,a.) ) dt t

=

N(r,a)

n(t,a)

j

0

dt t

0

and, under the conditions of inequality (7.9) ,

N (r, a)

J

(7.14)

N (r ,a.)

min f. =f, a.

J

J

j=l, .,. ,n

In the general case in the right hand side of (7.14) an

0 (1)

term has to be

added. We formulate Proposition 7.3:

Let

fez)

romorphic vector function in

0)

If

z =0

N(r,a)

= (f1(z), ... ,fn (z» CR

is no

and

a

a. - point of

J

min f. '$ a. J J

E

C

f. J

be a non-constant me-

n

for

j=l, ... ,n

,then

(7.15)

N(r,a.)

J

j=l, ... ,n (ii)

In the general case,

N (r, a)

min f. '$ a.

J

N(r,a.) J

+

00)

(7.16)

J

j=l, ... ,n

We now consider the proximity- and the characteristic functions. We have the obvious inequalities m(r,£.) J

m(r,£)

j=l, ... ,n

From the inequality n

L

16g x k

+

log n

k=l

we see that in the other direction mer,£)

1

+

2 10gn

We formulate this and the consequences of Propositions 7.2 and 7.3 for the Characteristic

T

in

Proposition 7. 4: function in

Let

CR

be a meromorphic vector

Then for

0)

m(r,I)

(ii)

If

o<

r < R

1

+

z =0

is not a pole for

f

(7.17)

2 10gn j=l, ... ,n

, then for n

s

T(r ,f)

;;>

T{r,f.) )

LT(r,fk )

+

1

(7.18)

2 logn

k=l (iii)

If

z =0

is a pole for

f

,then for

j=l, ... ,n

n

L T(r,fk )

T(r,f)

+ 0(1) .. (7.19)

k=l

Comparing Oi) and (iii) we see that (7.19) holds in all cases.

§8

The Order of Growth

We shall apply to 'vector valued Nevanlinna theory the following notions and results of Nevanlinna theory and of the theory of integral functions. Definition 8.1: Let

s(r)

(r > rO)

is said to be of 0 r d e r (of growth)

be a positive function·. Then rP

,or shorter of a r d e r

p

s(r)

,if

the limit superior P

= r++ oo

log s(r) logr

(0

:;0 p :;0

+00 )

51

is finite. P

is called of i n fin i t e o r d e r if this limit superior is

s(r)

= +00 If

then this definition is equivalent to the following properties: P+E: for each E: > 0 there exists r 1 > 0 , such that s(r) < r for all r > r1 and there exists a sequence of r - values tending to infinity, s(r) > rP-E: for which P < +00

In this sense then, the growth of s (r) is characterized by the compar P ison function t r if P < +00 • The definition of finite order gives no conclusion about the behaviour of the quotient s(r)

for large

r

Therefore one defines

Definition 8.2 t:

A function

s(r)

t y peT

or m a x i mum

of order

rP

is of min i mum , mea n

respectively, according to whether the limit su-

perior

is zero, posi ti ve an d finite, or is infinite. Now, let the function creasing. Let

j

j.1 > 0

s(r) --:j1 dr rj.1

Then for each

tNevanlinna [27]

52

s(r)

be of order

rP

and, for the moment, in-

be a number such that

<

E:

>

0

there exists

r1

>

rO

,such that for

+00

j

s(r)

+00

j

dt t

]1

+1

r

which shows that

p ;$

s( t)

- - dt ]1 + 1

r

Now assume that

]1

<

t

]1

. In this case we put

> p

lJ -P

-2

and see that there exists

such that

+00

s( t)

r -dt +1 ]1

J

=

<

t

j

t

P -]1 -1 +

E:

dt

<

r 1

This reasoning shows Proposition 8. 3 : s(r)

If

P

(8.1)

--dt t

]1

+1

]1

> P

,divergent for

s(r)

if and only if integral (8.1) converges for every

s(r)

(r > rO)

we say that

s(r)

]1

=

belongstotheconvergence

in the second case it belongs to the d i v erg e n c e s(r)

]1

>

is a positive function of finite order

integral (8.1) can be convergent or divergent for

If

is that the integral

(0 < P < +(0)

s( t)

is convergent for

= 0

to be of order

(r>rO>O)

j p

Necessary and sufficient for the positive increasing function

p

p

,then the

classoforder

p,

cIa s s t of order

p.

then the first part of the reasoning above Lemma 8.3 shows that T

0

• In the first case

is increasing and of convergence class of order

be of minimumtype i. e. ,

is of order

0 < p < +00 s(r)

must

= 0

For meromorphic vector functions in the plane

C

or in a finite disc

CR

tNevanlinna [27] 53

it is natural to define order and class exactly as in Nevanlinna theory , In the case of the plane e, g"

we give

Definition 8,4: The" (fl(z)"",fn(z»

r d e r"

0

in

The " lower

0

r d e r

=

is the order of

C

T(r,f)

f( z)

Le"

log T{r,f) logr

lim r++ oo

p

of a meromorphic vector function

p

X

II

of

£( z)

is the limit inferior

log T{r,f) logr

lim r++ oo

If

0 < p < +00

, then the" t y P e " and the " cla's s "of

respectively defined to be the type and the class of X = p

If

, then

sense of Borel

o<

f{ z)

E , F

g row t h

, such that

o<

11

in the

E < +00

F < +00 T{r,f)

lim

E

r++ oo then

r P

will be called of " v e r y

£( z)

lim r++ oo reg u 1 a r

F

g row t h " in the

sense of Valiron t, If the limit T (r, f)

lim r++ oo

rP

exists, then g row t h

f ( z) 11

tValiron [37]

will be called of " per f e c t 1 Y

in the sense of Valiron t

It'is clear that

54

are

T (r, f)

is called of " reg u 1 a r

. If there exist constants

£( z)

always,

reg u 1 a r

We can show Proposition 8.5:

Let

f.

and that of

max

f. J

. Then

be a meromorphic vector

denote respectively the order of

(8.2)

p.

denote respectively the lower order of J • then

(j=1, ... ,n)

max

and that of

f

(8.3)

/.. J

f_1(z), • . . ,fn (z)

in particular. if

f

is given by

p

.\ .

and

.:\.

Pj

J

j If

and

p

(j==1 •••• ,n)

J

==

p

== (f 1(z), ...• fn (z»

f( z)

function in the plane. Let

are of regular growth in the sense of

Borel, then also the vector function

is of regular

growth in this sense. Proof. From Proposition 7.4 we have the inequalities

n T(r,f.)

T(r,f)

+ O( 1)

)

LT(r,fk )

+ 0(1)

(8.4)

k==l

for

The left hand side shows that

j=l •... ,n

max j

(8.5)

p

p.

)

In the other direction we assume that

Now

max

>

p

choose a number

].I

p.

J such that

p

>].1

> max Pj

. From the right

inequality (8.4) and Proposition 8.3 we conclude +00

n

jL

k=l

T(t, f k ) t

].I

+1

dt

=

+00

to

> 0

,

to which is impossible. since by the same Proposition each integral

55

+00

I

k=l, ... ,n

is convergent. Thus we must have

max . J

this proves (8.2).

, and observing (8.4)

p.

J

(8.3) follows immediately from the left hand inequality (8.4).

Let

f

o<

be of order

lim r->-+oo

T(r,f.) J rP

+00

p <

. Then from inequality (8.4) we have n

T(r,£) lim r->-+oo

~

T(r,f k )

L

lim r->-+oo k=l·

~

rP

rP

(j = 1, ... ,n) . Here in the extreme left and right hand sides all terms vanish which belong to component functions of order are the component functions of order

P

. Thus if

f. , . . . ,f. J1 Jm , then we have the following ine< P

qualities between types m ~

T.

J.1

i=l

In particular, if

f. , . ) 1 If

f

L

~

T

f

is of order

(8.6)

T.

Ji

o<

P

<

and of minimum type, then

,£. are of minimum type. Jm is of convergence class, then we see from inequality (8.4) that

. . . ,f. are also of convergence class. If J at least She function among f . , . . . , f. Since according to

Nevanlin~!

transformation

J J

=

f

f., )1

is of divergence class, then

must be of divergence class.

£. is inJ is transformed by a linear

theory thrg characteristic of each

variable up to an additive bounded term if

L. f.

+00

cd. + S. JJ J y.£. + O. JJ J

f. J

(cx.O. - S.y.)

J J

J J

(8.7)

'" 0

we have Proposition 8. 6:

The order

meromorphic vector function

56

p

,and if

o<

p <

+00

the type

T

of a

in the plane rest the

same if some or all component functions

f. J

mations (8.7). Now, with

f( z)

If

and

p

f.

J

,f'(z». n

and

p

,

we can consider its derivative defined by

=

f'( z)

Pj

0=1, ... ,n)

f: J

are subjected to linear transfor-

,

P;

denote respectively the orders of

f , f'

,Jthen from Nevanlinna theory \t is known that

j = 1, ... ,n

=

Thus, applying Proposition 8.5, we have also P

=

pj

max

=

max

Pj

P

This shows Proposition 8. 7:

If

is a meromorphic vector

function in the plane, then the order f'( z)

P'

of the derivative

=

is equal to the order

P

of

f( z)

tWhittaker [601

57

3 Generalization of the Ahlfors-Shimizu characteristic and its connection with Hermitian geometry §9

o

The generalized Ahlfors - Shimizu characteristic

a E Cn

In Chapter 1, §3 we proved for

,

T(r, f)

0 < t < R

the i.dentity

211

j

1 211

'0 logllf(te1)-allde

+ N(t,£) = V(t,a) + N(t,a,f) + log lie

q

(a)11

o Putting

(9.1)

a::: 0 = (0, .. , ,0)

ponen t vector N(t,(l,f)

(l, £)

in (9.1) and replacing

it is clear that

f

by the

N ( t, a, (l, £) )

O

n +1

an d

com-

N ( t ,f) =

Using this and the explicit formula t

V(t,O)

j j

2~ ~s

:::

o

C

"oglif(,)11 dx , dy

(z

= x +i.y)

,

s

we obtain from (9.1) by differentiation that

,;, j"ogJ!

+

IIf(,)II'

dx' dy

Ct

d~ [N(t.f)

"j 1

+

211

logV, +

'0

IIf('e' )11

2

] (9.2)

de

o By integration from

t = rO

>

0

to

t =r

(r O < r < R)

we get

r

J 2~: J Alogv'" 1'0

112 dx

11£(,)

, dy

Ct

r

21f

J n(t,f)~n(O,f)

dt

r + n(O,f)log-

+

rO

1 2n

JlogY/I

+ II[(re I"e )11 2 de

o

rO

(9.3)

At

Z :::

0

we have the vectorial Laurent-development + z q +l c

f(z)

q+l

q ::: n(O,O,f) - n(O,f)

where

ros and poles of

f

at

is the difference between the number of ze0

Z :::

+ .• "

,and

IIc q II

'"

0

Concerning the development of

G) at

Z

:::

0

(column vector)

we distinguish three cases:

i)

q>

ii)

q:::O

0

iii) q< 0:

C) C) C)

:::

:::

:::

(~)

+

Go)

+

,q

zq(:q) +

zCJ +

CJ·

zq+l

z2

zq+l

GJ

(~

) q+l

+ .•.

+...

C

(0c q +l ) +... + Co )+

z

e)+.". c1

In these three cases respectively the expression

59

= behaves as

r0

-+

0

like

or

In the cases i) and ii) the point iii) the point " . ,0)

z:::

0

z::: 0

is not a pole of

f

In case i) we have

is a pole of

=

; in case ii) we have

1If(0)1!

.

f; in the case f(O) = 0 ::: (0,"

i) and

ii) can be

considered together so that we have logJ1 + IIf(0)1[2 finite at :::

)

Thus, if

z::: 0

l'~;

I

rO

Ct

r

=0

is not a pole of

'10g"\ +

lI£(z)II'dx

is

(cases i) and

ii) )

A

dy

f

is a pole of

lie q II f

+ o(r O)

if

(case iii»

we obtain from (9.3)

"

I 2')]

r

r

f

-n ( 0, f) log r 0 + log

z

if

+ o(r 0)

z ::: 0

n ( t , f)

~n ( 0 ,f) d

J

+

1 271

"e 2 de - logYI 1 + 11£(0)11 2 logYI 1 + IIf(re1)1[

o

rO so that, letting

t

r0

tend to zero,

-o(r o)

,

r

J 2~'tJ .'og/,+ 11£(')1I o

2 dxA dy

"

Ct

211 N (r, f)

+

J logYI 1

1 21T

II.

+ lIf(re l"e ) 2de

(9.4)

o If on the other hand

z::: 0

is a pole of

f

,then we get from (9. 3)

r

f 2~~ J .lDg~, rO

+

11£(,)11 2 dx

A

dy

Ct

211

r

J n(,.Ot(o.n d'

+ n(O,f)logr

1 + 21T

and letting

J

J logVl/

+ lIf(re l"6 )11 2 de

o

rO

r

"

r 0 -+ 0

2~~ J ",ogJ,

o

+

lIf{z)" 2 dx

A

=

dy

Ct 21T

N(r,f)

1 1T

r logYI 1 +

+ -2

J

"e 2 de IIf(rel)1I

-

log

IIcq II .

(9.5)

o Now the function

61

which appears in (9.4) and (9.5). behaves asymptotically very similar to mer,£)

; it can serve equally well as

proximation of

f

mer,£)

to infinity on circles

dC

as a measure for the mean ap. We can therefore introduce

r

the following modified proximity function

o

o

m(r,oo,f)

m(r,f)

with respect to infinity:

Definition 9. 1 :

21T

o

mer,£)

::

1 21T

f.j log

1 + \\f(re 1"e )11 2dE)

(9.6)

o if

Z ::

0

is not a pole of

f, and

21T

o

m(r,f)

::

,'. f 10gJ,

lIf(re")1I 2de

+

-

log

Ilc q II

(9.7)

o if

z:: 0

f

is a pole of

. Here

c

q cient vector in the Laurent development of

is the first non-vanishing coeffif( z)

at

Z ::

0

.

With this definition (9.4) and (9.5) can be written in this unified form: r

~(r,£)

+

N(r,f)

0

f 2~~ f o

AlogJI +

11f(,)1I 2dx

A

Ct

Now we already observed on p. 26 that in the scalar case

62

dy

.

(9.8)

and that

i JAlOg.,!'

+

1',(,)1' dx

dy

A

Ct is the spherical area of the Riemannian image of

Ct

under the mapping

f 1;

and in this case the sum in the left hand side of (9.8) is called the characteris tic function of AhHors and Shimizu or the spherical characteristic of

t

fl.

We now try to find in the present vector valued case the correct geometric meaning of the expression

i J . log.,!,

+

II f(,) II' dx

A

dy

Ct occurring in equation (9.8). We compute

i

tdog V\

i·4.

+ IIf( z) 112 dx

A

32 --10gJ1 + /If(z)11 2 dZd

z

=

i

( 1 + LkfkTk )( Lkfilk) -

"2

(1

=

dy

~ dz ~

A

dz

aalog(l

=

+

r.f{;

(Lkfkfk )( Lkfkfk )

dz

=

A

dz

(9.9)

+ Lkfkfk)2

tAhlfors [1]. Shimizu [54]

63

k = 1

Here each sum is extended from en

Now en £; pn

is sitting in

pn

to

as an open set, and we have the inclusion map

Pulling back by the inclusion we obtain the

II

k=n

the Fubini-Study metric on

F ubi n i - Stu d y

met ric

0

n

en

1\

-r.

pn

,

It is given

by ds 2 =

(1 + LkWkw k ) (Lkdw k 0 dwk"j- (LkWkdw k )0 (LkWkdw k ) ( 1 + Lk w~wk )2 (9.10)

Its Kahler form is

(1

+ LkWkw k )( Lkdw k

1\

dW k )-( LkWkdw k )

(1 + LkWkwk

= If

ds 2

"41 dd C log ( n ::; 1

-

1 + Lk ~ ~

1\

(

LkWkdw k )

l

)

(9.11)

for example, then

pI

is the usual spherical metric on

is the Riemannian 2-sphere

S2, and

e

dW10 dW1 - 2 (1 + wl~) which is a conformal (Hermitian) metric of constant Gaussian curvature

4

and the associated Kahler form dU l

=

1\

dV 1

::;

- 2 (1 + w 1w1 )

is the spherical volume form. We now return to the general case the pull-back of

64

to

is

n

~l

. (9.11) and (9.9) show that

*

( 1 + rkfkfk )( rkdfk " df k )-( rkfkdf k ) " (rkfkdfk ) i = u 2"

f w...

( 1 + rkfkfk

l

=

(9.12)

This shows that in the general vector valued case

i J AlogV'

+

11£(,)11' dx

~

1

the integral

=

dy

A

n

(9.13)

Ct is nothing but the vol u m e of the image of

(If (9.13) is divided by

Tr

in

Ct

pn

under the map

we obtain the nor mal i zed

vol u me).

We now return to equation (9.8). At this point, in view of equation (9.8) and the given geometric interpretation (9.13) of its right hand side, it is natural to introduce a new modified characteristic function

o

Definition 9.2

T(r,f)

=

o

m(r,f)

o

T (r, f)

by

+ N(r,f)

(9.14)

o where

mer,£)

We will call z u

is given in Definition 9.1 .

o

T (r ,f)

the" g e n era 1 i zed

c h a r act e r i s tic

II

or the

II

A h 1 for s

g e n era 1 i zed

- S him i -

s p her i cal

c h a r act e r i s tic ", since it agrees with the characteristic of AhHors Shimizu in the scalar case. Summarizing, we can formulate this result: Theorem 9. 3:

Let

Then denoting by en

f

uu

be a vector valued meromorphic function on C R the Kahler form (9.11) of the Fubini-Study metric on

,and defining the generalized spherical characteristic

o

.

T(r,f)

by Defi-

nition 9.2 , we have the formula:

65

r

o

T (r, £)

::

2~ j ~t

j

o

Ct

'10g/1 ,

,

o

This geometric interpretation of

T (r, f)

terpretation of the generalized characteristic

Ilf(,)11 2 dx

, dy

(f '" con st.)

(9.15)

.

gives also a quasi-geometric inof N evanIinna because

T (r , f)

of the following Proposition 9.4

T(r,f)

o

T(r,f) as

r

-+

R

T (r, £)

::

o

differs from

T(r,£)

only by a bounded term:

(9.16)

O( 1)

+

.

Thus, in many investigations

o

T

ean be used instead of

T

and vice

versa, without any changes of formulas. The estimate (9.16) can be seen as follows. We have

211 0

m(r, f)

::

21,

j

10g/1 , Ilf(re ie ) 112 de

d

0 where d if

::

z :: 0

pole for

logJ1 + Ilf(0)11 2 is not a pole for

f

16g Ilfll

, and

f

d

::

z :: 0

if

log lie II q

Since <

logJl + IIfl12

5

+

16g IIfll

logVz

it follows that mer,£)

;;;

0

mer,£)

+

d

;;;

mer ,£)

+

IogVz

is a

-

o

d

mer,£)

mer,£)

logyZ

-

d

This shows that the difference

o

mer,£)

-

mer,£)

o<

remains bounded for

r < R

and from (9.14) we see that the same holds

therefore for the difference

o

T(r,f)

-

T(r,£)

Proposition 9.5:

be a vector valued mero-

Let

morphic function. Then the order and type-class of

f( z)

are the same as

the order and type-class of the integral

j C

f

*ub

r

Proof.

j C

f

Suppose that

*

<

fDO

r

Then by (9.15) r

0 T(r,f)

<

j~

tk

dt t

+

0(1)

=

K k r Ilk

+

0(1)

rO This shows at once that the order and type-class of that of the integral in Proposition 9.5. Vice versa if

o

T (r, f) then

<

o

T(r,f)

cannot exceed

2r

fr' "b 10~ 2 C

~

1

f f· t dt

11

r

r

f Wo

0 T(2r,f)

~

K(2r)k

~

,

(r

>

r 0)

Ct

so that the order and type-class of the integral in Proposition 9.5 cannot ex-

o

ceed that of

T (r, f)

. This proves Proposition 9.5.

Since the integral representation (9.15) for

o

T (r, f)

has the same form as

,the same proof as that of Proposition 4.1 shows that

VCr ,a)

o

lim r++ oo

T(r ,f) logr

!

::

lim r++ oo

f / Wo C

(9.17)

r

We can apply this to rational vector functions. Using (9.17), (9.14) and (4.8) we conclude Proposition 9.6

If

is a vector valued rational

fez)

function, then the normalized volume of the image in is an integer, which is equal to the degree

1 11

lim r++ oo

f f\ C

§10

* 00) n(

pn

of

C

under

of f

* n(oo)

::

f

(9.18)

r

The generalized Riemann sphere

The complex Euclidean space

Cn

with coordinates

can be identified with the real Euclidean space by assigning w

_u

, where

The real Euclidean space

w. J

=

R 2n

w::

(w l' ... ,w n)

with the coordinates

=

can be viewed as the hyperplane rectilinear coordinates of R 2n +1

Let

S 2n

0

R 2n+1

in

, where the

are denoted by

be the sphere u 21 +. . . +

of radius

0.5

U

22n + ( u 2n +1 -

0.5)2

(0, ... ,0,0.5)

centered at

0.25

It is tangent to

R 2n

origin which is its south pole and has north pole p = (0, ... ,0,1) 2n+l " line in R through the fIxed pomts p = (0, ... ,0,1) and

at the The (u l " ..

is parametrized by a.(t)

::;

::;

o ;;;

(tu 1 ,···,tu 2n ,1-t)

The points of intersection of this line with

t ;;; 1

S2n

correspond to values

t

,

which satisfy the equation

2

2

t (u l

(0.5 -

+

+ ..• + t ::; 0

Its only solutions are

t)2

and 1

1 t

0.25

::;

::;

2

+ ... + u 2n

::;

1 +

where we have used the abbreviation

Ilull 2 :

=:

The first value

::;

t::; 0

corresponds to the point

1;. J

I; 2n+1

2

2

u 1 + . . . + u 2n '

corresponds to the north pole (1;1, ... ,1;2n,1;2n+l) (j:::1, ... ,2n)

::;

::;

of

p

S2n

, and the second given by (10.1)

(10.2)

69

As usual we define map

s

from

the point ing

u

R2n

s(u);; to

lIull

p

2

s t ere to

g rap h i c p r

0

0

j e c t ion

to be this

S2n - {p}

,which sends a point u E R2n to 2n (E., 1"'" f, 2n+1) E S - {p} on the line segment join-

. From (10.2) we compute

1

f, 2n+1

;;

1 -

=

f,2n+l

1 -

f,2n+1

so that from (10.1) (.

u. J

J

;;

1 -

(j=l, ... , 2n)

(10.3)

f, 2n + 1

Thus stereographic projection

f,

;; s(u)

has an inverse

u

= s

-1

U:) .

whic:h is given by (10.3).

= (u 1 ' ... ,u 2n ) and v;; (v l ' ... ,v 2n ) be two points , and respectively f,;; (f,1 .... 'f,2n+l) and n;; (n 1 , ... , · · S2n {p} D b h t elr lmages on . enote y

Now let in

R2n

n 2n +1 )

[u,v 1

u

:

•• +

;;

the Euclidean distance of

E,.

n

E S2n - {p}

. Using (10.1), (10.2) we

compute ;;

+ ( lIull 2 (!lvll 2 +l)-llvll 2 (Ilull 2 +1»)2

+ [[u[[\ [[v[1 2 +1)2 + [[v[l\ Ilul[2+1)2 - 2[[ul[21[v[12( Ilv[[2+ 1)( [[uI1 2 +1) (l+I[vl[2)2[[u[[2(l+[lu[[2) + (l+[lu[[2)2[[vl[2(l+[lv[12) ,2 2 2 2 2 2 - 2u'v(l+livli )(l+I[u[[ ) - 2[[u[[ I[vll (1Iv[1 +1)( liull +1)

~

2 + (l+[lull 2 2 - 2u.v -2[!u[1 2 IlvliJ ::: (l+[lu[1 2 )(l+[[vll 2 ) [(l+I[vl[ 2 ) Ilull ) Ilvll

2 [lIuli 2 + Ilvll 2 - 2u'v] ::: (l+lluli 2 )(l+lIvll) 2 2 2 ::: (l+IJull )(l+I[vli ) Ilu-vll

Thus we have obtained the formula [Iu-vll [u,v J

(l0.4)

:::

Remark. We conclude from (l0.4) that the Euclidean metric 2n+1 ds 2

L

::

j:::1 of

R2n+l

induces on

S2n 2n

ds 2

by the inclusion

duo 0 duo J J

Lj:::l

the metric n

duo 0 duo

J

J

(l+lluI1 2 )2

:::

Lj:::l

dw.0 dw. J J (l+llwl[2)2

S2n ~ R2n+l t.

Using the identification

en::: R 2n

section, we now define for any two points

mentioned at the beginning of this a, bEen

the expression

t Spivak [33]

71

[ a, b J

by putting lIa-bl!

=

[ a,b 1

(10.5)

=

[a,b]

We have shown that

images of a', b on the sphere diameter of

o

~

S2n

is

[a, b 1

will be called the" g e n era 1 i zed

dis tan cell or the of

a

and

b

under stereo graphic projection; since the

1

[a, b J

The number

S2n

g e n era 1 i zed

II

n = 1

,since for

We recall now that the symbol droff one-point compactification

c h

0

r d a 1

s p her i ,c a i d i s tan c e"

it agrees with the ordinary chordal dis-

CI :::

tance of points on the Gaussian plane

tends to

is the Euclidean distance of the

we always have

1

~

[b,a]

e

denotes the ideal element of the Alexanen

u

{oo}

. We will say that

a E en

,if the expression

00

a Ilal! tends to a unit vector as

lIali + +00

• Then clearly

1 lim b+ oo

[ a, b I

and it is thus natural to define the generalized chordal distance any finite point

a E en

to the IIpoint at infinityll

[ a,OO ]

of

to be the number

1 (10.6)

[ a,oo]

Further, since obviously lim

o

[ a,OO 1

a+ oo

we define [oo,ooJ

72

[ 00,00]

by putting

o

(10.7)

In this manner, the generalized chordal distance [a, b 1 n fined for any two points a,b E C U {co} and satisfies

o ;;;

[a, b 1

;;;

has been de -

1

It is now natural to call

S2n

the" g e n era 1 i zed

R i e man n

sphere" .

§11

The spherical normal form of the generalized first main theorem

In Chapter 1, §3 we proved for

a E Cn

identity (3.9)

2n

2~ I

log lifere i .) - alidl

V(r,a) + N(r,a) - N(r,£) + log Ilc (a)11 q

o here

( H.l)

log Ilc (a)11 q

Now, if

=

z = reiij>

log Ilf(O) - all

if

is not a pole for

£(0) '" a,CO w

==

f( z)

we have from formulas

(l0.5), (10.6)

IIf(reiij»

- all

1

so that

==

[f (re i

Using this the left hand side of (11.1) can be written 2n

2~ I o

2n

log [I(rei'),ald. + logJ1+lIal1 2 + /nj log _ _I_ _ d

o

73

and (11. 1) can be given the form 2rr

i. flOg

1 [f(re i
N(r,f)

+

d

:::

,00 ]

0

2rr

2" flog

1 d1>

+

V(r,a)

+

N(r,a)

+ logllc (a)11 q

[f (re i1> ) , a ]

0

( 1l.2) for

a E en We now introduce a function

Definition 11.1:

If

o

m(r,a)

a E en u {oo}

by the following formulas:

a;to f(O)

,

,

f(O) E en u {co}

,we put

2rr

0 m(r,a)

2~ flog

:::

1

1 d¢

(11. 3)

log

[f(re i ¢) ,a]

[£(0) ,a]

0 In the other cases we put 2rr 0 m(r ,a)

:::

2',

1

f,og

(11.4)

k

dcjJ

[f(rei¢),a] 0

where log

~0)112 J 1+lla11 2

:::

k

if

a ::: f(O) ;to co

Ilc (a) II q (11.5)

log Ilc II q

if

a ::: £(0)

As in Nevanlinna theory, we sometimes write

::: co

0 m(r,£)

for

0 m(r,oo)

. De-

finition 11.1 is then consistent with and more general than Definition 9.1 • The function

o

m(r,a)

pro x i mit y

will be called the" g e n era liz e d

s p her i cal

fun c t ion " or the " g e n era 1 i zed

A h 1 for s -

Shimizu

proximity

function

ll •

Using this function, equation (11. 2) can be written as

o

m(r,f) + N(r,f) (11. 6) even holds for

o

m(r,a) + N(r,a)

= a =

+ V(r,a)

,since we defined:

00

(11.6)

V(r,oo) - 0

Note that the sum on the left hand sioe is exactly the generalized ShimizuAhlfors or generalized spherical characteristic

o

T(r,f)

o

m(r,f) + N(r,f)

=

introduced in §9, Definition 9.2 . Note also that in (11. 6) both sides tend to zero as

r -+ 0

We summarize: Theorem 11.2 (First Main Theorem in Sp herical Formulation) : (fl(Z)""

eR

is a non-constant meromorphic function in

,fn(z»

w=f(z)=

If

, then the

identity

o

o

T(r,f) holds for

m(r,a)

o<

r < R

+

and

N(r,a) a E

+

(11. 7)

V(r,a)

en u {oo}

In applications of the first main theorem sometimes the spherical formulation is of advantage and sometimes the original form is better suited.

§12

The mean value representation of

o

T (r ,f)

r 2n (0 < r < R) can be regarded as a function on S2n . We denote by dS the 2n - dimensional measure of S2n . The total volume Is 2n l of S2n

By means of stereowaphic projection,the right hand side of (11. 7) for fixed

is

=

We consider the integral of for

a E

en , a;t

f( 0)

Z over

,f( 0) E

en u

S2n {oo}

. According to Definition 11. 1, the function

o

m

is given by

75

2n

o

m(r,a)

/nf

=

1

log _ _I_ _ d¢

log---

[f(rei¢),a]

(12.1)

[f (0) ,a]

o Now, 2n

J

dS2n 21n

j

1 _ _ _ _ dS 2n ,

log _ _I_ _ d¢

[f(re i ¢) ,a]

[f(rei¢),a]

S2n

0

and for reasons of spherical symmetry, the inner integral in the last expression is independent of the point f(re i ¢) . This point can be replaced e. g. by the point

f( 0) , so that the inner integral is equal to

r log _ _I_ _ dS 2n )

[f( 0) ,a 1

S2n which does not depend on

¢

. From this and (12.1) we conclude that

o

=

Thus, assuming that

f

is non-constant and by taking the spherical mean of

( 11. 7), we obtain

o

T(r ,f)

=

1 -2

Is nl

j

(N(r,a) +V(r,a»

S2n For

we have

a " f(O)

r N(r,a)

=

j ()

n(t,a) dt t

dS

2n

(12.2)

so that r

I

N (r, a) dS 2n

II dt t

:;

0

S2n

The function a

E

en

n (t, a)

n(t,a) dS 2n

S2n

as function of the variable

,which do not belong to the image of

f

a

; if

has value zero for all n > 1

therefore, it

vanishes up to a set of 2n - dimensional measure zero on the sphere

S2n

.

Further,

i

i~t i r

V(r ,a) dS'"

o

S2n

v(t,a) dS 2n

S2n

and we know that this vanishes identically for (12.2) can be written

+

Is nl

2n V(r,a) dS

. Using these facts,

n > 1 ,

if

S2n

0

T(r,f)

i

n = 1

=

l11

i

(12.3) N(r,a) dS 2

n = 1

if

s2 If we introduce the abbreviation

A(t,f)

1 -2 Is nl

i

by putting

v(t,a) dS 2n

if

n > 1

S2n A(t,f) if

n

=:

1

S2

77

then (12. 3) can be written

j r

o

T(r,£)::

A(t,O

dt

t

o in the case

n:: 1

this is a well-known formula of Nevanlinna theory. We

summarize this result in Theorem 12.1:

o

The generalized spherical characteristic

constant vector valued meromorphic function in

CR

T(r,£)

of a non-

can be represented by

the integral mean

if

o

T(r,£)

n > 1

,

(12.4)

::

~f

N(r,a) dS 2

n ::: 1

if

S2 Equivalent to this is the formula r

o

T (r, £)

f

:::

A(t,£) ---dt

(12.5)

t

o where

A(t,£)

denotes the spherical mean given by

1 f Is 2n l

v(t,a) dS 2n if n > 1

S2n A (t, f)

1 f
Is nl

.

S2n

( 12.6)

~f S2

78

n (t, a) dS 2

if n" I

Since by the properties of positive for ishes for logr

0 < t < R r =0

for

V(t,a)

or

n(t,a)

the function

,we conclude from this result that

o

A(t,f)

T(r, f)

is

van-

and is a positive, increasing and strictly convex function of

o<

r < R

,unless

f( z)

reduces to a constant vector. The

reader will remember that we already found properties of monotony and con vexity for the generalized Nevanlinna characteristic in §6,

Chapt~r

2 .

By comparison of (12.5) with (9.15) we obtain the following geometric interpreta tion of the sp herical mean

A ( t, f)

Corollary 12. 2

A(t,f)

~J

=

l'Ilog.J1 +IIf(z)11 2 dx

21T

dy

A

=

Ct (12.7)

We conclude this section by giving an application of the representation

(12.4). In the case

n > 1

we have

o

T(r,f)

=

so that

vo(r,a»)

dS 2n

=

o

(12.8)

T(r,f)

In the first main theorem in spherical formulation

o

o

m(r,a)

T(r,f)

we have for

a

~

f(O)

+

N(r,a)

+

V(r,a)

2IT 0 m(r,a)

1

flog

1 2IT

1 d<j>

log

[f( re i<j> ) , a ]

[f( 0) ,a]

0 and

;;; 0

N (r, a)

0 T(r ,f)

so that

0 T(r,f)

VCr ,a)

1

1 1 +

0 m(r,a)

;;;

VCr,a)

;;; 1 -

log

0 T(r,f)

[f CO) ,a]

1

1 +

0 T(r,f)

log [f(O),a]

27f

1 ;;;

0 T(r,f)

,', flog

1 d<j>

~

0

[f(re i ¢) ,al

0

Thus, if lim

o

T(r,f)

r+R

then we obtain from Fatou's Lemma and from (12.8)

_ V(r,a») dS 2n

o

_ V(r,a») dS 2n

~

o

T(r,f)

T(r,f)

so that _ v(r,a») dS 2n

o

o

T(r ,f)

Since the integrand here is non-negative, we conclude that almost every'l1 on

S2n

lim r+R Rfl

(

1 _ :(r,a») T(r ,f)

V(r,a)

=

1 -

lim r+R

o

T(r ,f)

=

o

i.e., we have Corollary 12.3 : V(r,a) lim r-+R

1

o

(12.9)

T(r,f)

holds almost everywhere, unless

o

lim T(r,f) r-+R

<

+00

or

Remark. In the Nevanlinna case the volume function

V(r, a)

n = 1

n :;:: 1

replaced by

relation (12.9) holds a. e. with N(r,a)

Corollary 12.3 will be considerably sharpened in Chapter 5 .

4 Additional results of the elementary theory §13

The genus of a meromorphic vector function

We recollect some results of Nevanlinna theory. In the theory of integral functions the notion of genus of a canonical product is defined as follows. Let zZ' . . .

zl'

be a finite or infinite sequence of non-vanishing complex numbers. If

the sequence is infinite, we assume that ~

be an integer

0

z

n

-+

00

as

n

-+

+00

•

Let

q

such that the series

is convergent. Let

u+

E(u,p)

=

u

T

Z

+ ...

E(u,O) = l-u

(l-u) e

( 13.1)

denote the prime factor of Weierstrass. If the sequence

is infin-

ite, it can be shown that the infinite product TTE(zz ,q) . \!

converges uniformly in each bounded region of the plane

C

thus an integral function which vanishes exactly in the points

and represents z

\!

. If the

sequence is infinite and the series

is divergent, then

q

~

0

is called the g e nus

of this infinite product. If

the sequence is finite, the genus of the corresponding finite product with q=O, which is a polynomial, is defined to be zero. Now let

f1(z)

be a (scalar) non-constant meromorphic function in the

plane. Let

z11(0),z21(0),...

and f 1 (z)

denote the zeros and poles of

z11(00),z21(00),...

z =0

outside the origin

respectively . Let q ~ 0

be an integer such that the integral

!

(13.2) r

q+2

is convergent. Then the series q+1

q+I

1

and

L:z ( 00 ) vI

both con verge. Let

k1

be the smallest integer

~

converges, and let

k2

be the smallest integer

~

0

0

such that the series

such that the analogue

z 1 (00 ) con ver ges. Then clearly k 1 ~ q, k 2 ~ q v Nevanlinna proved the following fundamental representation of f 1 (z)

series for the poles t

Theorem 13.1:

.

Under the above conditions the meromorphic function

f 1 (z)

has the representation

(13.3)

is an integer and

where

In NevanJinna theory the function

f 1 (z)

g e nus

is a polynomial of de gree

m 1 ~q.

g 1 of a not id. zero meromorphic

of finite order is defined to be the integer

tNevanlinna [27], p. 40 83

which is readily obtained from the representation (13.3). If

PI

denotes the order of a not id. zero meromorphic function

of finite order, then the following results of Nevanlinna

(ii) (iii)

a)

if

PI

Let

f1

be of in te ger order

g1 P1

are valid:

is the largest integer ~

1

< PI

; then the following holds:

If the characteristic function satisfies

0

lim r++ co

<

then b)

is not an integer, then

f 1 (z)

T(r,f I ) r

has genus

fl

:;;

+00

(13.4)

PI gI = PI

If

T(r,f I )

lim r++ co

r

0

(13.5)

PI

then one has to distinguish two cases, according to whether the integral +00

I

( 13.6)

is convergent or divergent. If this integral is convergent, the genus is

gI

= PI - 1

g 1 = PI - 1

84

. If this integral is divergent, the genus is

,provided that the series

1

L:--

and

zlll (0)

both converge; and the genus is

if at least one of these

series is divergent. Thus the genus

g1

of

f1

is determined uniquely by the order

alone, with the exception of the case where type of the integer order

~

PI

1

T(r,f 1 )

PI

belongs to the minimum

and the integral (13.6) diverges.

We can now apply these results of Nevanlinna to vector valued meromorphic functions in the following way. First we propose to define the genus of a vector function by Definition 13. 2:

L,et

f ( z)

= (f 1 ( z) , ... , f n ( z) )

function of finite order in the plane ¥ zero. Let g

g e nus

0=1, ... ,n)

gj 0

f

f( z)

n

C

be a meromorphic vector

; we assume that

denote the genus of

fj(z)

f l' ... ,fn are all . Then the

is defined to be the integer

=

g

From result (i) above we have the inequalities

g.

J

Since the order of

(j=1, ... ,n)

+ 1

f( z)

is

max p. J

p

13.2 ProEosition 13.3 :

The order

tor function

(fl,···,fn

fez)

the inequality ;;;

g

P

;;;

P

and the genus zero)

""

we obtain from Definition

g

of a meromorphic vec -

of finite order in the plane satisfy

g + 1

From this we conclude in particular Corollary 13. 4 (f1, .. ·,fn ¥

genus

g

If the order

zero)

P

of a meromorphic vector function

f( z)

of finite order in the plane is not an integer, then its

is the largest integer

< P

Now let

be an integer

p

g = p

either

;;; 1

. Then from Proposition 13.3 we have

g = p - 1

or

In view of the above results of Nevanlinna,assume first that p ~

mean or maximum type of an integer order

o

f( z)

is of

1 . Then

T(r,f) (13.7)

lim

<

r-+-+ oo

In Chapter 2, §7 we proved the inequalities n

T(r,f.) J

+ 0(1)

T(r,f)

$

( 13.8)

S

(j=1, ... ,n) Now, by Proposition 8.5,

and from (13.8) we see that

p n

T(r,f)

L

lim

=

lim k=l r-++ oo

r-+ +00

since

if

-0

p

jo

From this and (13.7) we see that there exists an index

such that

p.

JO

and

=p

T(r,f. )

o

)0

lim

<

+00

r++ oo

f.

is thus of mean or maximum type of the order

=

is

Next let

By the above

JO

= max g. = p

g f

g

;;;

p

,it follows that the genus

g of

in this case. • J beJ of an integer order p ~ 1 ,of minimum type and of

convergence class. Then

86

•

p

p.

Since by Proposition 13.3 always f( z)

p

JO

;Jlsult Hi)a) of Nevanlinna it follows that g. JO

p.

+00

T(r,f)

=

lim

0

j

and

rP

r-+ +00

T(r,f) r

dr

p+1

+00

<

rO Using inequality (13.8) we obtain +00

lim r-+ +00

T(r ,f.) J

T(r,f j )

j

and

0

rP

r

dr

P +1

rO By Proposition 8.5 there is an integer

result iii)b) of Nevaplinna

=

p -

1

(j=1, •.• ,n)

such thai

jo

ip follows that the genus j1

. Assume that there is an index

so that by Proposition 8.5 necessarily

and of

p

vanlinna we obtain

=

~ p

g

g. J1 • This shows that

Finally, let

f

is of order

f.

g.

of

JO

such that

. Then by result i) of Nevanlinna we would have

minimum type of order

p -

=

p

1 -

1

T(r,f) r++ oo

class. By result

in this case. ~

p

and

rP

j

T(r,f) r

P +1

dr

=

Applying inequality (13.8) we get

r-+ +00

T(r ,f.) J r P

and there is an integer

=

f.

iiil~)

o for which

0=1, ••• ,n)

p. JO

JO

g. J1

is of of Neg. J1

1 ,of minimum type and of di-

rO

lim

p. - 1

f. is of genus )1

+00

0

is

which contradicts the assumption

be of integer order

=

p.= p.

JO

p . By (13.9)

p.

conver~1nce

f.

vergence class. We have

lim

(13.9)

+00

is of minimum type and of JJ!onvergence clas/OBy

f.

From (13.9) we see that

<

=

p

and

+00

+00

T(r,f. ) JO

j

dr

+00

(13.10)

p+l

r

rO From the result iii)b) of Nevanlinna we conclude that least one of the two series p

1

= p. = P JO

g. JO

p

1

( 13.11)

and

z . (0)

z . (00)

vJO

JlJO

diverges, where

z . (0)

and

z . (00)

vJO

and poles of f. outs~J~ the origin. In . 1·]0 era1 mequa Ity max g. J

g

that

p

Now assume that for all indices

jo

respectively denote the zeros

this case we conclude from the gen-

g

p

' for which

p.

satisfied, both series (13.11) converge. Then by have

=

g.

( 13 .10)

i~Onot

or we have

if at

1

p -

for these indices

jo

satisfied for

=

replaced by

p

and (13.10) is

iii}b) of Nevanlinna we

• For indices

jl' for which

j 1 ,we have either

f.

p.

<

J}

p,

is of con vergence class. In the latter 1 b g. p y resu It Jl Jl iii)b) of Nevan linna. In the former case we have p.

f . J 1.IS

. case, SInce

g. Jl

;;;

p. JI

In this case, if

p.

and

jo

re~<

=

p

" t ype, J1 0 f mInImum we h ave

<

P

is an integer, it follows that

is not an integer, Jlthen also

g.

Let

p - 1

f = (fI, ... ,fn )

If the characteristic function satisfies

T(r,f)

then

f

has genus

p

p

:;;

;;;:

Co~bllary

(fI, ... ,fn 'Ii 0)

vector function in the plane, of integer order a)

by

]1

Thus we can summarize: Theorem 13.5:

;;;

g.

1

p - 1

13.4 .

be a meromorp hic

b)

1£

T (1', £)

:::

lim

0

rP

r-+ +00

then one has to distinguish two cases according to whether the integral +00

T(r,£)

f

l'

dr

p+1

( 13.12)

is convergent or divergent. If (13.12) converges, the genus If (13.12) diverges the genus

index

jo

g

is

f '- f.

such that (13.12) diverges for

P-1.

and at least one

JO

p

:::

provided that there is an

p

of the two series 1

g

p

1

and z . (00)

z . (0)

]JJO

diverges, where zeros and poles of

§14

vJO

z . (0)

]JJO

f.

JO

Some relations between

and

z . (00)

vJO

respectively denote the

; otherwise the genus

g::::

p - 1

M,m; N,n; V,v and A

In this section we assume for simplicity that the given vector function is meromorphic in the entire finite plane

e

We have seen that the generalized proximity function

m(r,a)

to measure the asymptotic convergence of the vector function a E en

£( z)

f

can be used to the point

. As in Nevanlinna theory it is important to study also the relation of

m(r,a)

to other expressions, which can serve a similar purpose. Such an ex-

pression is the generalized maximum modulus

M(r ,a)

for finite

max

Izl:::: l'

a

,and for

M(r,£)

== M(r,oo)

M(r,a)

defined by

1

II£( z) - aI

a max

Iz I::::r

II£(z)11 89

We obviously have

Here, if

a

+

~

m(r,a)

a

.1ogM(r,a)

3C

f(z)

lies a solution of the equation

r

a

==

the same inconvenience holds in the case ()C

a ==

, i f there are poles of

00

f

r

Let us first consider the case where

£(z)

is an i n t e g r a 1 vector

function; then this difficulty can not arise for T(r,f)

U {oo}

is finite, the right expression has the disadvantage of becoming

infinite if on

on

Cn

E

m(r,f)

==

a ==

co

•

From

N(r,f) == 0 •

we obtain 21T

2~ J

T(r ,f)

+

i6 16g \\f(re )lId6

log M(r,f)

(14.1)

o o~

Now, for

r < s

the generalized formula of Poisson

and

-Jensen-Nevanlinna gives 21T log I\f(re i ¢ )\\

1 2n

·o Jr logl\f(se 1 )\1

.2 2 s - r dO s 2 + r 2 - 2sr cos ( 6 -¢ )

o

L

s(z-z.(O»

Iz.(O)\<s

L

+

log

Iz.(oo)\<s J

]

]

2 -s -z.(oo)z

log

]

s(z-z.(oo» ]

2 -

i" j

s -1; z

log

illog

11£(1;)\\

do

s(z-1;)

\ I; \ < S

The sum over the zeros and the last integral being non-negative, we conclude

2n 1 2TI

J

s

10+g I'I f( se ie ) II s

2

+ r

2

2

- r

2

de

- 2sr cos( 0--1»

o s + r s - r

m(s,£)

so that

/

+

10gM(r,f)

s + r T(s,f) s - r

<

04.2)

We summarize (14.1) and (14.2) in the fundamental Theorem 14.1 isfies for

An integral vector function

0:;; r < :;;

T(r,f)

s

~

sat-

the fundamental inequality +

log M(r, f)

If in (14.3) we put

T(r,f)

f( z)

= kr

s

+

log M(r, f)

s + r T(s f) s - r '

~

(k > 1) k + 1

:;;

~

04.3)

we obtain

T(kr,£)

From this we conclude Theorem 14.2:

If

f( z)

= +

tion, then the order of

(f 1 (z), ... ,f n (z»

10gM(r,£)

is of positive finite order

p

,then

is an integral vector func-

is equal to the order of +

]ogM(r,f)

and

T(r,£)

T(r,f)

. If

f

belong to

the same type or convergence or divergence class. Remark. In connection with Theorem 14.1 we obtain exactly as in Nevanlinna t

theory the following results for non-constant vector valued integral functions: +

T(r,f)

10gM(r,f) < r .... +oo

if

t

o<

p < +00

e( 2p + l)lim r .... + oo

and if the right hand side is finite.

Hayman [16],p.19-20 91

log M(r ,f) (ii )

lim

o

=

K T (r, f) {log T (r, f) }

r++ oo

for any

The last inequality in (14.4) is apparently valid only if

f

K > 1

is integral.

Similarly to Nevanlinna theory it is interesting however that (14.4) remains essentially valid in the general case of a mer 0 m 0 r phi c vector function if + 10gM(r,f) is replaced by its mean value in the interval (O,r) ,the latter being a quantity, which is finite for each finite Theorem 14.3:

r

. We shall prove

A meromorphic vector function satisfies for each

t

r

> 0

the

inequality r

~ 116

C(k) T(kr ,f)

<

g M(t,f) dt

(14.5)

° where the constant

k

C

only.

depends on

k

can have any value

>

1 ,and where the expression

Again the proof rests on the generalized formula of Poisson-Jensen-Nevanlinna. Similarly to the previous reasoning we have for z = te i ¢

271 +

logllf(te

i¢

)11

2~ 116

g [[£(8e16 )[[

o +

Here for

t < r < s

the integral on the right hand side is less than

s + r m(s,f) s - r and in the sum

tsee NevanIinna [271 ,p. 25

92

so that for

0 < r < s s

r

j

s +r - - r m(s,f) s - r

lrig M(t.f) dt

j

L

+

Iz.(oo)l<s

o

]

log

2s

dt

jt - jZj(oo)11

0

(14.6) The integral

s

j

2s log-dt It - A I

J(A)

o ;:;

A < S

o needed in (14.6) is equal to s

S-A

s

f 10g2s dt

J log It -

0

0

J log hi

dT -

J0

-A A

= s log2s

d'I

log 'I dT

S-A

- J log 'I d'I

J

log 'I d'I

0

0

:::

log I'II J -A

S-A

0

= S log2s -

= s 10g2s -

AI dt

slog(2se) - AlOgA - (s-A)log(s-A)

The sum in the right side of (14.6) has been estimated by Nevanlinna as follows:

s

s

J( Izt,,)I)

:::

J J(A)

dn(A,f)

= J(s)n(s,f)

J n(A.,f)dJ(A) ),,=0

A=O Now

dJ(A.)

= dA.

s - A. log -A-

is positive for

o<

).. < s 2

and negative

2"s

for

< A< s

, so that

s

- J

~

n(A,f)dJ

s

-I

1.::0

~

n(A,f)dJ

n(s,f) (J(~) - J(s»)

s

2" and consequently

L

J(

Izt")I)

S

::

s n (s , f) J ( 2")

=. s n ( s, f) 10 g ( 4e)

Iz. ( 00 ) I< s J

Now put

rlk

where rk

f

n(s.f)Ioglk " n(rlk.f)

k

1

>

is an arbitrary number. Then rk

~t

<=

rYk

n(tt,f) dt

f

~

N(kr,f)

rlk

so that

L

J ( IZj ( 00 )

Ik

I>

log( 4e) r T(kr ,f)

loglk

Iz. (00) 1< s J

and s+r - - r m(s,f) s-r

~

1k+1

r T( Ik r,f)

~

1k-1

Introducing these bounds into (14.6) gives r

~f l~gM(t.f)

C(k) T(kr,f)

d'

o with

Ik log( 4e)

/k + 1 C(k)

+

::

Ik -

1

log/k

This proves the Theorem. Again as in Nevanlinna theory we have

94

Ik + 1 - - - r T(kr,f) Ik - 1

~

T(kr,f)

Proposition 14.4:

a E en u {oo}

For

the functions

n(r,a)

and

N(r,a)

have the same order and, in the case of positive finite order, the same type and the same convergence or divergence class. For any have further

+00

+00 11

1

n(t,a)

,1

:::

z. (a)

J

t

11+

1

dt

,21

:::

0

iz.(a)i> 0

J

we

0 < 11 < +00

N(t,a) --dt 11 + 1 t

(14.7)

0

in the sense that these three expressions are all infinite or are all finite and equal. Proof. Suppose that n(r,a)

<

Then r

N(r,a)

1

n(t,a) --dt

:::

0(1)

+

<

t

so that the order, type, convergence or divergence class of

exceed that of

n(r,a)

N(r,a)

N(r,a)

can not

. Vice versa, if

<

then the inverse statement follows from 2r n(r,a)log2

~1

n(t,a) --dt

~

N(2r,a) + O( 1)

t

r Further, we have

L !z.(a) 1> 0 J

r

r 1

11

dn(t,a) :::

z. (a) J

1 0

til

f n(t.a)dt

n(r,a) +

:::

rll

11

t 0

11+1

r

n(r,a)

=

j.l

+

.

I

r

n(r,a)

dN(t,a)

---+

rj.l

tll

\1N(r,a) rj.l

+

j.l

2I

o

N(t,a)dt t\1 + 1

o

If the last integral diverges, then we conclude from this relation that also the

other two quantities in (14.7) must diverge. If vice versa the last integral converges, then

N(r,a)

has at most convergence class of order

at most of minimal type of order

II

•

\1

and so is

Thus

N(r,a) -0

as

r + +00

and thus, from what we proved above, also n(r,a) as

-0

r + +00

This proves the Proposition. In the same way we prove Proposition 14.5:

The functions

v(r,a)

and

V(r,a)

have the same order

and, in the case of positive finite order, the same convergence or divergence class. For any

0 < j.l < +00

+00

+00

v( t, a)

---dt \1+1

I

we have further

=

t

V(t,a) --""'1 dt tj.l +

(14.8)

o

in the sense that the two integrals are both infinite or are both finite and satisfy (14.8). Let us return to Theorem 14.3. From (14.5) we see that the expression r

II

r

log + M(t,£) dt

o is at most of order

p

Now for

we have the inequalities

M(r,a)

1 max - - - . - - JzJ=r II£(z) - all

1 &

max JzJ:::r

(j:::l, ... ,n) [£.(z) - a.1 J J

Putting

1 M(r ,a.) J

:::

max Izl=r

I£.(z) - a·1 ] J

we can write M(r,a)

M(r,a.)

•J

(14.9)

(j=I, ... ,n)

In the following we assume for simplicity that each

f. ]

, j::: 1, ... , n , is

non-constant. If we apply inequality (14.5) in the case

n ::: 1

to the function

1 (14.10) f.(z) - a. J J we see that the expression r

-

r1

Jlog M(t,a.)J dt

(14.11)

+

o is at most of the order of the function (14.10). Now from Nevanlinna theory it is seen that the function (14.10), being a linear transform of

f.( z) , has the ]

same order as order

p.

J

£. ( z) . We conclude that the expression (14.11) has at most J . From this and (14.9) we see that the expression

r

r1 Jlog + M(t,a)dt o

97

is at most of order min

Using the above results, the meaning of

n (r, a)

and the generalized first

main theorem we can formulate Proposition 14.6:

Let

f( z) = (f 1 (z) , ... ,fn (z», (f l' ... ,fn non-constant)

be a meromorphic vector function of order

p

•

Put

p

*:

= min {p l' ... , P n}

Then the following holds: (i)

The expressions V(r,a), v(r,a), N(r,£), nCr,£), mer,£)

arrd

r

~J

16gM(t,f) dt

o are at most of order the type and class of (ii)

For

and, in the case of positive finite order, at most of

p

f

a E en

the expressions

N(r,a), n(r,a), m(r,a)

and

r

~I

16g M(t,a) dt

o are at most of order

p

* ;

the series

( 14.12) z. (a)

Iz.(a)l> 0 J

J

converges for every

E >

Now choose

E >

M(r ,f)

e

r

0

if

p

*

<

+00

and assume that for all

P+ E

holds. Then it follows that

98

0

the inequality

r

~I

r P+

16g M(t, f) dt

E:

1

P + s+

+

0(1)

o so that the left integral would be of order

f:

P + E:

•

From this contradiction

with Proposition 14.6 0) we conclude Corollary 14.7:

Let

be meromorphic of finite order

f( z)

be given. Then for infinitely many arbitrarily large values of M(r,f)

e

<

p+

r

and let

p

r

E: >

0

we have

E:

In the same way we obtain Corollary 14.8: p

*

Let

be finite. Let

large values of

E:

r

f(z)

be meromorphic, (fl"" ,fn non-constant) and

> 0

be given. Then for infinitely many arbitrarily

we have p* +

M(r,a)

e

<

E:

r

We now put r a(r)

I'

r1

==

(a ECnU{co})

log M(t,a) dt

(14.13)

0

Then r

I

r

16g M(t,a) t

ll+ 1

d(ta(t» dt

I

rO

til + 1

I r

a.(r) +

(ll + 1)

a.(t)dt 11 + 1 t

Thus, if the integral

99

r

f

l6gM(t,a} +1 dt t fJ

(14.14)

.

is convergent, then so is the integral

Vice versa, if the last integral is convergent, then for sufficiently large

r , since

ra(r)

+00

I

>

0

and for all

is increasing,

+00

a{t)dt >

£

E:

~

tJ.! + 1

ro(r)

r

I

a{r)

dt

]1+2

(J.!+l)r fJ

t

r

This shows that then (14.14) converges. Thus we have Lemma 14.9: E

en u {oo}

be meromorphic. Then for

Let and

J.! > 0

the integrals

+00

+00 + log M(t,a)

j

tJ.! + 1

dt

and

f

tJ.!+2 dt

rO

rO

t

I·

10gM(s,a)ds

0

are simultaneously convergent or divergent. So the order of r

rII

log + M(t,a)dt

o is equal to the limit inferior of the numbers

100

J.!, for which the integral

a

+00

+

log M(t,a) --~-dt t].l + 1

J

is convergent. Let

p

*

r::,

be finite and denote by

e:

(a)

the set of r - values such that

the inequality M(r, a)

<

e

r

(a E

en,

e: > 0)

p * < ].J < p * + e:

is not valid. Then for +00

+

+

!ogM(r,a)

J

r

].J+1

logM(r,a) dr

~

J

r

,J

dr

].J+1

n = p* + e: -

].l

~J

1

dr =

dr"

r::, (a)

h. (a) e: where

p* + e: -].J -

r

h. (a) e:

e:

0

and

<

n < e:

; by Lemma 14.9 and Proposition

14.6 (ii) the integral on the left side is finite.

a = 00

If

, the analogue identity holds if we replace

p

*

by

p. Thus

we can formulate Proposition 14.10 : Let

f(z) = (f1(z),. ..• ,fn (z»

function (f 1 , ... ,fn non-constant). 0) If p* isfinite,let h.(a)(e:>O) e: such that

be a meromorphic vector

denote the set of r - values

M(r,a) Then the total variation of

o<

such that (ii)

If

p

rn

in the set

r::, (a)

e:

rests finite for all

n

n < e:

is finite, let

t:,

e:

(e: > 0)

denote the set of r - values such

that M(r,f)

e

r

p+ e:

Then the total variation of that

r

n

in the set

r::,

e:

rests finite for all

n

silch

0 < n < e: 101

Proposition 14.11:

Let

be a non-constant mero-

morphic vector function. If for some

fl > 0

the integral

+00

j

T(r,f) r

fl+1

dr

rO > 0

rO is convergent, then the same holds for the integrals

j

V(r,a) - - ' 1 dr r fl +

j

+

logM(r,a) - - - - ; - - dr r

(14.15)

fl+1

and the series

1

Iz. (a) Ifl Iz.(a)l> 0

J

J

Proof. The convergence of the series follows from Proposition 14.4, observing that by the first main theorem the integral

j

N(r,a) r

fl+1

dr

is convergent. The convergence of the left integral (14.15) follows from the first main theorem. The statement that the right integral (14.15) is convergent follows from Lemma 14.9 and Theorem 14.3, in the case

a E en

by combining

the inequalities (14.9), (7.19) and the footnote on p. 97. This proves Proposition 14.11. Proposition 14.12:

Let

morphic vector function. If for given

102

be a non-constant mero].I

>

0

the integrals

+co

+co

+

log M(r,a) -----;-- dr

f

r

V(r,a)

)1+1

f

r

1l+1

(14.16)

dr

and the series (14.17)

are convergent for some value a E C n U {co} ,then they stay convergent for every a E C n U {oo} ,as does the integral +co

T{r,f)

f Proof.

r

)1+1

Since

( 14.18)

dr

m(r,a)

is majorized by

+

logM(r,a)

,the integral

+00

m(r,a)

f

r

1l+1

dr

rO is convergent for the given value of

a

. Since the series (14.17) is conver-

gent, also the integral +00

N(r,a) ---=--1 dr

f

rll +

is convergent by Proposition 14.4. Observing that the right integral (14.16) is convergent, we obtain from the first fundamental theorem that the integral (14.18) is convergent. The convergence of (14.16) - (14.17) for every

a

follows from Proposition 14.1l. Combining Lemma 14.9, Proposition 14.11 and Proposition 14.12 we have

103

Theorem 14.13:

Let

fez)

phic vector function. Let

= (f 1 (z), ... ,fn (z»

be a non-constant meromor-

denote the order of the mean value

01 (a)

r

~f

'6gM(t,a) dt

o denote the limit inferior of the exponents

Let

°>

,for which

0

the series 1

Ida)!> J

a

converges. Let

v(r,a)

denote the order of

(or V(r,a»

. Then

o(a) rests in variable for all

a E en

u

and is equal to the order

{oo}

of f( z).

p

From the generalized first main theorem + N(r,a)

m(r,a)

=

+ V(r,a)

T(r,f)

+ 00)

we obtain (

m(r,a)

lim r++ oo

N(r,a) +

+

T(r,f)

V(r,a) )

=

(14.19)

1

T(r,f)

T(r,f)

which shows that the limit inferior and the limit superior of each of the quo tients N(r,a)

m(r,a)

VCr ,a) and

T (r, f)

lies in the closed interval

n

m(r,a)

since

We also remark the following:

and

N(r,a)

min {p l' ... , p n }

are of order

is of order

p , or

,then for

and

, then by the first main theorem T(r,f)

f = (£ l' ... , fn)

is the order of

p

If

104

[0,1]

= max {p l' ... , p n } non-constant) and if p* = If

f

T(r,f)

T(r ,f)

a=oo

V(r,a)

V(r,a)

(£ l' ... ,

a E en ~

p.

must be of order

p,

is of order

Lemma 14.14: Let the functions

o<

increasing for der of

sk (r)

rO ;;; r < +00

sl (r) .

provided that

02

k::: 1,2

let

ok

denote the or-

o

(14.20)

01 < 02

Proof. Assume first that <

. For

be real, non-negative and

. Then

lim

E

sI (r) ,s2(r)

02

<

+00

. Let

e: > 0

be given such that

. Then

for

<

r ::: r

for a sequence

-+ +00 Cl

Therefore

r

<

and if we choose

E

>

Cl

0

such that

2e: < 02-01

is clear that (14.20) remains valid in the case Since

°2 ::: +00

m(r, a) , N (r, a) and m(r, a) + N (r, a) n a E C , we deduce from Lemma 14.14

for

p*

Proposition 14.15

Let

m(r,a) lim r-++ oo

r-+ +00

of order

a EC n

'"

p Q,

0

T(r ,f)

m(r,a) + N(r,a)

lim

lim r-+ +00

If the vector function

f R,

=

lim

T(r,f)

r-++ oo T (r, f)

tion

• Then for all

p

N(r,a) 0

V(r,a) -

are at most of order

be a meromorphic vector

function (f1, ... ,fn non-constant) such that

I

,then we have (14.20). It

<

f( z) I

2'

:::

0

T(r ,f)

has a non-constant integral component func-

' then by a theorem of Wiman

uniformly to infinity on a sequence of circles

Iz I =

rv

f Q, (z)

tends

with un boundedly 105

increasing radii. Thus the proximity function for any a E Cn on a sequence r -+ +00 v Proposition 14.16:

•

vanishes in this case

This proves f( z)

If the meromorphic vector function

stant integral component function

f Q.

of order

m(r,a) lim r-+ +00

m(r,a)

p

Q.

<

1

has a non-conthen

2"

VCr,a) + NCr,a)

=

·0

1

r-+ +00

T (r , f)

T(r,f)

for all The quotients arising in Proposition 14.15 and Proposition 14.16 will play an important role in Chapter 5. In concluding this section we discuss another problem. The generalized first main theorem in spherical formulation CTheorem 11. 2)

R

shows in the case tion

f( z)

for fixed

V(r,a)

that for a non-constant meromorphic vector func-

+00

a

+ N(r,a)

o

T(r,f)

<

From this we see that for a sequence

+ r k -+

(r < +(0)

00) +00

,

<

Here

ACt,£)

sum

vCr,a)

is given by (12.6) or (12.7). Hence for each fixed +

n(r,a)

is for a sequence

r k -+

+00

a

the

not much larger

than the average of this sum over the sphere S2n . This sequence however in general will depend on vCr,f)

=

Thus putting

a

sup a

E

{vCr,a) + nCr,a)}

(14.21)

Cnu{co}

it is reasonable to ask if necessarily {vCr,£) - A(r,£)} lim r-+ +00

<

+00

We are unable to prove anything as strong as this. But we can prove the following less strong result, which again extends the corresponding result tof Nevanlinna theory

t

Hayman [16]'p.14

106

If

Theorem 14.17:

fez)

= (fl(z), ... ,fn (z))

is a non-constant vector val-

ued meromorphic function, then with the abbreviation (14.21) we have vCr,£) 1

<

lim

e

(14.22)

g(x)

is a positive strictly increasing and con-

r >- +00 A ( r , f)

Proof. We need the following Lemma 14.18';':

Suppose that

vex function of quence

x.

-7

J

x

for

J

Here

h '(x)

derivative of

Xo

such that if

+'"

convex function of h'(x.)

x:.>

x

hex)

such that

Q

Then given

>

1

there exists a se-

is any other positive increasing and

hex)

g(x)

<

for

x "'

Xo

,we have

(j=l, 2, ... )

<

denotes the right derivative of

hex)

and

g'(x)

the left

g(x)

From the generalized first main t.heorem

o

T(r,£)

V(r,a)

+ N(r,a)

o

(14.23)

+ m(r,a)

we obtain for

o

o

T(r,£)

0

V(r,a)+N(r,a)-V(rO,a)-N(rO,a)+m(r,a)-m(rO,a)

or

o

0

0

0

V (1', a) +N (r, a)-V (r 0' a)-N (r 0' a) =T(r ,f)-mer ,a)+m(r o,a)-T(r O'£) . We now choose

1'0

for a suitable

i3

such t.hat

'0 i3

[f(w),f(O») [f (0) ,a]

If

[few) ,a] on

Iwl

;c

1.2

>

B

0

f( w)

f( 0)

"

on

for

Jwl

Jw! = 1'0

04.24) then

1'0

it follows that

[£(w) ,£(0)] -

[£(0) ,a)

so that by definition of

B -

":

!

B

=

!

B

o

m (Definition 11.1),

Hayman [l6],p.15

107

I 2'Tf

o

~ 2~

0

- m(r,a) + m(rO,a)

log

2

log 6

lie de [f(roe ) ,a)

o Thus in this case we have using (14.24)

I r

o

v(L,a) +t n(t,a) dt

= V(r,a)+N(r,a)-V(ro,a)-N(ro,a) < T(r,f) +log

2

i3'

(14.25)

[f(O),a]

On the other hand, if

1 log---[f(O),a)

then

2 S

log -

o

m,

and, using (14.23) and again the definition of

I r

v(t,a) +t n(t,a) dt

Thus 04.25) holds for

and all

a

o 2 T(r,f) + logS

~

V(r,a) + N(r,a)

<

.

Now the two functions

o

and

T ( r ,£) + 10 g

are positive, increasing and convex functions of

log r

2

S

for

log r > log r 0

and the second function is strictly increasing. Thus by Lemma 14.1S we can find a sequence that for

r . .". +00

depending on

]

r = r. ]

and all

o

T(r,f)

a

d r dr [V(r,a)+N(r,a)]

<;

e

Q

d 0 2 r dr [T(r,f) + logS]

i.e. v(r.,a) + n(r.,a) J J for

lOS

j = 1,2, ...

<

and all

but not on

e Q A(r.,£) ]

a

. We conclude that

a

, such

v(r) lim r-++ oo Letting

A(r,f)

Q > 1

tend to

1

gives the right hand side of (14.22). The left

hand side of (14.22) is correct, since + n(r,a)

and as such not larger than

A(r ,f)

v (r)

is the mean value of for all

r

v(r,a)

.

109

5 Extension of the second main theorem of Nevanlinna Theory §15

The generalized second main theorem

Let

be a meromorphic vector function in

CR' It

was shown in §9, Proposition 9.4 that the characteristic functions r

T(r,f)

m ( r ,f) + N (r , f)

an d

j j

o

-dt t

T(r,f)

o differ by a bounded term only; here liJ n ni - Study metric on C . So T (r, f)

-'IiI. f * V

'*'

Ct

denotes the Kahler form of the Fubias well as

o

T(r,f)

can be thought

of as measuring the volume of the image of the disc C under the mapping r n f( z) 111 C ,equipped with the Fubini - Study metric. Associated with the curve

f( z)

is its Gauss map (15.1)

which is a holomorphic curve in

pn-I

and is defined by projecting the de-

rived curve CO

into

pn-l

CR -

en

by natural projection, and by extending the result holomorphical-

ly into the poles of

f

and into the common zeros of

case of the original Nevanlinna theory useless, since

pO

n;:: I

fi""

,f~

. In the

of course,the Gauss map is

is a point. The Gauss map associates in particular with

each tangent plane at each point

f( z) E C n

point, which this plane defines in

pn-I

of the complex curve

f

the

; this notion of Gauss map generalizes the corresponding notion of differential geometry in R 3 ,where to each tangent plane of a surface,is associated the point, which the normal defines on the unit sphere. According to Chapter 1 (formulas (3.8) and (2.4)) the function

no

r

j

=

V(r,O,f)

v(t,O,f) dt t

°

with

j,

TI1

v(t,O,£)

2~ j

=

f w

Ct

1 where- w ·n

'log IIf(UII do

A

dT

Ct

is the curvature form of the hyperplane section bundle

Pn-l , measures t h e voI ume

(t nv"

° f)

" In

pn-l

0

f t h" e Image

0

Hover f t h e d"ISC

'V

Ct

under the mapping

f G(r)

So, if we define the function r

G(r)

V (r .0. n

"

" J

~t jV

o then

G(r)

-

G(r,£)

by putting

r

w

j2~; j

'log 11"(, III do, d,

"

°

Ct

Ct

(15.2)

measures the volume of the image of the disc

C

'V

f', if

under the Gauss map

n

~

2 , and vanishes if

corresponds in differential geometry of

R3

n

==

r

in

pn-l

1 ; (this volume

to the area of the spherical image

under the Gauss map).

A second geometric interpretation of the function

G (r)

can be obtained

as follows. DeC

In general, if on a domain h

where

is a positive Coo-function on ==

g

du

1\

2

(w == u + iv)

g (du + dv )

g dw0dw g

iJ;

2

a Hermitian metric assumes the form

D

then its volume form is

dv

and its Kahler form is /\

h

==

i

2" g dw

A

dw

so the volume and Kahler forms are equal. The Gaussian curvature

K

of the metric

h

is defined by

III

K

1 - 2g 1Il0g g

==

where t:,

d2 --2 dU

==

d2 --2 dV

+

=

d2

4

dW

dW

is the usual Laplacian. Hence 2 d log g

=

KJ/I

- i

1

c

"2 dd log g

dw /\ dw

dW dW We now define the Ricci form t i 27T

=

Ric lj!

a a log g

Ric J/I ==

of the volume form

ljJ

by putting

1 C 47T dd log g

(15.3)

Thus the formula Ric lj!

1 Klj! 27T

(15.4)

- -

==

is valid. We will now apply this. By the map f

Cn

ooC R -

the flat metric

= of

Cn

induces on

00

CR

the pseudohermitian metric tt

1\

which is Hermitian on

~CR

is =

tCarlson and Griffiths [15}

112

2

According to the above its volume form dSf=dA f

By (15.3), (15.4) the Ricci form of this volume form is 1 - -KdA

=

=

27f

(15.5)

f

where K

=

is the Gaussian curvature of the metric harmonic on

~CR

ds; . Since

log

Ii£' (w) II

~CR' K:;; 0 ,and

we remark that on

is sub-

Ric dA f

~ 0 as

a positive differential form. Since

vCt,D,n

"

~ f /" Ct

2~ f "oglll'culido

A

2" J dd"log 1ll'cw)11

d,

ct

Ct

the function (15.2) can now also be written as r

r

f ~: f "og 1II'cU lido

G(r)

o

A

d, " J:.. 27f

f fdt t

o

Ct

K dA f

Ct

(15.6)

From this interpretation we see that

G(r)

measures the growth of the total

curvature

of

f( z)

In view of the above the function G(r)

=

will be called the

G(r,£) lie u r vat u r e f u net i

0

nff

or

fiR

ICC

i

fun

C -

113

t ion II of the vector valued meromorphic function

f( z)

. This function is the

new ingredient that arises in our extension of the second fundamental theorem of Nevanlinna theory to vector valued meromorphic functions. Theorem 15.1 (Generalized Second Main Theorem): Let (z»

be a non-constant meromorphic vector function in

C n u{oo}

(k;:: 1, ... ,q)

~

q

be

3

= (fl(z), ... ,f

Hz)

C R . Let

a

k

n E

distinct finite or infinite points. Then

q

L

m(r,a k )

(15.7)

G(r)

+

k=1 or, in view of the generalized first main theorem, q (q-2) T(r,f)

+

G(r)

L

~

[V (r ,a k) + N (r ,a k) J

-

N 1 (r) + S (r) .

k=1 (15.8)

Here

G(r)

is the curvature function introduced above; and 2 N(r,f) -

N(r,f') + N(r,O,f') r n 1(t) - n 1 (0)

n 1(O) logr

=

+

dt

(15.9)

t

f

0

is the generalized counting function of all multiple finite or infinite points, where

n 1 (t) denotes the number of multiple finite or infinite points of in Izi ~ t , each such point counted with its multiplicity diminished

Hz) by 1

R = +

If

S(r) as r -+

r -+

+00

00

=

,

then the term O{logT(r,f)}

J dr

+

without exception if

outside a set J

+00

S(r)

<

O(Iogr) Hz)

114

0 < R <

has finite order and otherwise as

( 15.11)

+00

+00

( 15.10)

of exceptional intervals of finite measure:

J If

satisfies

,then the estimate

S(r)

holds as

O{J6gT(r,f)}

+

O{1og

without exception if

r -+ R

R~r}

f

( 15.12)

has finite order

log T(r,f) lim r-+R

p

log _1_ R-r r -, R

and otherwise as

outside a set

J

of exceptional intervals such

that

1 R-r

(15.13)

<

In all cases the exceptional set J is independent of the choice of the a k E-C n and of their number.

finite points

(Remark: If

R::

side of (15.10) is and if

f

and if

+00

O(log r)

f

is of finite order, then the right hand

by definition of the order; if

0 < R <

+00

is of finite order, then the right hand side of (15.12) is 1

O{logR_r} by definition of the order in this case). Proof.

We choose

p

~

p F( z)

::

L \)::1

2

distinct finite points

a

v

E

1

IIf(z) - av II

From the inequality + log xl

o<

we get for

+ + log x 2

r < R

2 'iT

2 'iT

2~ f o

(15.14)

16g F(rei ' )d,

,

m(r,O,£,) +

i, f

16g{F(rei ,) iif'(reil )lI)d, '

(15.15)

o 115

Put

o

min II} - a j II

=

i"'j

Let for the moment

II

E {l,

<

2~

2, ... ,p}

be fixed. Then we get in every point

where IIf(z) - all

II

(;;;

1 since

p

:i:;

2)

;:; Iia ll - a v II - lIf(z) - all II

>

( 15.16)

the inequality IIf(z) - a v II for

v '" II

. Therefore the set of points on

2~ ~

0 -

3C

340

(since

which is determined by

r

(15.16) is either empty or any two such sets for different

have empty

jJ

intersection. In any case 211

2~ J

2~

16g F(re i , ld,

o

P

L

II =1

..i.. 2p

!If-all 11< Iz[ = r p ;:;

1

211

L

II =1

1

+

log

J

Ilf(re i ¢ ) _ all II

d¢

..i.. 2p

IIf-allll< Izi = r Because of

~J

l~g_l_

211

Ilf-a jJ II <

2~ J

d¢ = m(r,a ll )-

II f-a jJ II

..i. 2p

Iz[

;:;

116

+

m(r,a ll )

-

1+ og ~ /)

1

d<jJ

log IIf-a llil

Ilf-allll;:;

[zl = r

p~2)

=

0 2p

r

it follows that 211 ~

+

,1,;gFlre'w)d'

21

P log 0

a so that by (15.15)

I 2IT

p

L

~

m(r,a,f')

m(r,a lJ )

1 211

J.l =1

Since

fez)

~

+ iq, ) II' + 0 log{F(re f (re i<j> ) II }dq, - P log

a

is non-constant,

f'(z)

does not reduce to the constant zero

vector, so that the generalized first main theorem (§3, Theorem 3.2) can be applied to

f' (z)

T(r,f') c~,

where

a =a

with

=

G(r)

+

,and gives + log lic~,11

+ m(r,D,f')

N(r,D,f')

is the first non-vanishing coefficient vector in the Laurent devel-

opment f' (z)

~'+1

=

z

,

+ •••

c~'+l

,

(c~,.,

0)

Using this we have p

T(r.f')

~

G(r) + N(r.D,£') +

L ]J

m(r,a ll

)

=1

211

-

2~ 1,;glFlrei.) lin rei! )Illd;

- pI;g 'l'

+ loglle,.11

o On the other hand, using (15.14), T(r,f')

=

m( r , f' )

+

N ( r , f' )

m(r,f')

+

N(r,f')

+

117

We now introduce the function N(r,a,f')

= N 1 (r)

2N(r,f)

f'(z) , i.e. the multiple points in which the function

to the multiple poles of the function of

N(r,a,f'), characterizes the

assumes a finite vector; the second term,

era liz e d f( z)

c

u n tin g

0

(15.17)

N(r,f')

consists of two components; the first,

distribution of zeros of fez)

+

£( z)

2N(r,£) - N(r,f')

. N 1(r)

fun c t ion

f

0

refers

will be called the

"~_

m u I tip I e poi n t

S"

since it can be written in the formr

=

f n,(t) :

+

n 1(0) dt

(15.18)

a where

=

n 1(t)

n(t,a,f')

+ 2n(t,£)

-

n(t,f')

ber of all multiple finite or infinite a - points of

is obviously the num-

f

Iz I ; ; t

in

point counted with its multiplicity reduced by one. Introducing omitting

, each aN1

we get,

T(r,f'), p

G(r)

L

+

Il

m(r,a ll

+

)

N(r,f')

=1

21T

-

2~ J

Itg{F(re;' >Ii£' (re;') II} d,

• log

lie;, II -

p It g

~

a 21T

mer ,f)

+

N(r,f')

+

21T 1

Jlog + a

Observing that

We now put

N(r,£)

we obtain p+l G(r)

+

L k=l

118

m(r,a k )

+

N1(r)

-

2T(r,f)

= T(r,£) -

mer,£)

,

2 'IT

-i, II;;g{F

(ce;1 ) II f'( re;; ) II Jdj

Ioglle

,

~,II

-

p I;;g

£j'

o 2 'IT 1 r + IIC(rei¢)11 -- log . d¢ 2n , IIf(re 1 ¢) II )

o and hence the inequality p+l G(r)

I

+

m(r,a k )

+

(15.19)

N 1 (r)

k=l where

2 'IT

,

2~ I

I.igiF (re i'liIf' (re il ) II Jdl

o ( 15.20) Using the generalized first main theorem (§3, Theorem 3.2) k k k k k = V(r,a ) + N(r,a ) + m(r,a ) + logllcQ,(a )11 + dr,a )

T(r,f) for

k = 1, ... ,p,

and using that

m(r,co):::: T(r,f) - N(r,oo) ,V(r,oo) =0,

inequality (15.19) can alternatively be written as

p+l (p-1)T(r,f) + G(r) + N 1 (r)

;;;;

I

[V(r,a k ) + N(r,a k ) J + Sl(r) + 00)

k=l

(15. 21) with p

00)

::::

L

(log IlcQ,.(ak)11 +

dr,a k »

k=l We will now try to find an estimate of the function

Sl (r)

given in 05.20). 119

Note first that an application of the inequality n +

n

L

log

L l6g x.J

x.

J

j=l

(x. ~ 0)

log n

+

(15.22)

J

j=l

gives 211

2',

I

16g{F (re II

>11£' (reII) II )dl

+ log P

,

,

o so that 211 211 1

i 211

p

11f'(re icp ) II

I log +

dcp

IIf(re icp ) \I

L

+

1

. Ilf'(re1cp)11 +

k=1 211 ' log IIf(reicp) _ akll dcp

o

o

+ log P

+ p 16g 2f

log II c ~,

-

II

(l5.23)

We now need the following important Theorem of Nevanlinna theory. t

Theorem 15.2;

Suppose that the complex scalar valued function

meromorphic in

CR' If

and

s

(0 < r < s < R)

mCr,f)

<

Co =

1;;(0)

r;( z)

is

is finite and not zero, then for all

r

the inequality

24 +3l6g Ic1 1 + 0

2l6g~

+

4l6gs

+

3l6g s : r +4l6gT(s,1;;) (15.24)

is valid. If

1;( 0) = 0

or

00, then

r;( z)

has around the origin a development of

the form 1;;( z)

=

tNevanlinna [27] ,p. 61 120

(c

K

'"

0)

(15.25)

In this case (15.24) is to be replaced by m(r,r)

I

34 + 516glK

<

I

316g[c1

+

+

7l6g~

+ 416g s

K

+ 1 + 310g -

+

+ 410g T(s, z;)

s-r

(15.26)

Since (IS. 26)is also valid in the first case, where (IS. 25) holds with

K

=0 ,

we will not apply (15.24) but only (15.26), which holds in the most general case. Substituting z;( z)

=

(a. E C)

f.(z) - a.

J

J

J

in (I5.26) and using the inequality T(s,f.-a.) J J

+ + 10gla.1

T(s,f.) J

+ log 2

J

we obtain Lemma 15.3:

Assume that

f.( z) - a. J

opment K.

f.( z) - a. J J then for all

=

c.

JK.

Z

<

s

K.+l

J + c.

JK.+

J

rand

admits around the origin the devel-

J

lz J

+ • • •

,

J

(0 < r < s < R)

+

34 + 5logjK.J J

+ 1 + 310g -

;oe

0),

the inequality

+ 1 310gc. JK. J

+

(c. JKj

+

+ 41ogJa.1 s-r)

+

+ 1 + 710g- + 410gs r

+ + 410g 2 + 410g T(s,f.)

]

(15.27)

is valid. From inequality (15.22) we conclude that

=

121

2n

2~ I [~16g Ir;'h; ]

1

2" log n

+

dl

o ,

C

.~_

L-

+

"'"' m(r'f a. ) J J I

L

where

denotes summation over those indices

f. =t= const.. Using (15.27) we obtain for J O
which

j

E

fl, ... ,n}

,for

a:= (a1, ... ,a n ) E

en and

2-rr

-

1

I( log +

2n J

II'f'(re i ¢ .

I

II£(re1¢)

-

)I!,

d¢

all

o

+

+

31
1 glIc:J

Kj

Using the inequalities

T(r,f.)

J

<

T(r,f)

+

00)

we obtain

(15.28) where "const." is a number, which depends only on the development of

122

f( z)

at the origin and on We now consider the case

0)

First let

f( z)

and have for large T(s, f)

R:: +00

and distinguish two subcases.

be of finite order

p

• In this case we put

s:: 2 r

r {e > 0)

<

and with (15.28) 211 1

r

+

211

J

log

Ilf'(rei
Ii {(rei )

_ all d

o <

If

(ij)

+ '4nlogr

0(1)

+

4n(p + c)Iogr

O(Iogr)

=

(15.29)

.as of infinite order we can use the Lemma of Borel t accord -

f( z)

T(r) :: T(r,f)

ing to which the increasing function

satisfies the inequality

1

T(r +

)

(15.30)

(k > 1)

<

log T(r) with perhaps the exception of a sequence of r - intervals

J(k)

of finite

total length. Let us call ordinary the segments where inequality (15.30) is

k

valid. We will apply the Lemma of Borel with

==

2

and

1 s

=

r+---

log T(r) then we have on the ordinary segments +

log T{s) and, if

2l6g T(r)

<

r > 1 , T(r) >e,

then 1

log s

Jog r

:=

+ log (l +

)

~

log r

+ log 2

rlogT(r) + 1 log s-r

::

+

+

log log T(r)

~

+

logT(r)

and hence finally

tNevanlinna [46] gives a proof on p.57. 123

211

2~

j

11f'(rei<jl)11

16g ---,--'---

d<jl

IIf(rei<jl) - all (15.31)

o 0(1)

<

4n log r

+

+

lln log T(r, 0

O{IogT(r,O}

+ O(Iogr)

excluding the extraordinary intervals. Note that in this analysis the exceptional intervals do not depend on a E C n . This proves part a) of Lemma 15.4:

fez) = (f1(z), ... ,f n (z» CR .

Let

vector function in If

(a)

R

= +co

be a non-constant meromorphic

,then

211

1

r

2n

J

+

Ilf'(rei<jl

log IIf(rei<j»

)!!

o {log T(r,O}

_ all d<j>

+ O(logr)

( 15.32)

o in the case of infinite order, with perhaps, the exception of a sequence

J

of

intervals of finite total length

Jdr

<

+00

J The exceptional intervals are independent of If the order of

fez)

a

is finite, then relation (15.32) is valid without re-

striction, so that in this case

j

2 'IT 1 21[

+

11f'(rei<jl )11

log

d<jl IIf(rei<j»

=

o (log r)

(15.33)

- all

o without restriction. (b)

o<

If +

R < +00

'

O{logT(r,O}

+

,then (15.32) holds with the right side replaced by

1 O{1og R-r}

in the case of infinite order,with perhaps,the exception of a sequence intervals such that

124

(15.34) J

of

+00

<

The exceptional intervals are independent of

a

.

If the order

log T(r,f)

=

p

of

f(z)

lim r+R

1

log R-r

is finite, then the estimate (15.34) holds without restriction and can

be written

1 O{Iog R-r}

(15.35)

In order to prove part b) of Lemma 15.4 we distinguish again two subcases. 0)

First let

f( z)

be of finite order

p

in

C R (0 < R < +00)

•

Then

we have T(s)

(R - s) -P-(;

<

for any

s

put

= R +r

2s s - r

(0 <

=

S

(c > 0)

< R) , which is sufficiently near to

R

. In this case we

so that

R - s

:::

R-r -2-

It follows that the right side of (15.28) has the form

which proves the second part of b) If

(ii)

mine

s

f( z)

is of infinite order in

C R (0 < R < +<») , then we deter-

by the condition

+

:::

1 log T(r,f)

An application of Borel's inequality (15.30) gives 2

T(s,f)

<

:::

T(~

R -r

+

1 ) log T (r , f)

<

(T(r,f)

k

,

125

log T(s,f)

klogT(r,f)

<

perhaps up to a sequence

I

d _I_ R-r

J( k)

I

::

J(k)

(k > 1)

of intervals such that

dr

<

2

(R-r)

+00

J(k)

Further, from ?

1

R'"'

- - - - l o g T(r,f) (R-s) (R-r)

::

s-r

::

2 (log T( r, f) + R - r} (R-r) .

---:0

we see that 1

1

:: 210g + 2log R R-r

log s-r

::

1 O{1og R-r}

+ log (log T (r ,f) + R - r}

+ O{loglogT(r,f)}

Together with (15.28) this proves the first part of b) in Lemma 15.4 . The proof of Lemma 15.4 is complete. The proof of Theorem 15.1 follows now from combining inequality (15.23) with Lemma 15.4 and with (l5.19) or (l5.21) . Remark.

Using the meaning of

N 1 (r)

we conclude from inequality (15.8)

as an important Corollary the inequality

(q-2) T(r,f)

+ G(r)

-

with

in

(15.36)

+ S(r)

I r

N(r, a)

where

k

+N(r,a )1

<

::

n(t,a)

Iz I :; t

n(O,a)logr

+

n(t,.) : n(O,.) dt

°

denotes the number of solutions of the equation

(15.37)

f(z) :: a

,each solution counted only once.

The generalized first main theorem shows ,in particular, that the sum 126

V(r,a)

N(r,a)

+

(15.38) T(r,f)

can not grow faster than

O( 1)

+

We can ask on the other hand,

if it is possible that the sum (15.38) grows only very slowly. The generalized second main theorem shows that, although this might be the case for some the sum of expressions (15.38) for any

3 different points

a,

a E en u {oo}

will grow at least as rapidly as T(r,f)

G(r,f)

+

+

N 1 (r)

except perhaps of a certain exceptional set

J

of r - intervals which is inde-

pendent of the points chosen. As a first application of the generalized second main theorem we can prove a generalization of the theorem of Picard. If non-constant meromorphic vector function in era liz e d

f(z)::: (fl(z), ... ,fn(z» e

e x c e p t ion a I

Pic- a r d

,then we will call val u e"

is a

"~-

any point

a

E

for which V(r,a)

N(r,a)

+

O(logr)

or equivalently for which v( +00 ,a) .+ n( +00 ,a)

Now assume that al

values

a2

q ::: 3

with

<

f( z) has 3 3 a E en u {oo}

different generalized Picard exceptional ' th eorem . Th en th e genera l'lze d secon d maIn

gives

T(r,f)

+ G(r,f)

<

Thus T(r,f)

O(logr)

<

or dividing by

+

T(r,f)

o {log T(r,f)}

O(Iog r) I

O{logT(r,f)}

+

<

T(r,f)

T(r,f)

outside the exceptional intervals. Here on the right side the second term tends to zero as

r

-+ +00

;

and if

f( z)

is transcendent, then also the first term

tends to zero. This contradiction proves Corollary 15.5 (Generalized Theorem of Picard)

(z»

Let

f(z)::: (f 1 (z), ... ,fn

be a transcendent meromorphic vector function in the plane

e

Then

127

there are at most V(r,a)

2

a E C n U {oo}

different points

+ N(r,a)

for which

O(Iog r)

or equivalently for· which v( +00 ,a)

If 1

f( z)

n( +00 ,a)

+

<

is entire or has only finitely many poles, then there is at most

such point Another application of the generalized second fundamental theorem is

Corollary 15.6 :

Let the non-constant meromorphic vector function be of finite t order in the plane

(fl(z)' ... ,fn(z»

C

f( z)

. If, for a positive

lJ,

the integral +00

!

V(r,a) + N(r,a)

(15.39)

--------~-----dr

r

converges for

!

IJ

+1

different

3

a

E

Cn

U roo}

,then the integral

T(r,f) + G(r,£) r

IJ

dr

+1

converges, so that

fez)

is at most of convergence class of order a E C n U {oo}

]J

,and

( 15.39) con verges for every

The proof follows from the generalized second main theorem, observing that

!

log r --dr ]J +1

<

+00

r

ro From the last Corollary we obtain the very important

tNevanlinna [271 ,p. 72

128

Corollary 15.7 (Generalized Theorem of Picard-Borel) : meromorphic vector function in the plane

C

f( z)

Let the non-constant

= (f 1(z) , ... ,fn (z»

Then there are at most

be of finite order p different points a E C n U {oo}

2

for which the sum V(r,a) is of order if

N(r,a)

+ <

v(r,a)

(or equivalently the sum

+ n(r,a»

p

f( z)

is entire or has only finitely many poles, then there is at most one such point a E C n The last considerations concerned meromorphic vector functions in the plane.

For meromorphic vector functions in a finite disc

CR

(0 < R < +(0)

similar

corollaries can be proved provided that T(r,f) (r -+ R)

+00

1

log R-r The most important application of the generalized second main theorem is the generalized Nevanlinna deficiency relation, which will be proved in the next section.

§16

The generalized deficiency relation

In view of the generalized first main theorem for each given of the expressions

m(r,a), V(r,a),

and

N(r,a)

(type, class) as

T(r,f)

exist at most

different points, for which the sum

2

a

the function

at least one

has the same order

. By the generalized theorem of Picard-Borel there

of smaller order (type, class). For any such value II

a

m(r,a)

II

V(r,a) + N(r,a)

is

generalized Borel exceptional

must then have the same order as

So far we can not say much with respect to the other points

a

T(r,f).

,apart from

a few results we obtained in §14. Combining the two generalized fundamental theorems we will in this and in the next section submit to a deeper study the behaviour of the quotients m(r,a)

V(r,a)

J'Hr,a) - N(r,a)

N(r,a)

G(r,f) , and

T(r,f)

T(r,f)

T(r,f)

T(r,f)

T(r,f)

In this entire section we assume that

129

lim T(r ,f) r+R

=

this is no restriction since in this case (16.1) only

R = +00

In the case means that

(16.1)

+00

is non-constant. (16.1) and the generalized first main the-

Hz)

orem show that V(r,a)

+ N(r,a)

+ m(r,a)

1

lim r+R

T(r,f)

from this we get

1

~

m(r,a) lim - - r+R T(r,f)

V(r,a) + N(r,a) ~

lim r+R

+

so that the sum in the middle must be equal to we define the number

6(a)

=

0 (a)

1

. For any

a

E

en u

{Go}

by putting

0 (a,f)

m(r,a) lim r-+R T(r,f)

6(a,f)

1

T(r,f)

V(r,a) + N(r,a)

=

1 -

lim r-+R

T(r,f)

(16.2) Then we have

o

~

~

6(a)

1

further, I)

(co)

=

m(r, f) lim r+R T(r,f)

1 -

N(r,f) lim r-.. R T(r,f)

(16.3)

The quotient (16.2) has already been considered in the Propositions 14.15 and

14.16. As on p. 126 we denote by a - points of

f( z)

in

Iz I ~

n(t,a)

t

=

n(t,a,f)

the number of distinct

and again we use the counting function

r

N (r, a)

N(r,a,f)

=

f o

N(r,f) We put further 130

N(r,co)

n(t,a) - n(O,a) dt + n(O,a)logr t

(16.4)

m(r,a) + N(r,a) - N(r,a) ==

B(a)

B(a,f)

lim T(r,f)

V(r,a) + N(r,a) (16.5)

1 r-+R

T(r ,f)

and N(r,a) - N(r,a) O(a)

O( a. f)

( 16.6)

lim r-+R

T(r,f)

then we have in particular N(r,£) B(oo)

1 -

(16.7)

lim r-~R

T(r,£)

N(r,£) - N(r,£) o( 00 )

o

~

B(a)

Given

R

to

(16.8)

lim r-+R

E: > 0

~

T(r,f)

o

1

~

~

O(a)

1

,we have from (16.6) and (16.2) for

r

sufficiently close

, N(r,a) - N(r,a)

>

(O(a) - s)T(r,£)

V(r,a) + N(r,a)

<

(1 - o(a) + s)T(r,£)

so that V(r,a) + N(r,a)

(1 - 0 (a) - O(a) + 2E:)T(r,f)

<

'and consequently B(a)

o (a)

+

(16.9)

flea)

As in Nevanlinna theory (i.e. n==l, fez) == f 1 (z» quantity

0 (a)

the point

a

of the values of

f

is relatively strong. Equivalently, sum

by definition (16.2) the

can be positve only if the asymptotic mean approximation to

V(r,a) + N(r,a)

on circles about zero with increasing radii is positive only if the growth of the

0 (a)

is deficient in the sense that it is relatively slow in

comparison with the growth of

T (r, f)

are clearly exceptional, the number

. Since points

0 (a)

a

with

o (a)

> 0

will be called the ffd e f i c i e n -

131

~If

of the quotient V(r,a)

N(r,a)

+

T(r ,f) or simply of the point

0 (a)

with

0

>

will be called the

a

,quite analogous to Nevanlinna theory. Points

will be called

"d e f i c i e n t

of m u I tip I i c i t Y

"i n d e x

is positive only if there are relatively many multiple by a multiple tion s

f ( z)

The quantity

If.

of

TI

a

e(a)

since

a - points of

a

e(a)

f

; here

a - point we understand a point such that the system of equa-

=a

f' ( z)

has multiple roots; these roots are zeros or poles of

and are thus countable in number.

e(a)

1

will attain its maximum

if the

relative density of multiple roots is large, and if their orders of multiplicity are z =

unbounded in the vicinity of 0 (a)

deficiency

o (a)

+

00

•

e(a) = I

For such a point with

the

must vanish since

e( a)

:;;

1

Remark. If we define as in Nevanlinna theory the number

nI(t,a)

by

setting n(t,a) nI(t,a) f( z)

-

n(t,a)

is the number of multiple solutions in

= a

,where a solution of multiplicity

We can then introduce the

TI

\)

Izl:;; t

of the equation

is counted only

(\) - 1) times.

counting function of multiple a -points

by putting

I

If

N I (r, a)

r

nI(t,a) - nl(O,a) --------dt

N(r,a) - N(r,a)

t

o for fixed

r

there are only finitely many

we have

=

L

NI(r,a)

a E Cnu {co} so that

132

a

for which

N I (r, a) '1= 0

,and

N 1 (r) lim - - r-+R T(r,f)

N 1 (r,a) lim - - - - r-+R T(r,f)

L

a E en u {oo}

Now, as an essentially new ingredient,as compared to Nevanlinna theory,we introduce the quantity

<5

G(r ,f) lim r-+R T(r ,f)

=

G (f)

( 16.10)

r -+ R

This non-negative number measures the relative growth as volume of the image of a disc 'V

f'

e

r

under the Gauss map

p n-1

:

(16.11)

compared to the growth of the characteristic fore the characteristic disc

e

T (r, f)

T(r,f)

f

into

quipped with the Fubini-Study metric. the

G a u ssm a p

or

It

It

. As we have shown be-

itself measures the volume of the image of a

under the map

as

r

of the

en

,where

is e -

!Ii n d e x

will be called the

Ric c i - i n d e x It of

en

f ( z)

<5

G

0

f

is positive

only if in a certain sense the growth of the Gaussian image is not essentially smaller than the growth of the image of

f( z)

In the following,as in Nevanlinna theory,we need on the rest term

S(r)

in the generalized second main theorem the condition

S(r ) \I

o

( 16.12)

T(r ,f) \I

for a sequence

r

. It is only then that

-+ R

\I

S(r)

plays the role of an

unimportant error term. Therefore we shall call the function

f

(for the generalized deficiency relation) if (16.12) holds.

is certainly ad-

missible if either

R = +00

,or in case

o<

R < +00

f

admissible t

,if

T(r ,f) \I

(16.13)

1

log - R-r \I

holds for a sequence -j-

r

\I

-+

R

outside the exceptional set of the second

Hayman [16] ,p.42

133

main theorem. This can be seen as follows: Suppose first that

= +00

R

• If

is a rational vector function, then

II f' (z) II 11f( z) II

so that in (15.20) this case. If

1if'(z)11 11f( z) - akll

o

---

f

= 0(1)

Sl (r)

o

as

; this shows that (16.12) holds trivially in

is not rational, then

T(r ,f) r

as

-+

+00

log r

o<

so that (16.12) follows in this case at once from (15.10). If

R

<

+co

and

if (16.13) holds, then (16.12) follows at once from (15.12). We can now prove an extension to vector valued meromorphic functions of Nevanlinna's deficiency relation. Theorem 16. I (Generalized Nevanlinna Deficiency Relation): phic vector function {a E en

u

f( z)

e(a) > O}

{co},

Let the meromor-

be admissible. Then the set is at most countable and summing over all such

poin ts we ha ve

L

[o(a) + e(a)]

I

+

a

2

+

Sea)

a

(16.14) here the quantities

0 (a),

e(a),

G(a)

and

<5

are defined by (16. 2) ,

G

(16.5), (16.6) and (16.10), respectively.

Proof. with

q+l

We choose a sequence different points

{r} satisfying (16.12) and apply (15.8) v q n q+l a, ... ,a E e , a = '" 1

q

(q-l)T(r ,f) v

+ G(r ) v

"

~

rVer ,a k ) + N(r ,a k ] v

+ N(r ,f) v

v

k=l -N1(r) +S(r) .

134

v

v

Observing that N(r,O,f')

+ N(r,£)

-

N(r,£)

and using (16.12) we get q

(q-l+ o(l»)T(r ,£) v

L

~

+ G(r ) v

k + N (r ,a )]

rVer v,a k )

v

k=1 + N(r ,f) v . a k - pomt o·f

Now an f'

f

of multiplicity 1

and so contributes only

to

is also a zero of order

f1

n(t,a k )

N(r ,O,f') v

-

n(t,O,f')

II -1

of

. Thus we may

rewrite the last inequality as

(q-l+o(l) )T(r ,f)

+ G(r )

v

v

~

+

where than

k

N (r ,a ) 1

+

N(r ,f) v

-

v

N (r ,o,f') v

0

N (r,o,f') refers to those zeros of f' which occur at points other k O and ignoring a - points of f (k = 1, ... ,q) ,Dividing by T(r ,f)

v

this latter term we deduce that k k V(r,a ) + N(r,a )

q

L

T(r,£)

q

1 r+R

N(r,f) lim r+R T(r,f)

+

lim k=1 r-+R

T(r,f)

L

[V (r , a k)

+

N(r , a k) 1

+

N(r , f)

k=1

G(r,£) q - 1 + lim r-+R T(r,£)

?

i. e.

q

L

[l -

B(a k )]

+

1

B(oo)

k=1 or

us

+

fl( (0)

This inequality shows that finite points

a

2

+

flea)

1

>

u -

for at most

fI.

. Thus the points for which

flea)

0

>

1

distinct

may be arranged

in a sequence, in order of decreasing which

EJ(a) = 1

flea) , by taking first those points for 1 ,then those, if any, for which EJ(a) > 2" ' then those of

the remainder for which

e(a)

>

the resulting sequence and putting

q

:3 '

{a

etc .. If

a (0)

=

00

,

(fI.)

}

_ (fI. - 1, 2, ... )

is

we deduce that

2

+

for any finite

1

,and hence if the sequence

{a (fl.)}

is infinite we con -

elude that +00

2

+

This proves Theorem 16. 1. Using the definition of

e( (0)

the right hand inequality of the deficiency

relation (16.14) can be written

L

e(a)

+

1

N(r,I) lim - - r+R T(r,I)

+

2

0G

with

where now the sum is to be taken only over any set of >

Sea)

0 • This gives with (16.9)

L

[0 (a) + 6(a)]

L

N(r,£) flea)

:;;;

1 + lim

r+R T(r,I) ( 16.15)

We deduce in particular Corollary 16.2 : If under the conditions of Theorem 16.1 the meromorphic vec-

136

tor function

f

has no poles in

N(r, f) lim r-+R T(r,f)

or more generally, if only

o 2

then the number

(16.16)

on the right side of the deficiency relation (16.14) can

1

be replaced by

CR

provided that the sums are extended over finite

a

only,

i.e.

L

[0 (a) + e(a)]

L

+

9(a)

( 16.17)

+

a E Cn For example (16. 17) is always satisfied for an entire vector function. From (16.15) we see further If

Corollary 16.3: in

CR

0

,then the

:;;

f( z) is an admissible vector valued meromorph.ic function inde~

N(r,f) 1 + lim r-+R T(r,f)

:;;

°G

of the Gauss map satisfies the inequality

( 16.18)

In particular, 0 if

f( z)

0 if

:;;

:;;

°G

2

is meromorp hic ; and :;;

f( z)

:;; 1

°G

is holomorp hic .

In the sum m(r,a) the terms

+ N (r, a)

+ V(r,a)

m(r, a)

and

V(r,a) + N(r,a)

tically very differently as convergence of

f(z)

to

weak, so that the deficiency

r -+ R a

behave for fixed

. For the "normal" points

,which is measured by ° (a)

m(r,a)

a a

asymptothe mean

,is relatively

will vanish, i.e.

137

m(r,a) lim r-+R T(r ,f)

"-

0

whereas V(r,a) + N(r,a) lim r-+R If

I)

(a)

1 T(r,f) V(r,a) + N(r,a)

is positive, then the growth of

weak; in general the number V(r,a)

I)

(a)

is relatively

is a measure for the growth of the sum

+ N(r,a)

If in particular for a transcendent meromorphic vector function in

point

a

C

the

is a generalized Picard exceptional value, then the deficiency

attains its maximum value

1

=

logr

since

o(T(r,f)

0 (a)

. Therefore

Theorem 16.1 contains the generalized theorem of Picard (Corollary 15.5), ac-

2

cording to which there are at most

distinct such values

a

On the other hand, for a meromorphic vector function in

C

o (a) = 1

the equality

which is equivalent to V(r,a)

lim r-+ +00

+

N(r,a)

o

T(r,f)

does by no means imply that the point

a

is a generalized Picard exceptional

value. The notion of deficiency allows to distinguish between possibly countable infinitely many exceptional points as compared to only

2

such values in the

generalized Theorem of Picard-Borel. From Theorem 16.1 we also note that there are at most a

2

distinct points

for which

2 - 0 I)

(a)

>

G

3 and in particular at most

2

points with

Corollary 16.2 we see that there is at most

o(a)

o(a)

1

2

>

"3

analogously from

finite point

a

o (a)

>

such that

1 - 0G >

2 and in particular at most

1

One can pose the general

138

finite point such that

1

"2

Problem:

Given sequences

°G ;:

and a number

0

,such that

f(z) ::: (f1(z), .. ·.,fn (z»

is there a vector valued meromorphic function 51,

::: e51,

o(a )

°

and with

G

the Ricci-index of

,

f( z)

and

o(a) ::: 6(a) ::: 0

for

a

with

¢

{a 9-} ,

?

In scalar NevanIinna theory this deep problem, of which only partial solutions have been known earlier, has recently been given a positive answer t We now introduce an important new concept, which has no significant counterpart in scalar Nevanlinna theory, by setting

I

:::

Y(r ,a) lim r+R T(r ,f)

-

m(r,a) + N(r,a) ( 16.19)

lim

:::

T(1' ,f)

In view of the first main theorem we have always

and in particular

('\ Y( <',,)

(n=l) it is obvious that deed the quantity 0y(a)

:::

I

0y(a)

0y(a)

In the case of scalar Nevanlinna theory

1

for any

a

E

e

u

{co}

so that in-

is of no significance in this case. In general,

is positive only if the volume function

Y(r,a)

grows more slowly

than the maximum possible growth permitted by the first main theorem. For this reason the point

0y(a) a

will be called

will be called the

with respect to

"v

0

1u m e

f

!Iv

0

1 urn e

de fie i e n c y"

of

,and a point such that

d e fie i e n t".

A first application of this notion will be given in the following. We shall say

that a point the equation

a 51,

E

f(z)

en u = a

{oo}

9-

has multiplicity at least have multiplicity at least

, if all roots of . We have

tDrasin [I8]

139

1 !(, - [T(r,f) - V(r,a)1 m!(,

and if

T(r ,f)

+ O( 1)

is unbounded

V(r,a) + N(r,a!(,)

1

_

)lim m!(, r-+ R

!(, V(r,a )

(l - -

lim r-+R

T(r ,f)

T (r ,f)

so that !/,

1 m!/,

(l - - ) 0

e(a )

If

!(,

V

(a )

is admissible the deficiency relation (16.14) shows that

f( z)

'" (l - - 1 )OV(a !/, ) L.. Let us call a poin t for

f( z)

a!/'

E

if the equation

has multiplicity at least

(16.20)

2

m!/,

Cn

U {oo}

f( z) m!/, ;;; 2

= a

m u 1 tip I ell !(, has only mUltiple roots, i. e. a

IIC 0

!(,

m pIe tel y

. For such a point we have

1

2" and we conclude from (16.20) Corollary 16.4: CR

Let

n (0 < n

. Then given 4 - 28

f( z)

be an admissiblp. meromorphic vector function in ~ 1) ,

there are at most

G

(16.21)

n distinct points

0)

a

a

E

Cn

U {oo}

is completely multiple for

In particular, if

8

G

= 2

conditions of Corollary 16.4.. 140

such that the following conditions hold both: f( z);

,then there can be no point which satisfies the

f(z)

If

is entire then we see from (16.15) that instead of inequality(16.20)

we can write

1

+

where now the sum is extended over finite

a

Q,

only. In Corollary 16.4, if

is en tire, the bound (16.21) can be replaced by

f( z)

n if only finite completely multiple points are considered; if

oG = 1 ,there are

no such points in this case.

§17

Further results about deficiencies

In the first part of this section we compute the quantities

o (a) , e( a) , e(a) ,an d

0 V (a)

of §16 for the instructive example of rational vector functions ... ,f (z» n

In this case we know from §4 that d

= n* ( 00 )

::;

*( 00 )

f( z)

has degree

+ n ( +00 ,00)

and that the following formulas are valid: T(r,f)

= mer,£) +N(r,f)

mer,£)

= *(00) logr

N(r,£)

= n(+oo,oo)logr

m(r,a)

= tea) logr

* = n(oo)logr + 0(1)

+ 0(1) + 0(1)

+ 0(1)

(a E C n U roo})

From this we compute tea)

o(a)

=

-*n(oo)

v( +00 ,a) + n( +00 ,a)

=

1 -

* n(oo)

(17 .1)

so that in particular

141

*( 00 8(00)

)

-*n(oo) en u {oo}

We see that in

the generalized deficiency

8 (a)

o

up to the_

finite or infinite point a

=

o a

=

:

lim f( z) z-+ 00

'"

(a =

if, and only if one of the component functions

U)

f.

J

has a pole at in-

finity) . r -+ +00

Since as

-?-

---

* n(oo)

T(r, f)

n(+00 , a)

N(r, a)

n ( +00 , a)

N(r,a)

---

-?-

*

T(r,f)

nero)

we have -

n(+oo,a) e(a)

n( +00 , a) (17.2)

* new)

Now remember that the first main theorem m(r,a)

+

N(r,a)

+ V(r,a)

:::

+ 0(1)

T(r,f)

can be written *(a)logr

+ n(+oo,a)logr

+ v(+oo,a)logr

=

*

n(oo)logr

+

0(1)

so that we have the relation

*(a)

+

n ( +00 , a)

+ v ( +00 , a)

*

:::

n ( 00 )

* ( 00 )

:::

+ n ( +00 , 00 )

of Proposition 4. 2 . We compute further, *(a) +n(+oo,a)

v( +00 ,a) 1 -

----:;;*--

n ( +00 , a) 8 (a) +

*

n(oo)

n(oo)

*

(17.3)

n(oo)

and v( +00 , a) + E>(a)

1-

n(+00 , a)

*(a) + n ( +00 , a)

*

* n(oo)

n(oo)

=

142

- n( +00 , a)

8 (a)

+

e(a)

(17.4)

o

In particular we have for

'* a

a

n ( +00 ,a) O(a)

*

n(co) o

a E C n U {oo}

so that for

=

o (a)

=

O(a)

- {a} - f(

=

0y(a)

00

C)

B(a)

= 0

Specializing (17.1) - (17.4) to entire rational vector functions, we have *(a)

o (a)

=

(a E C n )

0

0(00)

= 1

( 17.5)

0

(17.6)

*(00) n(+oo,a)

-

n(+co,a)

=

Sea)

S( 00 )

=

*( 00 )

v( +00 ,a)

*(a)

1 -

o (a)

*( co ) so that

n( +00 ,a)

+ n ( +00 ,a)

=

+ *( 00 )

*( 00 )

n(+oo,a) (a

E

Cn )

, and of course

*( 00 )

v( +00 ,a) +

n( +00 ,a)

*(a) + n( +00 ,a) -

1 -

B(a)

*( co )

=

fi( +00 ,a)

=

0 (a)

*( 00 )

+ 8(a)

=

0y(a)

=

(a E C n )

8(a)

B( (0) = 1

(17.8)

We note that for

=

o (a) What

does

L

e(a)

=

B(a)

=

0

the generalized deficiency relation tell us

(O(a)

+

8(a»

+

2

aECnu {oo} if

f( z)

is a non-constant rational vector function?

For rational non-constant vector functions this becomes

143

o

n( +00 ,a) -

*(a) ;r-

+

n( +00 ,a) ;:;

+

*

2

,

(17.9)

n(oo)

n(co)

a E enu {co}

n( +co ,a)

n( +00 ,a) >

where as above

o a

z ___ co

For an entire rational vector function n ( +00 , a) -

L a

E

this simplifies to

n(+00 , a) +

(17.10)

1

*( 00 )

en

n ( +00 , a) >

Example 1.

n(+00 , a) As an example we consider the entire rational vector function

3 , z 5)

f( z)

(z

Here every point

a E

e2

not of the form zo *- 0

not assumed. In every point

(z~,z~)-pointofmultiPlicity point of multiplicity

3

We have

5

where

f( z)

*(00)::

the function

l,andin

fez)

zO::O

2:0

has an

it has an

n(r,a):: n(r,a) n( +00 ,a)

1

:::

up to the point

2

1

+

f' (z)

(3z

N(r,O,f')

:::

4

, 5z )

2logr

3

<5

G

5" directly as follows. From

T(r,£') m(r,O,f')

we obtain using the first main theorem, G(r,£)

::: V(r,O,f')

:: 210gr + 0(1)

On the other hand, T(r,£) 144

510g r

+ 00)

::: 410gr 0(1)

+ 0(1)

is

a:::(O,O)-

, so that the deficiency relation

i.e.

On the other hand we can compute 2

e , a:::

a:: (0,0)

(17.10) is

5"

E

Let us first examine the deficiency relation (17.10)

::: 3 ,

n( +00 ,a)

3 5 a::: (zO' zO)

so that in fact 2

=

5"

For completeness let us compute the other equidistribution quantities for this example. We have

a (co) = e( co) =

course

= mer,£)

T(r,£)

= 510gr

1 . For finite

a

+

0(1)

,so that of

we distinguish the following

cases.

0) Here

aEC

2

= 0(1)

m(r,a)

N(r,a)::: 0

= 510gr + 0(1)

Y(r,a)

=

0y(a)

=

0

and by the first main theorem

,so that

=

= a (a)

e(a)

Sea)

(ii)

Here

m(r,a)

theorem

0(1)

4

1

(iii)

Here

=

a (a)

0(1)

, and by the first main

so that

1

=

0

a (a)

=

Sea)

(0,0) 0(1)

n(O,a)

V(r,a)

first main theorem

=

+

5"

4 + 1 -5-

m(r,a)

a yea)

= log r

+ O( 1)

=

5"

1 a

N(r,a)

= 410g r

Y(r,a)

B(a)

,

1

2

5"

= 3

=

1 _ 2 + 3 -5-

5" = .0

= 3 ,

210gr

N(r,a)

+ 0(1)

B(a)

Sea)

= =

, and by the

= 310gr so that

1

2 + 1 -5-

3 - 1

-5-

=

2

5"

2

5"

As compared to scalar NevanIinna theory, where we have to consider the value distribution quantities

a (a)

Sea)

B(a)

we have in the vector valued theory the additional quantities The relations between these quantities in the vector valued theory are more

145

complicated than in the original Nevanlinna theory, and it is useful to examine a little the interdependence of some of these quantities, in particular under special assumptions. a = co

For the point 15 ( co )

e(co)

15

V

(co)

V(r,co) = 0,

we have from the definitions and from

=

1-'- N(r,£) 1 - 1m ~-R T(r,£) r-+

=

1 -

=

1

N(r,f) lim r-+ R T(r,f) N(r,£) - N(r,f) B( co )

lim T(r,f)

and

Proposition 17.1: function in

CR

be a meromorphic vector fez) = (f1(z), ... ,fn (z» Then the following conclusions hold.

Let

=

0(00)

N(r,£) lim r-+R T(r,£)

1

N(r,f) lim r-+R T(r ,f)

o e( co )

(ii)

e(oo)

= 1

N (r, f) lim r-+R T(r,f)

=>

= 1 ,

=

B( co

)

0

,

= 0

N (r, f)

=

,

0

B( co

= lim

)

r-+R T(r,f)

N (r, f) - N(r,f) ( iii)

B( 00

) = 1

lim r-rR

=>

1

e( 00)

N(r ,f)

N (r, f)

0(00)

= 0

, lim

= 0 , lim

r-rR T(r,£)

(iv)

(v)

146

0(00)

e( 00)

= 0

= 0

= 1

T(r,f)

=>

N(r,£) lim r-rR T(r ,f)

= 1

=>

N(r,f) lim r-rR T(r,f)

= 1 ,

0(00)

= 1.

r-rR T(r,f)

= 0 ,

B( co )

= O.

( vi)

e( 00 )

N(r,a) lim - - r+R T(r,f)

= 0

:;;

N(r,a) lim - - r+R T(r,f)

We also note Proposition 17.2

be a non-constant mero-

Let

morphic vector function in

CR

. Then the following inequalities are valid for

aECnU{oo} N(r,a) lim r+R T(r,f)

N(r,a) lim r+R T(r ,f)

(17.11)

N(r, a)

N(r,a) lim - - r+R T(r,f)

(17.12)

lim r+R T(r,f)

Proof of (17. ll) . m(r, a) lim - - r+R T(r,f)

m(r,a) + N(r,a) lim r+R

T(r, f)

- m(r,a)

m(r,a) + N(r,a)

=

+

lim T(r,f)

r+R

T(r,f)

The right side is :;;

N(r,a) lim r+R T(r,f)

~

and

N(r,a) lim r+R T(r,f)

This shows (17 .ll); (17.12) is shown analogously. From Proposition 17. 2, from the inequality

e(a) + 6 (a)

:;;

or from

Sea)

the definitions we deduce Proposition 17.3:

Let

morphic vector function in

fez) = (f 1 (z), ... ,fn (z»

CR

VCr ,a)

0)

6 (a)

1

be a non-constant mero-

. Then the following conclusions 'hold.

lim r+R T(r,f)

=

0 Sea)

N(r ,a) lim - - r+R T(r ,f)

=

1,,' e(a)

=

=

0

,

0

147

48

N(r,a) Oi)

o (a)

::::

0

N(r,a)

lim

'*

~

lim

<

°V(a)

r-}-R T(r,f)

r .... R T(r,f)

N(r,a) ( iii)

lim

---

;:

0

o (a)

oyea)

'*

r .... R T(r,f)

B(a)

O(a)

::::

V(r,a) (iv)

B(a)

::::

1

N(r,a)

lim

'*

0

Jim

0

r-+R T(r,f)

0

r+R T(r,f)

N(r,a) 1 - lim r-}-R T(r,f)

;;;

~

0 (a)

N(T,a) 1 - lim r .... R T(r,f)

N(r,a)

N(r,a) ( v)

B(a)

;:

0

;;;

lim

=>

:5

6 yea)

lim r .... R T(r,f)

r .... R T(r,f) N(r,a) ( vi)

lim

---

;:

0

oyea)

=>

(:I( a)

r .... R T(r,f) 6(a)

( vii)

6(a)

1

=>

B(a)

;:

;:

N(r,a) lim r+R T(r,f) VCr ,a) lim r-}-R T(r,f)

1

N(r,a) lim - - r .... R T(r ,f)

;:

V(r,a)

lim

( viii)

0

0

;:

6 yea)

;:

O(a)

;:

1

o

r .... R T(r, f)

o (a)

N(r,a) ;: I-lim - - - , r .... RT(r,f)

N(r ,a) 0)(a) ;: I-lim - - -

r-.. RT(r,f)

0

(ix)

<')V(a)

::

0

o(a)

~

O(a)

::

::

0

B(a)

0

N(r,a) lim - - r-+R T(r, f)

0

::

o

In §4 we proved for rational vector functions that the limit V(r,a) lim r-+ +00

lim

::

v(r,a)

v( +00 ,a)

log r

exists. The same statement and proof holds for arbitrary meromorphic vector functions the only difference being that the limit

v( +00 ,a)

may now be

+00

In exactly the same way we prove that the following limit exists, in the sense that it is either a non-negative number or is

+00 :

N(r,a) lim r-+ +00

::

lim n(r,a) r-+ +00

log r

n(+oo,a)

Using this remark and the fact that T(r,f) +00

(as

r -+ +00)

log r for any transcendent meromorphic vector function in the plane, we obtain the next 2 propositions.

Proposition 17.4:

Let

f( z) :: (f I (z) ,. .. ,fn (z»

a

morphic vector function in the plane. Let

E en

be a non-constant mero-

u {oo}

and assume that 1

of the following 2 conditions is satisfied: (i)

( ii)

VCr ,a) lim r-+ +00 T (r , f)

f( z)

o

is transcendent and

v ( +00 , a)

<

+00

Then it follows that

o (a)

N(r,a) ::.1 - lim r-+ +ooT(r, f)

N(r, a) , B( a)

::

1 - lim r-+ +ooT(r, f)

Remark. The assumptions of Proposition 17.4 are satisfied in the Nevanlinna

149

case

n = 1

for all

a ECU{oo}

f(z) = (f1(z), ... ,f n (z) morphic vector function in the plane. Let a E Cn

Proposition 17.5:

Let

be a non-constant meroU {oo}

and assume that 1

of the following 2 conditions is satisfied: (i)

(ii)

N(r,a) lim r+ +00 T (r , f) f( z)

=

o

is transcendent and

+00 ,a)

n(

< +00

Then it follows that 8(a)

e(a)

=

0

The last Proposition shows that in many cases the deficiency qual to the volume deficiency

<5

(a)

is e-

I)V(a)

Remark. If the conditions (ii) of Proposition 17.4 and of Proposition 17.5 are both satisfied, then

V(r,a) + N(r,a)

O(Iogr), so that the point

a

is an exceptional point in the generalized sense of Picard; in this case it follows that <5 (a)

8(a)

We now introduce"lower"andllupper"

1 11

c

0

u n tin g

d e f i c i e n c i e s"

by putting N (r, a) lim r+R T(r,f)

N(r,a) lim r+R T(r,f)

N(r,a) lim r+R T(r, f)

N(r,a) lim r+R T(r, f)

and we introduce the following subsets of

Cn

Definition 17.6 E( <5)

150

=

{a E C n , I) (a) > O}

E(8)

.

= =

{a E C

If for example

n

{a E en, O( a) > O}

E( 0)

, t\N(a) > O}

is an entire rational vector function, then we see

fez)

from the investigation at the beginning of this § that f ( ' C) = E ( () N) = E ( () j\j) ::: E ( i:I N) ::; E ( t\ j\j)

For any

f

n:: 1

in the scalar case

we have

From Proposition 17.2 we see that

c

c

E( 6 V)

n > 1

If

(17.13)

,then

have

2n - dimensional Lebesgue measure zero since these sets are subsets of f(" C). If

relation

n:O; 1

is admissible and if

f( z)

E(,),

and

E( 0)

,then by the generalized deficiency

can be at most countable.

E(e)

From (17.13) and the deficiency relation we deduce If

Proposition 17.7:

is admissible and if

f( z)

is countable. On the other hand, if

E(6 V )

is countable, then

E(

I)

N)

f(z)

E( 1I N)

is countable, then

is admissible and if E(6 V )

is countable.

If condition (ii) of Lemma 17.5 is satisfied, then condition

0) is also satis-

fied and we deduce from Lemma 17.5 and the deficiency relation Proposition 17.8:

Let

Assume that for all

f( z)

a E en

be a meromorphic vector function in the plane. one of the following 2 conditions is satisfied:

N(r,a)

o

lim r->-+oo T(r,£)

(ii) Then the set

f( z)

6 V (a)

6(a)

E(

is transcendent and

I)

V)

::

E(E-)

::

EJ(a)

:: E(6)

e(a)

n ( +00 , a) ->- +00 ::

0

holds for all

is countable, and the deficiency rela-

151

tion can be written in any of the following 3 identical formulations:

I:

(i)

°G

+

°V(a)

~

1

+

N(r,f) lim r++ooT(r,f)

~

1

+

N(r,f) lim r->-+ooT(r,f)

1

+

N(r,£) lim r-+ +00 T (r , f)

a E en

I: °(a)

(ii)

+

°G

+

°G

a E en

I:

( iii)

Sea)

a Ee n

~

Assumption 0) of this proposition means that there are not too

Remark.

many a - points for all finite

a

; this will frequently be the case. One can

ask on the other hand what happens if there are many a - points for some particular

a

1 -

. From the first main theorem we obtain the inequality V(r,a) + m(r,a)

N(r,a) lim - - r+RT(r,f)

lim r+R

;;: T(r,f)

V(r,a) lim r-+RT(r,f)

+

It shows that we must have

V(r,a) lim - - r-rRT(r, f)

o (a)

=

if there are sufficiently many

o

a - points so that the expression

N (r, a) lim - - r-+ RT(r, f)

=

assumes the maximum possible value In the case f

n

R

= +00

1 .

we assume in the rest of this

§ that all

are non-constant. It follows then that T(r,f.)

~

J

o<

In the case T(r ,f.) J

152

,.

......:;,.

+00

as

R < +00 +00

r-+ +00

j = 1, ... ,n

we aSSUme for simplicity always that as

r .... R

j '" 1, ... ,n

o(a)

Using the estimate T(r,f)

T(r,f.) J

j ::: I, ... ,n

O( 1)

+

and the first main theorem of Nevanlinna theory we obtain the following 2 estimates for the same quantity I)N(a) for a::: (a 1 , ... ,an) E C n ; j ::: 1, ... ,n k ::: 1, ... ,n .

0)

=

I) N(a)

N (r, a) lim r>RT(r,f)

N(r,a k )

'S lim

T(r,f k ) lim r-+R T(r,i)

~

r-rR T(r,f)

T(r,f k ) lim r+RT(r,f.) J

~

Oi)

I) N(a)

:::

N(r,a) lim r-+RT(r,f)

N(r,a) ;;; lim r-+RT(r, f.) J

:; lim

(17.14)

N(r,a k )

r-~RT(r,

;;; lim

f.) J

T(r,fk )

(17.15)

r-+ RT(r, L) J N

Replacing replacing for

lim

by by

N lim

we obtain the analogous estimates for

I) N(a)

; and

we obtain the analogous estimates for

lI N (a)

and

lI N(a)

For

we obtain

a I)

N

(00)

:::

N(r,f) lim r+RT(r, f)

and similar estimates for

>

N(r,f k ) lim r-+R T(r,f)

I) N( (0) , 1I N (00 )

In the same way we obtain for

(17.16)

and

N(r,a k ) - N(r,ak ) + 0(1)

N(r,a) - N(r,a)

0)

O(a)

:::

lim r+R

:;; lim r-+R

;;; lim

T(r,i) N(r,a k ) + 0(1)

r-+R ;;; lim

T(r,f)

Sea)

T(r,i)

T(r,f k )

r+R T(r,f)

N(r;a) - N(r,a) (ii)

fI N( 00 )

O(a), (a E C n )

;;; lim

T(r,f k )

( 17.17)

r+ R T (r , f. ) J

N(r,a) - N(r,a) ;;; lim

::: lim

T(r,f)

T(r,f.) J 153

S lim

N(r,a k ) ~ N(r,a k ) + O( 1)

N(r,a k ) + O( 1)

S lim

S

r->-R

T(r,n

J

T(r,L)

J

T(r,rk) lim r->-RT(r, L) J (17.18)

and N(r,f) ~ N(r,£) e( co )

= lim

(17,19)

T(r,£)

r->-R

T(r,n

In particular we see from (17.18) that 8(a)

j

O(a.,f.) J J

= 1, ... ,n

(17.20)

For the volume deficiency we obtain for

the estimates m(r,a k ) + N(r,a k ) lim ---,-'- - - - - r-~R T(r, f)

m(r,a) + N(r,a) (i)

?

lim r+R

T(r,n

= lim

T(r,f k )

T(r,fk) lim r->-RT(r, L) J

:<;

r->-R T(r,n

m(r,a) + N(r,a) (ii)

lim r->-R

07.21)

m(r,a) + N(r,a)

lim

5

T(r,£)

T(r,L)

J

. m( r , ak ) + N ( r , ak ) ? 11m r-~R T(r,f.) J ,) (a)

For the deficiency

we obtain the following estimates for

m(r,a) (i)

o(a)

= lim

~

r+-RT(r,f)

m(r,a) (ii)

,) (a)

(17.22)

= lim

S

r->-RT(r ,£)

m(r,a k ) lim r->-RT(r,f)

m(r,a) lim r->- RT(r, L) J

<

~

a E en :

T(r,f k ) T(r,f k ) lim S lim r->-R T(r, f) r->-RT(r, f.) J (17.23) T(r,f k ) m(r,a k ) S lim lim r-T R T (r , f.) r->- RT(r, f.) ]

]

( 17.24) mer,£) ,) ( co )

= lim - - r+RT(r ,f)

154

N (r, f) I - lim - - r->-RT(r, f)

S

N(r,f.) I-lim J r+RT(r, f)

(17.25)

m(r,f) lim r+RT(r, £)

>

m(r,f k ) lim r+R T(r,O

(17.26)

In particular we see from (17.24) that 6(a)

(17.27)

6(a.,f.) J J

2

We have further for

a E

en

V(r,a) +N(r,a)

en

El(a)

N(r,a) -N(r,a)+ m(r,a)

=

= 1 - lim T(r,£)

r+R

lim T(r ,£)

r-~R

N(r,a) - N(r,a) + m(r,a) lim r->-R

~

;;;

lim r-+R

T(r,f.)

J

N(r ,a k ) - N(r,a k ) + mer ,ak ) + O( 1)

T(r,f k ) lim r-+ RT(r, f.)

T(r,f.) J

J

(17.28) N(r,a) -N(r,a) + m(r,a) El(a)

(ii)

= lim T(r,f)

r+R

;;; Jim r-+R

T(r ,f k ) lim r-+RT(r ,£)

N(r,a k ) + O( 1) + m(r,a k ) T(r ,£)

T(r,f k ) lim r+ RT(r, f.)

~

(17.29)

]

e( 00)

N(r,£) 1 - lim r-+ RT(r, f)

:;;

N(r,f.) 1 - lim J r+R T(r ,£) N(r,£) - N(r,£) + m(r,f)

N(r,f) e( 00

)

= 1 - lim - - r+ RT(r, f)

(17.30)

lim r+R

T(r,f)

N(r,f.) - N(r,f.) + m(r,f.) ;;; lim

r+R

J

J

J

(17.31)

T(r,f)

In particular we see from (17.28) that

155

e(a)

j :;: 1, ... ,n

e(a.,f.) J ]

(17.32)

These inequalities show thai the relative growth of the component functions f.(z) ]

and in particular the number T(r,fk ) min lim j,k r+RT(r,f.) J

(17.33)

has a very strong influence t on

o(a), e(aL

0v(a),

e(a),

0N(a),

(aECn~

from what we have just seen follows that these quantities must all vanish if (17.33) is zero. For meromorphic vector functions in the plane we could have deduced already from Proposition 14.15 that identically

provided that

p

Proposition 17. 9:

*

p

<

•

We formulate

Let

be a mcromorphic vector function in the plane such that f( z)

be a meromorphic vector function in

T(r,f.) -? +<» J

as

r + R

T(r,fk ) min lim j, k r+RT(r,f.) J Then we have for all

CR

(j:;: 1, ... ,n)

p

*<

(0 < R :> +00)

p

•

Or let

such that

and

o a

E

en

07.34)

(n i: 2)

(17.35) In particular, if

f( z)

is admissible in addition, then the deficiency relation

reduces to the following estimate of the index 1 +

N(r,f) lim - - r+RT(r,f)

0G

of the Gauss map: (17.36)

tThis was first recognized by H. Wellstein (1973, private communication).

156

This Proposition shows that the deficiency relation is of interest mainly for meromorphic vector functions not satisfying 07.34). So in the case of rational vector functions all component functions

f.

are of the same order zero, and

J

07.9) or example 1 show that the deficiency relation is , in fact, non-trivial in this case; each quantity 0 V(a), {-)(a) 2 values of a E C ,and the deficiency

e(a)

and

o (a)

is positive for certain

is zero for

a .i= 00

• In order

to give a non-rational example, where the component functions have the same order, we propose the following example of an entire vector function (n = 2). ( e z , e 2z)

f( z)

Example 2.

Here each component has order I . We put e

2rcosa

so that

+

e

z

= re ia

and compute

4rcosa

IT

2

,~ Ilog(ezrcosu

m(r,f)

'e'rcosu)d" ,

0(1)

11

2 IT

2

/, Izr COS" da ,

0(1)

Zr + o( 1) IT

11

-2 N(r,f) = 0

Since

T(r,f)

=

we have also 2r

+ o( 1)

1[

so that the order of

f( z)

is 1 as it should be, and

6 (00) = 1

. For

aEC

Z

we distinguish three cases

with

i)

Such a point further

a

is not- assumed by

m(r,a) = 00)

f( z)

so that

N(r,a) = 0

. We have

so that by the first main theorem

157

Zr + 00) 'IT

=

V(r,a) Thus we obtain

=

= 0

8 V (a)

= 0 (a)

B( a)

= 8 ( a)

ii)

Zo Zo

Here form n(t,a)

fez)

is a root of the equation + Zk'ITi

, where

Izi s

of all roots in

= t

n(t,a)

k

, All other roots are of tho

=a

is an integer', This shows that the number satisfies

t

+ 00)

'IT

so that N(r,a)

=

+ O(Jogr)

r 'IT

Further,

= 00)

mer ,a) r

'IT

V(r,a)

so that by the first main theorem

+ OOog r)

This gives

1

B(a)

'2 iii)

= 8 (a)

o

= Sea)

(0,0)

a

This point is not assumed by

N(r,a) = 0

so that

f( z)

have

m(r,a)

=

da

1 ~g'

IZ r cos a

Ve

'IT

'2

158

~

+e

4r cos a

+

00)

,Further we

3~

1 21T

=

I

2

I og e -2rcosa. d a.

2r

=

0(1)

+.

1T

+

0(1)

71

2 V(r, a) = 0(1)

so that, by the first main theorem,

. Thus we have in this

case

=

= 1

15 (a)

15

V

13(a)

(a)

=

e(a)

In particular the quantities

°

reach in

a

= (0,0)

their

maximum possible value 1 , What does the deficiency relation tell us about B(oo)

+ 13«0;0»

L

+

B(a)

6G

? We have 2

+

a E C 2_{(0,0)}

1

1

+

°

+

+

2

0G

oG =

°

so that the index of the Gauss map is zero. We can confirm this latter result by a direct computation:

=

f'( z)

T(r,f')

N(r,(O,O),f') so that

G(r,£) =

=

°

m(r,(O,O),f')

= 2r

= 2r + 0(1) 71

+

0(1)

71

= V(r,O,f')

Ger,£) lim r++ooT(r,f)

= m(r,f')

= 0(1) =

,and indeed

°

What can be said about the generalized Nevanlinna deficiencies is (a,£)

15 (a) =

beyond the general deficiency relation? Applying a selection of known

results on Nevanlinna deficiencies to the component functions

f.(z) of £(z) J we will now give a few rather simple conclusions about generalized Nevanlinna deficiencies. As was shown by R.Nevanlinna a meromorphic function

(;(z)

of order

p (;

159

can have

2 distinct values

a

, b E e u{oo}

such that r; is a positive integer or if PI:;

r;

I)(b ,r;) = 1 only if Pr r; '0 Assume now that for a meromorphic vector function fez) = (fl(z), ... ,fn n n ( z) ) for 2 distinct points a=(a1, ... ,a n ) Ee , b = (b 1 , ... ,b n ) E C 6(a) = 8(b) 1 we have maximum deficiency Then each component

f. (z) must have the same order by Propop. = P as f( z) J J 6 (a) :;; 6(a.,f.) ;;:1, 6(b)
function

=1)(b,f)=l n

We deduce that there must exist jo

an index

2 distinct complex numbers

such that

o (b.

o(a. ,f. )

JO

JO

,f. )

JO

P = +00

1

P

is a positive inte-

be a meromorphic vector

Let

function in the plane. Assume that for =

and

This proves

Proposition 17.10

6(a)

a. , b. JO JO

JO

From the above result of Nevanlinna we see that either ger or

n

I)(b)

= 1

(a

2 points

a , bEen

*" b)

Then all component functions

must have the same order p,

which is a positive integer or is infinite. Further, under the assumption we have

=0

6G

,

(, (c) = 0 O(c)

. (c E en u {oo} - {a, b })

=0

(c E enu{oo})

Next we apply the following Theorem 17.11 : If

o :5

P r;

when

1:;

>

and 0

r;( z)

is a meromorphic function of order

, then

ar

160

, where

when

p

'0

ular a meromorphic function

tNevanlinna [27] ,po 51

r;

= 0 or I) (a , 1:;) :~ 1 - cos "TTP r; 1:; I:; is the only deficient value of r;( z) . In partic-

6(a,l:;) >0 I:;

P

r;( z)

of order zero can have at most one defi-

. Clent va1ue t .

We deduce Let £(z) = (f 1 (z), ... ,f (z» , (£1""'£ non-constant) n n 1 be a meromorphic vector function of order p in the plane, where 0;;; p < 2"'

Proposition 17.12

a = (aI' ... ,an) E en

Assume that for some 6 (a»

1 - cos

value of . ,n

f( z)

, and each

function

f( z)

P > 0

when

lTp

also,

• Then

6 (a)

a

>

0

when

p

= 0

or

is the only finite deficient

a. is the only deficient value of £.(z) for j = 1, .. J J has order p • In particular a meromorphic vector

£. (z) J of order zero can have at most one finite deficient point. 6(c)=0

Under the assumption we have

(c*a,oo) N(r,£)

cos TIP

+ lim - - - , (0 < p < .!.) r-+ +00 T (r , f) 2

+

N(r,£) <

Proof.

Since

a

1

+

lim r-++ooT(r,f)

, (p = 0)

is deficient, each

f. ( z) has order p by Proposition J when 17.9. The assumptions and inequality (17.27) show that <S(a.,L) > 0 J J P = 0 or 6 (a.,L) ~ 1 - cos TIP when p > 0 . Theorem 17.ll shows J J that for each j=l, ... ,n the value a. is the only deficient point of J f.( z) . Thus inequality (17.27) shows that a = Car'" ,an) is the only J f( z) finite deficient point of ~

{a} ~ ~ ~ n (aI' ... ,an) E C

Now let

,~=1,2,

...

be the set of finite deficient points

for the vector valued meromorphic function

This set is either finite or is countably infinite. For each each

~

~

f( z)

j = 1, ... ,n

and

we have by (17.27) the inequality

~ Thus for each

t

a

6 (a: ,f.)

J

= 1, ... ,n

J

we have

Hayman [l6],p.ll4 161

L

o(a.R. ,£.) J

J

eEOC

0 (c,£.)

J

here the last sum is extended over all finite deficient points

c

of

We deduce the inequality

L

a(a)

min

.

L

f.( z)

J

O(e,£.) J

c E C

Using the estimates (17.20) and (17.32) instead of (17.27) we can do the analogue reasoning for the set of points set of points such that

EJ(a)

>

a

such that

O(a) > 0

,or for the

0 •

Summarizing and using Proposition 17.9 we formulate Proposition 17. 13

Let

be a meromorphic vector function in the plane. Then the following inequalities are valid.

L

a (a)

L a (c,£.)J

min

~

c EC

L

::;

e(a)

L

min

cEC

L

EJ(a)

~

min

e(e,f.) J

L

EJ(c,f.) J eEC

here the left sums can be positive only if all

have the same order.

Next we apply t

Theorem 17. 14:

Suppose that

in the plane, where

o<

A

1:;

< +00

l;( z)

is meromorphic and of lower order 1 . Then for a >3 we have

tFuchs (23), Hayman [l6) ,po 90, Weitsman [59)

162

L6(a,r,;)(J, where

<

depends on

(J,

and

We deduce using Proposition 17.13 Proposition 17.15:

Let

(f1"" ,fn non-constant)

be a meromorphic vector function in the plane and let function of lower order

A.

J

have 6 (a)(J,

L

L

;;;;

6 (c,f.)(J, J

A. < +00

J

be a component J 1 Then for (J, > "3 we

f.( z)

A«(J"A.) - 6 (00,f.)(J, J J

<

cEC

a EC n A( (J" A.) J

where

o<

such that

depends on

(J,

and

A. J

only.

Next we apply t

r;( z)

Theorem 17.16 : If

is an integral function of order

=

L

0

(0 ;;;; P

r;

P r;

,then

<.!.) =

2

6(c,r;)

l ' - sm

< =

TIP

r;

(.!. < 1) 2 < = P r; =

We deduce using Proposition 17.13 and Proposition 17.9 Proposition 17.17:

Let

f( z) = (f 1 (z) , ... ,fn (z»

, (f l' ... ,fn non-constant)

be a meromorphic vector function in the plane. Assume that tegral component function

f.( z)

J

of order

p

(0:;; p :;; 1)

f( z)

has an in-

. Then we have

t Edrei and Fuchs [22]

163

I a

E

~

O(a)

(

en

:::

p -~ .!) 2

(0

0

(.!2

1 - sin'lp

< 0"

p

~

1)

In particular an integral vector valued function of order

p :0

1

2"

can not

have any finite deficient value.

Remark.

Already Proposition 14.16 shows that an integral vector valued

i

function of order p < can not have any finite deficient value. . t Nevanlinna has shown that if a number p > 0 is given, which is not an integer, then there exists a positive number morphic function

r.;( z)

of order

p

such that for every mero-

K(p)

the inequality

N(r,c,l;) + N(r,d, 1;) >

T(r, s)

r-++ co

c·,d(ci=d)

is valid, whatever the complex numbers

arc.

For functions of order less than 1 the best possible value of

Remark. is ti-

=1

K(p)

For

p >

1

if

K (

the best val uc of

K (

p) = sin 1Tp

p)

if

1

-2

.-~ ,'.)

< 1

satisfies the inequalities ttt

Isimp I 2. 2 p +

iI

sin lTp

IT

I

p

where Isin1Tp I q + Isin1TP IT

p

Isin 1TP

I

q + 1

t

Nevanlinna [27], p. 51

tt Edrei and Fuchs [22] tttEdrei and Fuchs [21]

164

I

(q < p ;;; q +

(q +

2"1

21 '

q ~ 0 , integer)

< P ;;; q + 1 , q ~ 0 , in te ger) .

K

(p)

We now apply

t

Theorem 17. 18:

r,( z)

If

is an integral function of finite order

whkh

is not a positive integer, we have I

<

-

K(p

r,

)

We deduce using Proposition 17.13 and Proposition 17.9 Let Hz) = (f 1 (z), ... ,f n (z» , (fl, ... ,fn non-constant) be a meromorphic vector function in the plane. Assume that Hz) has an in-

Proposition 17.19

f. (z) J

tegral component function integer. Then we have

L

Remark.

of finite order

p

which is not a positive

0 (a)

Proposition 17. Ii presents a sharp form of Proposition 17.19 for

integral vector functions of order less than I . From Proposition 17.18 we see in particular that if

L

o (c,l',;)

I

=

. t egra 1 f'unc t·Ion f or an In

I',; () z

0

f f··t In! e or d er

PI',;' th en

.In t egertt. P I',; . IS an

In addition it is known that in this case A = p ,where A is the lower ttt r, r, r, -I tt order of (: ,and that all the deficiencies are integral multiples of P (: Further, all the deficient values a~ of I',;(z) are asymptotic values, i.e. v ttt 7,( z) ..~ a r, for z .,~ 'along suitable paths y v 00

Using these results, Proposition 17.19, Proposition 17.13 and the deficiency relation we deduce that if I)

t H

(a)

=

1

Hayman [16] ,p .104

Pfluger [491 tttEdrei and Fuchs [20], [21]

165

holds and if

f.(z)

is an entire component function of finite order

J

of lower order the order of multiples of

p -1

p = A. is a positive integer, which is equal to J ,and all the deficiencies a~ of f.( z) are integral ,then

A.

J f( z)

and

p

~

Further, all the deficient· f.( z)

asymptotic values of

v~lues

\)J a. of ]

and the inequality

J

f. (z)

J

arc

N(r,f)

L

+

O(a)

r-++ooT(r,f)

is valid. If

f( z)

ciencies of of

is entire,it is probable that under the assumptions also all defiare integral multiples of

f(z)

p-l

and that the deficient values

are asy mptotic values for the meromorphic vector function

f( z)

dz)

Further, if

is a meromorphic function of finite order

=

o(c,1;;)

f( z) and if

2

cECU{oo} then it is known that the number

#( ;:;)

of deficient values does not exceed

2p t.We deduce from this and from Proposition 17.9: r;; Let fez) = (fl(z), ... ,fn (z» be a meromorphic vector function of finite order

p

and such that

L

2

0 (a)

f. must have order p ,and the deficiency J 0(00) = 0 , 8(a) = 0 (aE CnU{oo}) , 0G = 0 . From the

Then all component functions relation shows that

above and from the inequalities

L

2

6 (a)

L c E C

number

tw eltsman . 166

J

2

the 6(00,f.)=00 1, ..• ,n) and that for each j = 1, ... ,n J ~ 2p . This shows that the #(f.) of deficient values of f. is J J ~ (2p)n is #(f) of deficient values of f( z)

we see that number

0 (c,f.)

[58)

It should be noted that the last deductions concerning

(5 (a)

are of less

importance than the corresponding results in scalar Nevanlinna theory, since information on

<5

(a)

does not give full information on the important quantity

VCr ,a)

lim r .... +00 T ( r , f) but only on N(r,a) + V(r,a)

1 -

lim r-+ +00

(5(a)

T(r,f)

167

Appendix Rudiments of complex manifolds and Hermitian geometry In this section we select a few basic ideas, which are related to the Hermitian geometric interpretation in Chapter 1, §2, Chapter 3, §9 and Chapter 3, §15 . Details will be found mainly in the books of Chern [5], Griffiths and Harris [l4] Wells [41], Morrow and Kodaira [26], Kobayashi and Nomizu [21], Vol. 2, and in the articles Chern [9], Griffiths [26] and [28], and otherwise scattered throughout additional literature given in the Bibliography. A. Complex manifolds. Let

cot~ngent

(w l' ... ,w m)

w.

J

or the complex basis

dw. = duo + idv. 1 J 1 span the

=

be the coordinates on the complex Euclidean space C m ~ R 2m

u. + iv. 0=1, ... , m) J J The real basis {du., dv.} J J

w

dw. = duo - idv. J J J

space to a point. Its dual is the tangent space with the

real basis d _d_ } {-,,-, " aU.

aV.

J

J

and the dual complex basis d

i_d_ )

dW.

dV.

+

dW.

J

J

A(w)

written \' L

dA -d- dw.

Wj

J

+

I

dA

-dw. d W. J J

or

dA

=

dA

+

dA

after introducing differential operators

168

d

and

d dV.

J

J The total differential of a complex valued function

dA

.

1-

d

by

on

cm

can be

d It

\' d It L -d- dw.

=

d It

J

Wj

I -dw.

=

dA

dW.

J A

COO function

J is called holo-

on an open set

morphic if 0

d It

i. e., if it is holomorphic in each complex variable separately. A complex manifold

M

is a manifold

V, ... }

such that

{U,

=

V_Cm

w

0

U

-1

is biholomorphic where defined. A funcis called holomorphic if

M tPV(U n V) c C m

holomorphic on system on

0

on an open set

A

that

provided with a distinguished open covering

U_Cm

tion

M

and coordinate charts

C

for all

V

It

0

-1

A holomorphic coordinate

U C M is a collection of functions w = (wI"'" w m ) , -1 -1
tPV(U n V) , respectively for each

V. A mapping

is

f

M-

N

such and of com-

plex manifolds is called holomorphic, if locally it is defined by expressing the coordinates of the image point as holomorphic functions of those of the original poin t.

f

is called an immersion, if

m

= dim

M :;; n = dim N

,and if the

Jacobian matrix is of rank m everywhere. An immersion is called an imbedding, if it is one-one, i.e., if

f(x)

= fey)

Examples of complex manifolds are: .,. ,w 3.

2.

m

,

x,y E M implies x =y C m with global coordinates

1.

WI'

A I-dimensional complex manifold is called a Riemann surface.

Complex projective space

pm

is a complex manifold ; the natural projec-

tion 71

is holomorphic.

pm

is compact since there is a continuous surjective map S2m+I in C m + I to pm em can be pictured as

from the unit sphere sitting in

pm

as an open set by the inclusion

Cm _

pm

given by

'V

- - -... , (l,w I ,·· .,w m ) Let

fl

(w O = 0) as the directions in which we can go to infinity in C m

denote the hyperplane at infinity with the equation

considering

fl

obtain an identification pactification of

Cm

fl ::=

pm-I

Thus we can picture

obtained by adding on the hyperplane

pm

. By we

as the com-

fl at infinity.

169

Let

M

be a complex manifold,

p EM

a holomorphic coordinate system around M

a tangent space to

1.

at

any point, and . There are 3 different notions of

p

p M

TR

is the usual real tangent space to ,p considered as a real manifold of dimension 2m T

T

and

R,p of real valued

2.

spanR { au. J

We can write

where

M

w. = u. J J

is +

iv. , J

}

J

COO functions in a neighborhood of

c

T <25> C,p R,p R P . We can write

T

d v.

p

can be viewed as the space of R -linear derivations on the ring

T

M at

and

d

=

R,p

at

span c {

C,p

p

is called the complexified tangent space to

d dU.

d

=

dv.

J

)

()

spanc { dW. J

a dW.

J

T

can be viewed as the space of C-linear derivations in the ring of C,p complex valued COO functions on M around p

T'

3.

=

P

M

is called the holomorphic tangent space to the subspace of

TC

phic functions (i.e.

,p f

at

p

. It can be viewed as

consisting of derivations which vanish on antiholomorsuch that

f

is holomorphic).

d

T'

spanC {

P

----=- } dW.

J M

is called the antiholomorphic tangent space to

T

being given as the real vector space C,p have the operation of conjugation sending

T

J

p, and we have

T'0T' p p

C,p

d dW.

at

and

to aw. J

T'

P

=

T

R,p

tensored with

T'

p

B. Hermitian metrics . An Hermitian metric on a complex manifold

170

C, we

M

is

given by a positive definite Hermitian inner product

0

T'

w

T' -

C

w

on the holomorphic tangent space at coordinates

w

w

for each

w E M such that for local

M the functions

on

( _d_ dW.

J

are

COO. In terms of the basis (T" w

0 -T"w ) *

T

the Hermitian metric

{dw j

* " (8)

w

T

0

dw k }

for

* ""

w

is given by

*"

(dw 1 ,.·. ,dw n ) of Tw we can locally construct a coframe for the Hermitian metric, i. e.

By applying the Gram-Schmidt process to the basis for each

w

m forms

I

k

a' k dw.

J

J

1, ... ,m

of type (1,0) such that ds

2

I

¢.

J is an orthonormal basis for

This means that

*"

T

* ' ,in

w

by on T" w w w The real and imaginary parts of an Hermitian inner product on a complex

terms of the inner product induced on

T

vector space give an ordinary inner product and an alternating quadratic form, respectively, on the underlying real vector space (see Chern [5] ,p .10).

T

R,w

0T

R,w

is thus a Riemannian metric on

-R

M, called the induced Riemannian metric of

the Hermitian metric. When we speak of volume or distance on a complex manifold with Hermitian metric, we always refer to the induced Riemannian metric. 1m ds

2

T

R,w

0T

R,w

--R

171

being alternating, represents a real differential form of degree 2;

- 2"1 1m

=

w

ds

2

ds 2

is called the associated (1, I)-form or the Kahler form of the metric pJicitly we write

=

0..

where

J

+ i

u.

J

,

6. J

13. J

are real differential forms. Then

=

I«a. +i6.)0(a.-i6.»

=

I(0:.0a.

j

J

J

J

j

J

J

J

+ i

13.013.)

+

J

J

L (-0:.0 j

J

S. J

+ 13.00..) J

J

The induced Riemannian metric is Re ds

2

=

I(0.·00.. J

j

+ 13.06 J.)

J

J

and the associated (1,1) -form or Kahler form is w

=

1 2

- - 1m ds

=

I

2

6.

J

J

1 2" Ij =

2 I

(Cl. 0 J

A

13. -

s.0 a.) J

J

J

~j

For example, the Euclidean metric on

em

is given by

m

I

=

dw.0 dw. J J

j=l Writing

w. = u. + iv. ,we see that J J J

Re ds

2

L (du.0

=

j

J

duo + J

is the standard Euclidean metric on ds 2 is w

172

2"

I

dw. A dW. J J

dv.0 dv.) J J R 2m

~ em

,and the Kahler fOJ

For the metric

I

=

cp.0-;P.

(p. =

Ct.

)

J

J

the volume element associated to djJ

Re ds t,(~

m

/\

J

2

+

is.

J

is by definition

S

m

On the other hand we have

L

(d

Ct./\

J

S.

J

so that the mth exterior power is til

m

m! . <11 /\ 131 /\

/\Ct

The Euclidean v~lume element of dp

=

u 1 /\ v 1 (JJ

=

J\U

J\

m

/\v

em

m

m! . d]l

=

m

is, for example,

=

i

"2 dw l

J\

clw

i

1

J\'"

/\

-dw 2 m

J\

dw

m

m

m!

Now let

SCM

be a complex sub manifold of dimension 1. The (1, 1) -form

associated to the metric induced on

S

by

ds 2

what we have just said to the induced metric on

is

S

(}J

is

and applying

we have the Wirtinger

Theorem for the dimension 1 :

vol( S)

= S

This theorem shows that the volume of a complex submanifold pressed as the integral over

S

S

can be ex-

of a globally defined differential form on

M

a fact, which is rather different from the real case. For example, for a smooth arc t ~ (u (t) v( t) ) in R 2 the element of arc length is given by 1

(u'(t)2

+

v'(t)2)

"2

dt

which is in general not the pull back of a differen Hal form in

R2

173

C.

The Fubini-Study metric. The unitary group U consists of the linm m ear transformations of C ,which leave invariant the expression <

In

Cm

of

m

W,W

>

IIWI1 2

:::

a unitary basis or unitary frame is an ordered set

vectors such that

o ;;

:::

Cm

The set of all unitary frames for F(pm-l)

A,B,C,D

;; m-l

( 2)

constitutes the frame manifold

. Choosing a reference frame

F0

,any frame

F

is uniquely of

the form

F

T . F

:::

The correspondence If

W

~T

F

Cm

F -

tesimal displacement)

T E U

o

dW

m

gives an identification

F(pm-l)

~

U

m

is a smooth function, the differential (or infinimay be resolved into its components relative to the

frame vectors dW where the

::: < dW, Wk

are I-forms on

>

For example, the frame vectors smooth maps

W.

J

F

~

U

m

themselves may be considered as

W.

J Expanding the differential (or infinitesimal displacement) the basis vectors in the frame

F

leads to the equations

dW.(F) J

in terms of

( 3)

where ::

< dWA,W B >

are differential forms in

U

m

From the orthogonality relations (2) we get by differentiation 174

( 4)

< dW A,W B >

9AB Since

+

< WA,dW B >

+

=

¢BA

WA(T . F)

=

0

( 5)

0

TWA (F)

for any fixed

a basis for the left invariant Maurer-Cartan forms in that under infinitesimal displacement, the frame transformation with coefficient matrix

¢ AB

-F

T U

,the

m

¢ AB

give

(3) and (5) say

undergoes an infinitesimal

. They are the structure equa-

tions of a moving frame. Taking the exterior derivative of (4) we get using (3)

=

-

-

<

<

= so that by (5) ( 6)

which are the Maurer-Cartan equations of the unitary group

U

m

. From (4),

(3) and (2) we get

=

=

=

(7)

where the multiplication of differential forms is understood in the sense of ordinary commutative multiplication. From (7) we get

(8)

175

w E Cm-O

Now let

. The vector

w (9)

lIwll has length 1 . From (9) we compute Ilwll dw

-

w d Ilwll

IIwl1 2 so that

1 --2 [[w[1

=

<

1 - - 2 < w , dw> [[wl[ =

1

-d[[w[[ I[w[[

dw , w>

1

- - < dw , dw>

[[w[[2

_l-d[[w[[ [Iw[[

- 11:11 4

[lw[1 dw , w d Ilwll >

<

1 1 - - - 4 < wdllw[1 ' [[wlldw > + - - 4 < wd[lwll ' wd[[w[[ >

[[wit

IIw[[

1 - - 2 < dw , dw > -

[[wit

dJlw[' <

dw , w > -

d[[w[1 - - 3 < w"

dw>

[[wit

[lw[[3

(d [Iw[[) 2 +

[[w[[2 This gives for (8)

1

1

IIw[[Z

[[w[[4

- - < dw , dw > - - - < dw , w > < w , dw >

(10)

Remark.

The last expression could be abbreviated by

=

176

1 [[w[[4

Z

I[w

1\

dw[1

1

( 11)

~

From ihis calculation we conclude that we can define a Hermitian metric in pm-I

by the formula

==

~(LkWkWk)( J:kdw k 0

():kWk Wk)-2

dw k ) - (L k w k dw k

)0 (LkWkdW k ){ (12)

In fact, the right hand side shows that the metric is Hermitian and the left side shows that it is positive definite. This is the Fubini-Study metric of pm-I In terms of the left hand side of (12) the associated Kahler form can be written m-I 2

I C==I

m-I

'c..") C==I I

==

<poc!\ <Pac

<poc!\
w here for the latter expression we have used (5). Now from (6), usin g

4)00

!\
:0

-

!\

Gl OO

==

0

,we have m-I

m-I

I

I

c==o

C==l

Thus the Kahler form can be written

w

==

ii

m-I

I

C==l

oc!\

( 13)

ca

so that the Kahler form is closed and the metric Kahler. As was shown in Chapter 1, §2

we can also find an expression for the Kahler form from a calculation

in terms of the right side of (12); we obtained that w

i d 310g

~

ilwll

In the special case

m == 2

d d C log

w

can be written as

ilwll

(14)

(12) is the Fubini-Study metric of

Explicitly, we obtain from the right side of (12)

177

=

(w I dw 2 - w 2dw I ) €)(w 1dw 2 - wZdw I )

=

-

(wIw I + w Zw 2 )

Z

We will now express this in the inhomogeneous coordinate ( 15)

With d~

==

we can write

=

=

Thus the Fubini-Study metric on mann sphere

S2

2

is just the natural metric of the Rie -

of constant curvature 4 . Its Kahler form is dl;;

w

pI

d~

1\

(l +

1;

do

1\

dT I;; =

- 2 I;; )

0

(16)

+ iT

which is the spherical volume element. This can also more quickly be obtained as follows.

I;;

By using on

w

==

is the coordinate on the open set U1

i

the lifting

a a log (1

. dl;;

w = (1 ,I;;)

+ 1; ~)

==

1

4"

d~

1\

1 _ _ _---;;-

==

"2 (l+I;;r;) - z S2

The volume of

is +00

21T

==

tdtdq,

J J (l+1; -1; ) 2

o 178

0

==

1T

U1

(w 1 '" 0)

==

we obtain from (14) -

""log(l + 1; I;; )

i

"2 dl;;

1\

d1;

in

pI

D.

Differential forms.

forms of degree forms. Since

p on 2 d = 0 ,

Let

AP(M,R)

M ,and

denote the space of differential

ZP(M,R)

the subspace of closed p-

d(AP-\M,R)) c ZP(M,R)

. The quotient groups

ZP(M,R) dAP-I(M,R) of closed forms modulo exact ones are called the de Rham cohomology groups of M . Similarly if

AP(M)

and

ZP(M)

denote respectively the spaces of

complex valued p-forms and of closed complex valued p-forms on

M ,we

have the corresponding quotient

= If

M

is a complex' manifold the decomposition T *' (M) w

of the cotangent space to

M

®

T

*; w

at each point

o

wEM

p *, 1\ T (M) w

(8)

gives a decomposition

q *, I\T '(M» w

p+q=r Therefore the space of r-forms can be written

o p+q=r where p

{ 1jJ EAr (M) : 1jJ( w) E 1\ T

is the space of r-forms of type (p, q). For

*' (M)

w

(8)

q *: I\T (M)forallwEM} w

we have

and we can define operators

179

a

=

d

where

a

+

In terms of local coordinates

w == (w l' .. , w m)

a form is of type (p, q) if it can be written

L

=

#I=p #J==q where for each multiindex

" ...

= The operators

a 1jJ( w)

a

a

and

are given by a

=

L

I,J ,j

a 1jJ(w)

aw.

a 1jJIJ(w) dw. aw. 1

L

E. Vector bundles.

11

E-

:

E

dW 1

1\

dW J

1\

1\

dWr

1\

dW J

1

Let

over

1jJIJ( w) dw. J

]

I,J ,i

vector bundle

==

I

M

M

be a

COO differentiable manifold. A complex

consists of a space

E

and a projection map

M ,such that: with

i)

There is an open covering {U, V, ... } equivalent to U x Ck by a COO map
On

n V»

1I-

l (U)

- u xc k

we require

(x , gUV(x) t,v) where

u

nv -

ck

COO functions, called the

are

- 0

transition functions. The transition functions necessarily satisfy the identities

=

I

I

is called a trivialization of

E

is called trivial if it is of the form

180

( 17)

for all for all

over M

x

Ck

U

x E U

nV n w

(18)

. A complex vector bundle on E

is caned a line bundle if

M

k :: 1 .

the fiber dimension

E

is called holomorphic if

M

is a complex

manifold and if the transition functions are holomorphic. We give a few examples without details. 1.

Let

M

be a complex manifold, and let

gent space to

M

at

x

xEUCM

. For

T

x and

be the complex tanU_em a
(M)

coordinate chart, we have maps

xEU

for each
, hence a map

*

-

T (M) x

U XEU

U x

e 2m

giving T(M)

=

T (M) x

U xEM

the structure of a complex vector bundle, called the complex tangent bundle. 2.

For each T

x

we have a decomposition

xEM

®

T' (M)

(M)

x

The subspaces

T~ (M)

x

{T' (M) C T (M)}

x

form a subbundle

x

T'(M) C T(M)

it has the structure of a holomorphic

called the holomorphic tangent bundle vector bundle.

3. E

*-

If

M

E -

M

is a complex vector bundle, then the dual bundle

is the complex vector bundle with fiber

E

* ::

x

(E)

x

*

the

tri viali za tion s 7f

-1

(U)

induce maps

E~ E

*

-

U x

::

U

has transition functions

E

e k * ;;

*

U x

ek

the structure of a manifold. If

x

{gUV}

, then

* -M E

E-M

is given by the transi-

tion functions t

-1 gUY

=

t

gvu 181

4.

Similarly, if k

ber dimension

E-M,F-M and

~

are complex vector bundles of fi-

and with transition functions

{gUV}

and

{hUV} , respectively, then you can define the following bundles: E (!) F

, given by the transition functions

=

E

0

F

, given by the transition functions

=

iuv

T *(M)

5.

T

E

h UV gUY C\ \& T(M)

GL(C k x C t )

E

*

*' (M)

the complex cotangent bundle; the holomorphic and t,he antiholomorphic cotangent bundle; p /\

T

*' (M) ® q/\

*'

T '(M)

6. A holomorphic vector bundle with fiber dimension 1 is called a holomorphic line bundle. We will now give an example of this in detail. F.

The universal bundle

Euclidean coordinates on n-l dinates on P . Let

on

J

pn -1 . Let

WI' ... , w n

denote

en and also the corresponding homogeneous coorn-l n n-l P x C be the trivial bundle on P , all

fibers being identified with en. We will define a holomorphic line bundle n-1 n-1 n JP , called the universal bundle J . J is a subbundle of P x e n-1 n-1 P ; its fiber J over each point wE P is the line w {AW}A C C n represented by w, i.e., J

=

w

AEC}

{A(w1,···,w) on

We show now that there exists in fact a line bundle with these fibers. Let denote the disjoint union of all

J

. Then any point

w

VEJ

J

can be repre-

sented (not in a unique manner) in the form v

where 1T

:

182

= ,

and

AE C

Moreover, the projection

:::

w. -' O}

U.

Putting

1

-1

1f

(w 1 " " ' wn)

:::

E

P

n-l

we see that

1

(U.)

,\,

AEC,W.-'O}

1

1

Now if

then we can write

v

in the form

wI w :\ . ( ---, ... , 1 , ... , -E )

v

J W. 1

and

A.

AW. E C

:::

1

W. 1

i ih

is uniquely determined by

1

. We can define the

v

mapping -1 11

(U.) -

U. xC

1

1

by setting
:::

1

n

1

The mapping

the fibers of

U. x C

«w 1 '· .. ,w n )

'V

is bijective and is linear [rom the fibers of

, \) 1l- 1 (U.')

1

to

Suppose now that

1

then we have 2 different representations for

v

and we want to compute the

relationship. We have
A.)

:\. )

1

1

J

where A.

:::

1

A.

AW.

1

1

:::

AW.

J

Therefore A. A

1

:::

:::

W.

A. -1. W.

1

J

w.

i.e.

A.

1

2A w.

J

j

Thus if we put w. gij

:::

1

w. J

183

. g . g = 1 . We deduce that J ij jk ki given the structure of a line bundle by means of the trivializations

then it follows that

g

can be

the transition functions

w.

u. n u. -

1

w. J

1

GL( 1,C) = C - 0

J

1T

G. is a

A section

Sections.

s

UCM

COO map U-E

s

such that

s(x) E E

a collection

of sections of sl'" .,sk is a basis for E for all

. A frame for

x E U

for all

x

Mover

E

A frame for E

over

U

over

U

E

over

such that

U

XEU

x

of

E - Mover

of a vector bundle

is essentially the same thing as a trivialization

Given

a trivialization, the sections s. (x) 1

=

{e,} the canonical basis of 1

, we can define a

form a frame. and conversely, given a frame tri vializa tion

¢U O

)

L A.1

in

s.(x) 1

Given a trivialization

of

we can represent every section tor valued function s(x) If

a

L

=

Ck ,

E

x

E

s

over

of

= (a 1 , ... ,ak )

E

U. it is important to note that over

U

uniquely as a

by writing

-1 ai(x) .
is a trivialization of

E

over

V

the

and

corresponding reoresentation of

= so

L cr.(x) 1 184

. e.

1

=

COO vec-

i.e.

a

=

Thus, in terms of trivializations

c!>~~

{

E

sections of

U

E

a

U

a:

ek

x

U U a:

over

}

correspond to collections

of vector valued COO functions such that a

:::

a

a, B , where the

for all

{CP} a A section

s

gaB

are transition functions of

E

of the holomorphic bundle U-E

s

holomorphic if

E

UCM

over

relative to

is said to be

is a holomorphic map, a frame

is called holDlJlorphic if each

s.

is; in terms of a holomorphic frame

1

a section

sex)

I

:::

a.(x) . s.(x) 1

1

is holomorphic if and only if the functions H.

The hyperplane section bundle

of the universal bundle fiber over line

w E pn-l

J

a.

1

are.

H _

pn-l

is the dual

H

::: J

*

,i. e., it is the holomorphic line bundle whose

corresponds to the space of linear functionals on the

{Aw}A' It has global sections (f}(pn-l,H)

given by the linear forms

A(w)

::: a1w 1 + ••• + an wn on en . Such a form A(w) determines a divisor, which is given by the hyperplane A ( w) ::: 0 • In more detail, let be constants and

a I ,··· ,an

A(w)

linear form

1T

en - 0

in

the projection

en - 0 _

pn-l

has in the local coordinates in

The 7[-l(U.) 1

the expression A(w)

w.(a1.r;

:::

1

1

1

+ ••. + 1 + '"

+ a

n

ith where

w.

:::

-1 w.

= 1, .... ,i-l,i+l, ... ,n

1

Denoting the expression in parentheses, which is essentially the linear form at the left hand side in "non-homogeneous!! coordinates in

a.

1

:::

+ 1 + .,. + a

n

U.

1

,by

.r;n)

1

ith 185

we see that in

1T

-1 (U.

nU.) J

1

w. w. cr.

w.cr. J J

1 1

so

{cr. } 1

C. 1

defines a section

s

Jcr. Wi J

=

:::

.1:;

j

1

cr. J

in the line bundle whose transition functions

are w.

:::

J w.

1

Because of this origin the latter bundle is called the hyperplane section hundle n-1 * of the universal bundle J H of P . It is the dual J I.

Divisors and Chern class. In the last example the hyperplane

is defined by U. 1

nU.J

cr. = 0 1

in

U.

1

= (w.

0) , and the transition functions in

;t

1

A ( w) = 0

can be written cr.

1

:::

cr. J

More generally, a divisor D on a complex manifold

M

is defined to be a

locally finite formal linear combination D

:::

M. It may be thought of as a col-

of irreducible analytic hypersurfaces of lection of holomorphic functions

m.

U.-C 1

1

such that

m. :::

1

m. J

u.1 nu.J

are non zero holomorphic functions in to be the zeros of the functions

g .. 1J

gjk

m.

1

m.

m.

m. J

mk

in

-2.. J .

gki

U.

1

mk m.

for all i, j . In

Ui

D

n U j n Uk

is defined we have

1

1

It follows that

m.

1

gij

m. J

are transition functions of a line bundle associated to the divisor

186

D

[D]

. It is called the line bundJe

In view of the above then we can say that the hyperplane section bundle H n-l is the line bundle which is associated to the divisor of a hyperplane in P . It can be shown that the line bundle

is trivial if and only if If

M

D

[Dl

M

on

is the divisor of a meromorphic function.

is compact we have Poincare'duality between

H 2m -q(M, Z)

D

associated to a divisor

In particular, a divisor

D

on

M

H (M, Z)

carriesqa

and

fundamental

homology class

We may consider

{D}

Then the divisor

D

as an element in the de Rham group is said to be positive if

closed positive (1,1) form w

=

i

2

This means that locally

Cu

h .. dw. IJ

i,j

where the Hermitian matrix

is represented by a

{D}

dw. J

!\

1

(h .. ) 1)

is positive definite.

It can be shown that collections

{gij}

and

{gij}

of transition func-

tions define the same line bundle if and only if there exist functions

([)\U i )

A.

1

E

satisfying

A.

1

I:-

(19)

gij

J

The transition functions sent a Cech

l-cochain on

{ g .. E M

1)

171*(u. n U.)} of V 1 ) .171*

with coefficients in

V

E -

;

M

repre-

the relations (17),

(18) mean that ($ ({g . .}) = 0 , i.e., {g .. } is a Cech cocycle. Moreover, by 1J IJ (19) two cocycles {g .. } and {g:.} define the same line bundle if and only if IJ -1 IJ their difference {g ... g:. } is a Cech coboundary. Consequently the set of IJ IJ . 1 d) holomorphic line bundles on M is the Cech cohomology group H (M,U) . A

The coboundary map H 1(M,([)*)

o

---+-,

H 2 (M, Z)

arising from the cohomology sequence of the exponential sheaf sequence

defines the Chern class E

c(E)

8({g .. }) IJ

of a line bundle. If the bundle

carries an Hermitian metric in its fibers, with. the curvature matrix

Chern has shown the important theorem that

c(E)

is represented in the de 187

Rham cohomology group

188

Bibliography

1.

Books

1

AhHors L. V. and Sario L., Riemann Surfaces, Princeton Univ. Press, 1960.

2

Andreian Cazacu C.

Theorie del' Funktionen mehrerer komplexer Veranderlicher, Hochschulbiicher fUr Mathematik 77, Deutscher Verlag d. Wiss., Berlin, 1975.

3

Borel E.

Lecons sur les fonctions meromorphes, GauthierVillars, 1903.

4

Cartan H.

Formes differen tielles, Hermann, Paris, 1967.

5

Chern 5.5.

Complex Manifolds without Potential Theory, 2. ed., Springer, New York, 1979.

6

Dieudonne J.

Foundations of Modern Analysis, 4. print 1963, Academic Press, New York, 1960.

7

Flanders H.

Differential Forms with Applications to the Physical Sciences, Academic Press, New York, 1963.

8

Fulton W.

Algebraic Curves: An Introduction to Algebraic Geometry, 4. print, Benjamin-Cummings, Reading, Mass., 1978.

9

Godbillon C.

Elements de topologie algebrique , Hermann, Paris, 1971.

10

Godement R.

Theorie des faisceaux, Actualites Sci. et Indus. No 1252, Hermann, Paris, 1958.

11

Goldberg 5.1.

Curvature and Homology, Academic Press, New York, 1962.

12

Greenberg M.J.

Lectures on Algebraic Topology, 4. print, Benjamin, Reading, Mass., 1977.

13

Greub W.

14

Griffiths P. and Harris J., Principles of Algebraic Geometry, Wiley, New York, 1978.

Halperin S. ; Vanstone R., Connections, Curvature, and Cohomology, Vol. 1, Academic Press, 1972.

189

15

Gunning R.C.

Vor lesun gen tiber Riemannsche FI achen, Bibliograph. lnst., Hochschultaschenbucher 837, Mannheim, 1972.

16

Hayman W.K.

Meromorphic Functions, At The Clarendon Press, 1964, reprinted with Appendix 1975.

17

Hermann R.

Vector Bundles in Mathematical Physics, Vol. 2, Benjamin, New York, 1970.

18

Hirsch M. W.

Differential Topology, Springer, New York, 1976.

19

Yano K. and Bochner S., Curvature and Betti Numbers, Annals of Mathematics Studies 32, Princeton Univ. Press, Princeton, N.J., 1953.

20

Kobayashi S.

21

Kobayashi S. and Nomizu K., Foundations oJ Differential Geometry, Interscience Publ., New York, Vol. 1 (1963), Vol. 2 (1969) . .

22

Kostant B.

Lectures in Modern Analysis and Applications III, Lec t ure Notes in Ma thema tics 170, Sprin gel' , Berlin, 1970.

23

Lefschetz S.

Topology, 2. ed., 2. print., Chelsea Publ., New York, 1965.

24

Lelong P.

Fonctions plurisousharmoniques ct formes differenticlles positives, Gordon et Breach, Paris, 1968.

25

Massey W.S.

Singular Homology Theory, Springer, New York 1980.

26

Morrow J. and Kodail'a K., Complex Manifolds, HoH, Ri.nehart and Winston, New York, 1971.

27

Nevanlinna R.

Le Theoreme de Picard-Borel et la Theorie des Fonctions Meromorphcs, Gauthier-Villars, Paris 1929.

28

Nevanlinna R.

Eindeutige analytische Funk tionen, 2. A ufl. , Springer, Berlin, 1953.

29

Ronkin L.l.

Introdu~tion

30

Sario L. and Noshiro K., Value Distribution Theory, Van Nostrand, Princeton, N.J., 1966.

190

Hyperbolic Manifolds and Holomorphic Mappings, Dekker, New York, 1970.

to the Theory of Entire Functions of Several 'Variables, Translations of mathematical monographs 44, Amer. Math Soc., Providence, R.I., 1974.

31

Simms D.J. and Woodhouse N.M.J., Lectures on Geometric Quantization, Lecture Notes in Physics 53, Springer, Berlin 1977.

32

Singer I.M. and Thorpe J.A., Lecture Notes on Elementary Topology and Geometry, Springer, New York, 1967.

33

Spivak M.

A Comprehensive Introduction to Differential Geometry, 2. ed. Vol. 2, Publish or Perish, Berkeley, 1979.

34

Sternberg S.

Lectures on Differential Geometry, 3. print., Prentice-Hall, Englewood Cliffs, N.J., 1964.

35

Stoll W.

Invariant Forms on Grassmann Manifolds, Annals of mathematics studies 89, Princeton Univ. Press, Princeton, N.J., 1977.

36

Vaisman I.

Cohomology and Differential Forms, Dekker, New York, 1973.

37

Valiron G.

Lectures on the General Theory of Integral Functions, Chelsea, New York, 1949.

38

Valiron G.

Fonctions entieres d 'ordre fini et fonctions meromorphes, Monograhies d ilL 'Enseignement Mathematique", No 8, lnst. de Math., Univ. Geneve, Geneve, 1960.

39

Warner F.W.

Foundations of Differentiable Manifolds and Lie Groups, Scott and Foresman, Glenview, Ill., 1971.

40

Weil A.

Introduction a 1 'etude des varietes kiihleriennes, Hermann, Paris, 1958, Nouv. ed. corr., 1971.

41

WellsR.O.

Differential Analysis on Complex Manifolds, Springer, New York, 1980.

42

Westenholz C.

Differential Forms in Mathematical Physics, North Holland Publ., 1978.

43

Weyl H. and J.

Meromorphic Functions and Analytic Curves, Annals of Maths. Studies 12, Princeton Univ. Press, 1943, Kraus Reprint, New York, 1965.

44

Weyl H.

Die Idee der Riemannschen Fliiche, 3. Aufl., Teubner, Stuttgart, 1955.

45

Wittich H.

Neuere Untersuchungen tiber eindeutige analytische Funktionen, Springer, Berlin, 1955.

46

Wu H.H.

The Equidistribution Theory of Holomorphic Curves, Princeton Univ. Press, Princeton , 1970.

191

2.

Articles

1

AhIfors L. V .

Beitrage zur Theorie der meromorphen Funktionen, 7. Congr. Math. Scand., Oslo 1929, 84-91.

2

AhHors L. V .

The theory of meromorphic curves, Acta Soc. Sci. Fenn. A 3, No 4, 1941.

3

Atiyah M.F.

Sheaf theory and complex manifolds, notes recorded by Mr. I. G. Macdonald, Univ. of Exeter.

4

Beckenbach E.F. and Hutchison G.A., Meromorphic minimal surfaces, Pac. Jour. of Math. 28, No 1, 1969, 17-47.

5

Bochner S.

On compact complex manifolds, J. Indian Math. Soc. 11, 1947, 1-21.

6

Borel E.

Sur les zeros des fonctions entieres, Acta Math. 20, 1897, 357-396.

7

Bott R. and Chern S.S., Hermitian vector bundles and the equidistributio_n of the zeros of their holomorphic sections, Acta Math. 114, 1965, 72-112.

8

Cartan H.

Sur les zeros des combinaisons Iineaires de p fonctions holomorphes donnees, Mathematica (Cluj) 7, 1933, 5-32.

9

Chern S.S.

Characteristic classes of hermitian manifolds, Ann. of Math. 47, No 1, 1946, 85-121.

10

Chern S.S.

The integrated form of the first main theorem for complex analytic mappings in several complex variables, Ann. of Math. 71, No 3, 1960, 536-551.

11

Chern S.S.

Complex analytic mappings of Riemann surfaces I, Amer. J. Math. 82, 1960, 323-337.

12

Chern S.S.

Holomorphic curves in the plane, Differential Geometry, in Honor of K. Yano Kinokuiya, Tokyo, 1972, 73-94.

13

Cornalba M. and Griffiths P. , Some transcendental aspects of algebraic geometry, Proc. Amer. Math. Soc. Summer Inst. on Algebraic Geometry, 1974.

14

Cowen M. and Griffiths P., Holomorphic curves and metrics of negative curvature, J. Analyse Math. 29, 1976, 93-153.

15

Carlson J. and Griffiths Ph., A defect relation for eq uidimensional holomorphic mappings between algebraic varieties, Ann. of Math.(2), 95,1972,557-584.

192

16

Deligne P.

Griffiths P.A. ; Morgan J. and Sullivan D., Real homotopy theory of Kahler manifolds, Invent. Math.29, 1975, 245-274.

17

De Rham G.

On the area of complex manifolds, Global Analysis, papers in Honor of K. Kodaira ed. by D. C. Spencer and S. Iyanaga, 1969, 142-148.

18

Drasin D.

The inverse problem of the Nevanlinna theory, Acta Math. 138, 1977, 83-151.

19

Edrei A.

Solution of the deficiency problem for functions of small lower order; Proc. London Math. Soc. (3) 26, 1973, 435-445.

20

Edrei A. et Fuchs W.H.J., Valeurs deficientes et valeurs asymptotiques des fonctions meromorphes, Comment. Math. Helv. 33, 1959, 258-295.

21

Edrei A. and Fuchs W. H. J . , On the growth of meromorphic functions with several deficient values, Trans. Amer. Math. Soc. 93, 1959, 292-328.

22

Edrei A. and Fuchs W. H. J . , The deficiencies of meromorphic functions of order less than one, Duke Math. J. 27, 1960, 233-249.

23

Fuchs W.H.J.

24

Greene R. E. and Wu H. , Analysis on non compact Kahler manifolds, Proc. Sympos. Pure Maths. Vol. 30, 1977, 66-100.

25

Griffiths P. A .

The extension problem in complex analysis II : Embeddings with positive normal bundle, Amer. J. Math. 88, 1966, 366-446.

26

Griffiths P. A .

Hermitian differential geometry, Chern classes and positive vector bundles, Global Analysis, Univ. of Tokyo Press, Tokyo, 1969, 183-251.

27

Griffiths P.A.

Differential geometry and complex analysis, Proc. Sympos. Pure Math. 27, 1974, 127-148.

28

Griffiths P. A .

On Cartans method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry, Duke Math. J. 41, 1974, 775-814.

29

Griffiths P. A .

Complex differential and integral geometry and curvature integrals associated to singularities of complex analytic varieties, Duke Math. J. 45 No 3 427-512.

A theorem on the Nevanlinna deficiencies of meromorphic functions of finite order, Ann. of Math. (2) 68, 1958, 203-209.

193

30

Griffiths P.A. and King J., Nevanlinna theory and holomorphic mappings between algebraic varieties, Acta Math. 130, 1973 145-220.

31

Hadamard J.

32

Hoffman D.A. and Osserman R., The geometry of the generalized Gauss map, Memoirs of the American Math. Soc., Providence, R. 1., 1980, 105pp.

33

Jensen J.L.W.V.

Sur un nouvel et important theoreme de la theorie des fonctions, Acta Math. 22, 1899, 359-364.

34

Kodaira K.

On cohomology groups of compact analytic varieties with coefficients in some analytic faisceaux, Proc. Nat. Acad. Sci. USA 39, 1953, 865-868.

35

Kodaira K. and Spencer D. e. , Groups of complex line bundles over compact Kahler varieties, Proc. Nat. Acad. Sci. USA 39, 1953, 868-872.

36

Kodaira K. and Spencer D. e. , Divisor class groups on algebraic varieties, Proc. Nat. Acad. Sci. USA 39, 1953, 872877.

37

Kodaira K.

On a differential-geometric method in the theory of analytic stacks, Proc. Nat. Acad. Sci. USA 39 1953, 1268-1273.

38

Kodaira K.

On Kahler varieties of restricted type, Ann. of Math. 60, No 1, 1954, 28-48.

39

LawsonH.B.

Lectures on Minimal Submanifolds, Vol. 1, Publish or Perish, Berkeley, 1980.

40

Lazzeri F.

Some concepts of algebraic geometry, Schriftenreihe des Math. Inst. der Univ. Munster, Ser. 2 Heft 2, 1970, 53pp.

41

Lehto O.

A majorant principle in the theory of functions, Math. Scand 1, No 1, 1953, 5-17.

42

Lelong P.

Fonctions entieres de type exponentiel dans en, Ann. Inst. Fourier Grenoble 16, No 2, 1966, 269318.

43

Levine H.I.

A theorem on holomorphic mappings into complex projective space, Ann. of Math. 71, No 3, 1960, 529-535.

44

Nakano S.

On complex analytic vector bundles, J. Math. Soc. Japan 7, No 1, 1955.

194

Essai sur I 'etude des fonctions donnees par leur developpememt de Taylor, J. Math (4), t. 8, 1892, 101-186.

45

Nevanlinna F. and Nevanlinna R., Dber die Eigenschaften analytischer Funktionen in der Umgebung einer singuHiren Stelle oder Linie, Acta Soc. Sci. Fenn. 50, No 5, 1922, 1-46.

46

Nevanlinna R.

Zur Theorie der meromorphen Funktionen, Acta Math. 46, 1925, 1-99.

47

N evanlinna R.

Eindeu tigkeitssat ze in der Theorie der meromorphen Funktionen, Acta Math. 48, 1926, 367-391.

48

Otsuki T. and Tashiro Y. , On curves in Kahlerian spaces, Math. J. Okayama Univ. 4, No 1, 1954, 57-78.

49

Pfluger A.

Zur Defektrelation ganzer Funktionen endlicher Ordnung, Comment. Math. Helv. 19, 1946, 91-104.

50

Picard E.

Sur une propriete des fonctions entiE~res, C .R. Acad. Sci. Paris 88, 1879, 1024-1027.

51

Pohl W.F.

Extrinsic complex projective geometry, Proc. Conf. on Complex Analysis, Minneapolis 1964, Springer, 1965, 18-29.

52

P6lya G.

Bestimmung einer ganzen Funktion endlichen Geschlechts durch viererlei Stellen, Mathematisk Tidskrift, 1921, 16- 21.

53

Rabinowitz J.H.

Positivity notions for holomorphic line bundles over compact Riemann surfaces, Bull Austral. Math. Soc. 23,1981,5-22.

54

Shimizu T.

On the theory of meromorphic functions, Jap. J. Math. 6, 1929, 119-171.

55

Seebach J.; Seebach L.A.; Steen L.A., What is a sheaf?, Amer. Math. Monthly, 77, 1970, 681-703.

56

Stoll W.

Die beiden Hauptsatze der Wertverteilungstheorie bei Funktionen mehrer komplexer Veranderlichen I, Acta Math. 90, 1953, 1-115. II, Acta Math. 92, 1954, 55-169.

Weierstrass K.

Zur Theorie der eindeutigen analytischen Funktionen, Werke Bd. 2, Berlin, 1895, 77-124; repro from Abh. Konigl. Acad. Wiss., 1876, 11-60.

58

Weitsman A.

Meromorphic functions with maximal deficiency sum and a conjecture of F . Nevanlinna, Acta Math. 123, 1969, 115-139.

59

Weitsman A.

A theorem on Nevanlinna deficiencies, Acta Math. 128, 1972, 41-52.

_ 57

195

60

Whittaker J. M.

The order of the derivative of a meromorphic function, J. London Math. Soc. 11, 1936, 82-87.

61

Wu H.

Normal families of holomorphic mappings, Acta Math. 119, 1967, 193-233.

62

Wu H.

Mappings of Riemann surfaces (N evanIinna theory) Symposia Pure Maths. 11, Entire Functions and Related Parts of Analysis, Amer. Math. Soc., Providence, R.I., 1968, 480-532.

63

Wu H.

Remarks on the first main theorem in equidistribution theory III, J. Diff. Geom. 3, 1969, 83-94. IV, J. Diff. Geom. 4, 1969, 433-446.

64

Yang P.

Curvature of complex submanifolds of C n , Proc. Sympos. Pure Math. 30, 1977, 135-137.

65

Ziegler II.J. W.

Ein Beitrag zu meromorphen Funktionen, 1965, 3 7pp ., unpublished.

66

Ziegler H.J. W.

Uber das Anwachsen der Totalkrummung der Flachen, die yon Systemen meromorpher Funktionen erzeugt sind mit Anwendung auf allgemeinere MinimalfHichen, Wurzburg, Diss., 1969, 98pp.

67

Ziegler H.J. W.

Uber das Anwachsen der Totalkrummung meromorpher Flachen, Congres lnt. Math. Nice, 1970, Les 265 communications individuelles, 183.

68

Ziegler H.J. W.

Nevanlinna'sche Wertverteilungstheorie, meromorph~furven in Cn und holomorphe Kurven in pn (C), Steiermarkisch Mathemat. Sympos. StiftRein, 1972, 26pp.

69

Ziegler H.J. W.

Deformation und Nevanlinna Theorie holomorpher K urven, J ahrestagun g Deutsch. Math. Ver., Hannover, 1974, 109.

70

Ziegler H.J. W.

Untersuchungen uber holomorphe und meromorphe Kurven nach Methoden der Wertverteilungslehre, Wurzburg, Habil. -Schr., 1974, 209pp.

Address of author Mathematics Department University of Siegen D-5900 Siegen West Germany 196

Table of symbols

Cn

1

H2(pn-l, Z)

12

w

1

13

< , >

1

00 OCR

II

1

f*

13

C

1

v(r,O)

14

CR

1

C +00

1

00

2

(lG

2

z.( 0)

8

z. ( 00 )

8

p n-l

9

]

J

a

16

f(z) - a

16

z. (a)

17

n(r,a,f) = n(r,a)

18

V(r,a) = V(r,a,f)

18,19

v(r,a) = v(r,a,f)

19,20

N(r,a) = N(r,a,f)

20

N(r,f) = N(r,oo)

20

J

11

'V

w

9

(l

10

a

10

d

10

(r ,a)

21

+ log

22

m(r, f) =m(r,oo,f)

22

m(r,a)

22

T(r,f)

22

10

00 CR

24

F

10 10

w

10

V(r,oo) 0 T(r, A)

24

F

dC 'V

11

* *

C

J J

*

H C

1 (I1)

25

*( 00 )

28

10

*(a)

28

11

n(+oo,oo) = n( +00 ,f)

28

11

n( +00 ,a)

29

12

* n(a)

29

12

v( +00 ,a)

30

12

v(r,oo)

30

197

m(r,f.) =m(r,oo,f.) ] J n(r,f.) = n(r,co,f.) J ] N(r,f.) = N(r, 00 ,f.)

45

Ric 1ji

45

N l(r)

114,118

45

n 1 (r)

114,118

T(r,f.)

45

N(r,a)

126,130

n(r,a)

126,130

J

]

J

112

m(r,a.) = m(r,a.,f.) J J J n(r,a.) = n(r,a.,f.)

46

o(a)

46

0(00) = o(oo,f)

130

N(r,a.) = N(r,a.,f.)

46

N(r,f) = N(r,oo)

130

J

J

] J

J J

= 0 (a,f)

130

B(a) = B(a,f)

131

O(a) = e(a,f)

131

n 1 (r,a)

132

57

N 1 (r,a)

132

0 0 m(r,f) =m(r,oo,f)

62

0G= 0G(f)

133

'il

64

p

51,54

T

52

A

54

f' (z)

0 T(r,f)

65

R2n

68

S2n

69

[ , J

70,71,72

1

72

0 m(r,a)

74

Is 2n l

75

A(r,f)

77

E(u,p)

82

[<x> , 00

g

85

M(r,a)

89

M(r,f) = M(r,oo)

89

M(r,a.)

97

J

p*

v(r,f) G(r) = G(r,f)

98 106 111,113

K

111

I::.

112

198

.0V(a) = 0V(a,f)

139

Index

admissible, 133 a -point, 2, 16 a -point at infinity, 28 associated 2 -form, 10 characteristic class, 12 Chern class, 12 Chern form, 11, 12 class, 54 completely multiple poin t, 140 complex Euclidean space, 1 complex projective space, 9 convergence class, 53 counting deficiency, 150 counting function, generalized, 20 counting function of multiple a -points, generalized, 132 counting function of multiple points, generalized, 118 curvature form, 11, 12, 111 curvature function, 113 deficiency, 131, 132 deficiency, counting, 150 deficiency relation, generalized, 134 deficiency, volume, 139 deficient point or value, 132 degree, 29 divergence class, 53 excep tional value, see deficien t point, 132 and volume deficiE exceptional value, generalized Borel, 129 exceptional value, generalized Picard, 127 first main theorem, generalized, 23, 26, 75 Fubini-Study metric, 10 Fubini-Study metric on C n , 64 function, meromorphic vector valued, 1 Gaussian curvature, III Gauss map, 110, Ill, 133 Gauss map, index of the, 133 generalized AhHors-Shimizu characteristic, 65

generalized generalized generali7.ed generalized generalized generalized generalized generalized generalized generalized generalized generalized generalized generalized generalized generalized generalized generalized generalized genus, 82,

Ahlfors-Shimizu proximity function, 74, 75 Borel exceptional value, 129 chordal distance, 72 counting function of multiple a -points, 13~ counting function of multiple points, 118 first main theorem, 23, 26, 75 genus, 85 Nevanlinna characteristic function, 22 Nevanlinna deficiency relation, 134 Nevanlinna proximity function, 22 Picard exceptional value, 127 Poisson-Jensen-Nevanlinna formula, 8 Riemann sphere, 73 second main theorem, 114 spherical characteristic, 65 spherical distance, 72 spherical proximity function, 74 theorem of Picard, 127, 138 theorem of Picard-Borel, 129 83, 85

Hermitian geometry, 9 Hermitian metric, 11, 64, 111 holomorphic curve, 12 holomorphic line bundle, 11 homogeneous coordinates, 9 hyperplane section bundle, 12 ind~ of multiplicity, 132 index of the Gauss map, 133, 137 index, Ricci, 133 inhomogeneous coordinates, 9

Kahler form, 11, 64 Kahler metric, 10, 11 local affine coordinates, 9 lower order, 54 meromorphic function, vector valued, 1 multiple point, 118 multiplicity, 1, 2, 16, 28, 139 multiplicity, index of, 132 normalized volume, 14, 19, 65 order of growth, 51, 54

200

perfectly regular growth, 54 Picard, generalized theorem of, 127 Picard-Borel, generalized theorem of, 129 plurisubharmonic function, 37 poin t, completely multiple, 140 point, multiple, 118 pole, 1, 2, 28 pole at infinity, 28 pseudohermitian metric, 112 rational vector function, 27, 141, 143 regular growth, 54 Ricci form, 112 Ricci function, 113, 114 Ricci-index, 133 second main theorem, generalized, 114 stereo graphic projection, 70 total curvature, 113 transcendent, 27 type, 52, 54 universal bundle, 11 vector valued meromorphic function, 1 very regular growth, 54 volume, 14, 19, 65 volume deficiency, 139 volume deficient point, 139 volume element, 13 volume function, 19 volume, normalized, 14, 19, 65 zero, 1, 2