Vector valued Nevanlinna theory

C(z)

tends to a fi-

(tell) dO

r < t < p,

2n r

T(t,f)

D(teio) d$

2n

(6.2)

0

where

is given by

D

D(z)

The function p

C(z) - BP(z) + N(p,f)

=

is harmonic in each point of the circular ring and thus satisfies there Laplace's equation D (z)

Dt AD

=

Dtt

+

D

+

After multiplication by

IzI

<

0

=

2

t

r<

t

this can be written

t2

2

d D

(dlogt)2 hence

+

D00

dO

=

=

0

2n r

d2D(teio ) 0

(dlogt)2 0

From this we conclude that the mean value in the right hand side of (6. 2) is a linear function of log t . We now compute this mean value for t = p and the function C (z) has for IzI = p t = r . On the circular boundary boundary values logllf(z)II . Since B (z) vanishes there, we have there P D(z) = logIlf(z)II + N(p,f) , so that 2 7T

2n

1D(pe)

dO

T(p,f)

=

0

On the boundary finition of

IzI = r

we have

JD(re) d 0

=

logllf(z)II

,

so that by de-

D (z)

2,r n

C(z)

2,r

2n

=

2n

Zn 0

JB(re1)

+N(p,f)

0

41

Using (6. 1) we obtain from this 2n 1

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

D (rein) do

22 7r

=

T(r,f)

0

These considerations show that for the boundary points t = r and t = p of the interval r < t < p equality holds in (6. 2). The right hand side of (6.2) being linear in logt , it follows altogether that T(t,f) is a convex This proves Proposition 6. 2 . function of log t Remark. If f(z) is an integral vector valued function, then Proposition 6.1 and Proposition 6.2 imply that .

m(r,f)

=

T(r,f)

is increasing and convex in log r for r > 0 . For a non-constant meromorphic vector function f in the plane generalized characteristic T(r,f) increases unboundedly with r erwise by the generalized first main theorem the sum m(r,a)

+ V(r,a)

+ V(r,a)

,

,

the

for oth-

+ N(r,a)

would stay bounded for every a E Cn , which is impossible since tends to infinity for r 3 - . We conclude: If for a meromorphic vector function f in the plane the sum m(r,a)

C

N (r, f (0))

+ N(r,a)

rf

+oo , then f reduces to a conrests finite for a given point a as stant. Vice versa, for a constant vector function f(z) __ f(0) , the sum m(r,a) + N(r,a) + V(r,a) is constant and bounded for every a up to 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 T(r,f) , m(r,f) , or m(r,a). ) For the expression w = (w1 .... I w we denote by E Cn I w 1i s n

11 *1 s

=

max{jw1l,

.

.

.

,1wnl}

Then we obtain 2

11wlls

a?

11w112

n Ilwlls

log!lwlls

log IIwII

<

logllwlls

logllwll

<

0

<

logllwll

Z logn

+

logllwlls

2logn

+

logllwlls

- logllwlls

(6.3)

,

2logn

This shows that up to a bounded term, logllwll can be replaced by the latter is often easier to compute than the former. We can define s a modified proximity function ms(r,f) by putting

log;wll

2,r r

1

ms(r,f)

logllf(rel )h

2 7r

do

0

Ts(r,f)

and a modified characteristic function

T(r,f)

by putting

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

=

Then we have

`I'(r,f) - Ts(r,f)

<

0

T(r,f)

which shows that

m(r,f) - ms(r,f)

=

and m(r,f) , reSo in most considerations T(r,f) m (r, f) instead of m(r,f) . Furs

Ts(r,f)

and

log n spectively, differ at most by can be used instead of T(r,f) and ther, from (6.3) we obtain ,

2 logn

or

m(r,f)

1

1

1

2 log n

log

+

log

llwlls

log

<

llwlls

Ilwll

from which the following two inequalities are obtained 1

1

log

<

IIwIIs

1 log n 2

+

log

1

1 ,

log

Nil

If we now define a modified proximity function by putting

<

llwlls

IIwII m

s

log

(r, a)

for finite

a E Cn

2 7r

1

ms (r, a)

.

=

2,r

0

log Iif(re") - a llsdO 43

then we conclude the inequalities

ms(r,a)

12

logn + m(r,a)

m(r,a)

,

<

ms (r, a)

so that

ms(r,a) - m(r,a)

0

Thus in most applications Finally we prove

<

I logn

(a E Cn)

m(r,a)

,

(a E Cn)

can be replaced by

m

s

(r, a).

are subjected to a linear homogeneous transformation A with constant complex coefficients and non-vanishing determinant, then the following inequalities hold with K a suitable constant: Proposition 6.3

:

f 1, ... , fn

If

Ims(r,Af) - ms(r,f) I

log (nK)

- Ts(r,f) I

IT

log (nK)

<

Im(r,Af) - m(r,f)

2 logn

+

log (nK)

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

A fk(z)

(Af(z)). k=1

be this substitution and n

2:

ak(Af(z))k j

k=1

its inverse. Put K

=

mix{ IA. I

j,

IIAf(z)IIs

44

o

I

,

la]11

nK Iif(z)IIS

nK I1Af(z)IIS

f(z)IIS

so that, using the inequality

log (x1 x2)

log x1

<_

logllAf(z)IIs

=

log (nK)

+

logllf(z)Ils

<

lob' (nK)

+ logllAf(z)IIS

+

log x2

loglIf(z)IIs

or

logljAf(z)IIs - logllf(z)IIs I

log (nK)

From this we see that

ms(r,Af) - ms(r,f) I

<

log (nK)

The inequality IT

s

(r,Af)

- T s (r,f)

log (nK)

follows from this and the definition of T s . For the original proximity function we have m(r,Af) - m(r,f) I = Im(r,Af)-ms(r,Af)+ms(r,Af)-ms(r,f) <

Im(r,Af)-ms(r,Af) I + ImS(r,Af)-ms(r,f) I

logn + log (nK)

<

z This shows Proposition 6.3 .

§7

The Connection of

m(r,fl)

and

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

and

N(r,f)

with

T(r,fl),

N(r,fl)

For the component functions w. = f.(z) of the meromorphic vector func1 tion w = f (z) we have the usual notions of Nevanlinna theory t 1

n(r,f.) = n(r,m,f.)

N(r,m,fT(r,f.) = m(r,f.)

+ N(r,f.)

tNevanlinna [27] 45

m(r,aj,fj)

m(r,aj)

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

J

n(r,aj) = n(r,aj,fj)

,

etc.

J

Note that some of these quantities may not be finite or well-defined, if f .(z) reduces to a suitable constant. We now establish connections of these quantities with the generalized quanT(r,f), m(r,f), m(r,a), N(r,f) and N(r,a) tities To begin with let f1(z) be a (scalar) meromorphic function in CR For each z0 in CR we can define an integer 7Tz (f 1) 0 by putJ

.

0

ting 0

if

is not a pole of

z0

f 1(z)

the multiplicity of the pole if

pole of

f1(z)

z0

;

is a

.

The term in the following sum being positive only at finitely many places in , and by definition of n(r,f.) we can write zI < r

n(r,fj) If now CR

f (z)

j = 1 , ... ,n

F 7Z(fi)

_

=

(f l (z) , ... , fn (z))

we define for each n (f)

z

z

in

CR

max j=1,

,n

is a meromorphic vector function in an integer irz(f) = 0 by putting

7rz(f.)

(7.2)

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

7rz(f)

n(r,f)

L.

zI`r

(7.1)

Izi < r

and, regarding (7.3)

'rz(f )

By (7.2) and (7.3)

n(r,f)

max

_ Izl;5r

j=1,...,n

nz(f J

(7.4)

The sum on the right of (7.4) is

1 n(fj)

max

j=1,.. and

n

n

L j=1

E Trz (fj .) 1zliir

From this, (7. 1) and (7. 2) we obtain the inequalities n <

n(r,f.)

<

n(r,f)

j=l,...,n ,

E n(r,fk)

(7.5)

k=1

which connect the "little" counting function of poles of f with the corre sponding little counting functions of the component functions. For a in Cn we can now treat similarly the a - points of f . Again, let al E C and let f1(z) be a (scalar) meromorphic function in CR , which now is assumed to be not identically equal to a1 . Then we put for z0 in

CR

if

=0

m

f 1(z0)

a1

vz(fI-al) 0

the multiplicity of the zero of if fI(z0) = a1 , f1(z) f- al

v(f1-a1)

Note that we do not define

if

f1 = al

f 1(z) - a1

. Then if

fj(z) 4 aj

we have

n(r,a.)

_

E vz(f.-a.)

(7.6)

Izj-_r If now

f(z)

=

is a non-constant vector valued mero-

(f l(z),...,fn(z))

morphic function, then we have for

a E Cn

E min

n(r,a)

zlgr j=1,...,n

vz(fj-a.)

f. :t aj i

min j=1, ...,n

f. J

s5t

a. ]

E zJ5r

vz(f.-a.)

_

n(r,a.)

min

(7.7)

,n

j=1, f.

2f a.

J

J

47

We can summarize these simple facts in Let Proposition 7.1 f (z) = (f 1(z) , ... , fn (z)) function in CR . Then for 0
be a meromorphic vector

n

n(r,fj)

n(r,f)

s

1:n(r,fk)

,

j=1, ....n

(7.8)

k=1

If

f

0ar
is non-constant, then for

a E Cn (7.9)

n(r,a

min

n(r,a)

and

aj

fj

j=1,

n

.

In order to establish similar relations between the counting functions of poles, let first z = 0 be no pole for f(z) so that simply

N

,

rr N(r,f)

n(t,f)

=

t

dt

0

is also no pole for each the definition of N(r,f .) we can write Then

z=0

(j=1, ... n) , and by (7. 8) and

fj

J

n

J:N(r,fk) , j=l,...,n

N(r,f)

N(r,fj)

.

(7.10)

is=1

Secondly, let

z=0

be a pole for

r

N(r,f)

n(t,f) - n(0,f)

n(t,f) dt

=

+

t

t

r0

r'_r0> 0

dt + n(0,f)logr0

0

and the same holds if f is replaced by hand inequality of (7.8) we conclude that

48

. Then we can write for

f

r0

f. J

(j=1, ... n)

.

Using the right

r

n

N(r,f)

n(t,f.)

Z

<

dt

0(1)

+

t

j=1

r0

n !__..

N(r,fj)

0(1)

+

(7.11)

j=1

For the other direction we first note that

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

L

7T

(f)

E 'r(f)

-

ZI
IZI
E (,rz(f) - irz(fi))

=

i10(f) - r0(fi)

=

=

n(O,f) - n(O,fi)

Izl!5 t

so that, using the left inequality of (7.8), rr r0 r

n(t,f.)

N(r,f)

3

t

n(t,f.) - n(0,f.)

dt

+

r0 For

r>r0>1

0

n(0,f)logr0

we have

N(r,f)

dt + n(0,f)logr0

t

>

n(0,fj)log r0

,

so that

N(r,f .)

?

J

for

0
we can in general only write

1

N(r,f.)

>

+

(n(0,f) - n(O,f.))logr0

=

N(r,fj) + 0(1)

.

We summarize:

Proposition 7.2 (i)

If

Under the conditions of Proposition 7.1 the following holds:

:

is not a pole of

z=0

, then

f

n

N(r,fj)

<_

N(r,f)

N(r,fk)

<

(j=1,...n)

.

(7.12)

k=1

(ii) If

z=0

is a pole of

f

, then 49

n

N(r,fj)

0(1)

+

N(r,f)

<

N(r,fk) + O(1)

,

(7.13)

k=1

where the 0(1) on the left hand side can be omitted if r 1 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 N(r,a) for First, if z = 0 is no aj - point of fj for j=1, ... n n(0,a) = 0 , and we have

finite a then also

.

r

r

n(t,a.)

N(r,a.)

_

J

n(t,a)

N(r,a)

dt

dt

t

t

J

J

0

0

and, under the conditions of inequality (7. 9),

N(r,a)

s

N(r,aj)

min

(7.14)

f.Et a. J

]

j=1,

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

f (z)

romorphic vector function in (i)

If

z=0

N(r,a)

(f 1(z), ... , fn (z)) CR and a E Cn

is no

_

aj - point of

j=1,

(ii)

j1,..., n

N(r,a.)

min fj

for

fj

be a non-constant me-

, then (7.15)

a. .

n

In the general case,

N(r,a)

min fj Fit aj

N(r,a.)

+

0(1)

(7.16)

J

j=1,...,n

We now consider the proximity- and the characteristic functions. We have the obvious inequalities

m(r,f.) cn

5

m(r,f)

j=1,

,n

From the inequality n

E log xk

log (x1 + ... + xn)

+

(xk ? 0)

logn

k=1

we see that in the other direction n

m(r,f)

1 m(r,fk) k=1

log n

+

We formulate this and the consequences of Propositions 7. 2 and 7. 3 for the Characteristic T in Proposition 7. 4 : Let f (z) function in CR . Then for

=

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

be a meromorphic vector

0
(i)

m(r,f)

L.

Z log n

m(r,fk)

(7.17)

k=1

(ii)

If

z=0

is not a pole for

f

j=1, ... ,n

, then for n

ZT(r,fk)

T(r,f)

T(r,fj)

+ 2 logn

(7.18)

.

k=1

(iii)

z=0

If

is a pole for

f

, then for

j=1, ... ,n

n

T(r,f.) + 0(1)

<

T(r,f)

<

L T(r,fk)

+ O(1)

.

(7.19)

k=1

Comparing (ii) 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.

be a positive function. Then s(r) (r > r0) is said to be of o r d e r (of growth) rp , or shorter of order p , if Definition 8. 1

:

Let

s(r)

the limit superior p

lim r+ +m

log s(r) log r

(0<_p < +m)

51

s(r)

is finite.

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

p = +m

If

then this definition is equivalent to the following properties: 0 there exists r 1 > 0 , such that s(r) < rp+c for r > rI ; and there exists a sequence of r - values tending to infinity,

p < +m for each e>

all

rp-e

s(r) > In this sense then, the growth of s(r) is characterized by the compar ison functiont rp if p < - . The definition of finite order gives no confor which

clusion about the behaviour of the quotient s(r)

rp

for large

r

.

Therefore one defines

Definition 8.2t: A function s(r) of order rp is of m i n i m u m , m e a n or m a x i m u m t y p e r respectively, according to whether the limit superior s(r)

lim

r-> -

rp

is zero, positive and finite, or is infinite. Now, let the function s(r) be of order rp creasing. Let p > 0 be a number such that

and, for the moment, in-

s(r) +m

r_u_+_' dr

<

r0

Then for each

c>0

tNevanlinna [271 52

there exists

rI > r0 , such that for

r > r1

dt

s(r)

s(t)

to +1 dt

s(r)

u ru

r which shows that

r . Now assume that

p
and see that there exists

e

<

r1 > 0

u

>p

.

In this case we put

such that

tp+E

s(t)

t o +1 dt

<

to r1

r1

+1

dt

jtp -u -1 + E dt

=

+

<

rl

This reasoning shows

Necessary and sufficient for the positive increasing function (r > r0 > 0) to be of order p (0 < p < +c) is that the integral

Proposition 8.3 s(r)

:

s(t)

J +1 dt

(8.1)

r0

0 p , divergent for p= 0 if and only if integral (8. 1) converges for every

.

s(r) u

>0

is of order .

s(r)

(r > r0) is a positive function of finite order p , then the p=p In the first case integral (8. 1) can be convergent or divergent for we say that s(r) belongstotheconvergence classoforder p, If

.

in the second case it belongs to the divergence classtoforder p. is increasing and of convergence class of order 0 < p < +m then the first part of the reasoning above Lemma 8.3 shows that s(r) must If

s(r)

T=0 . be of minimumtype i.e., 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 case of the plane e.g., we give Definition 8.4 : The " o r d e r "

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

in

i.e.,

log r

r-> +m

o r d e r"

The " 1 o w e r

T(r,f)

f(z)

log T(r,f)

lim

P

In the

of a meromorphic vector function

p

is the order of

C

.

A

of

f (z)

is the limit inferior

log T(r,f)

lim

log r

r-> +m

0 < p < +W

If

, then the "type " and the " c l a -s s

of

f(z)

are

respectively defined to be the type and the class of T(r,f) A=p If , then f ( z ) is called of " r e g u l a r growth IT in the 0<E< sense of Borel If there exist constants E , F , such that

0
.

T(r,f)

lim

E

rP

r-> +m

r-> +m

then f ( z ) will be called of " v e r y sense of Valiront. If the limit lim r->+m

exists, then

lim

T(r,f) rP

f(z)

It' is clear that

54

F

rp

r e g u l a r g r o w t h " in the

will be called of "perfectly regular

rowth" in the sense of Valiron t

tValiron [37]

T(r,f)

0
p m +-

always.

We can show

Proposition 8.5

Let

:

f (z)

=

function in the plane. Let p and p and that of f (j=1, . . , n) . Then

.

J

.

.

be a meromorphic vector

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

denote respectively the order of f is given by

J

p

max p.

=

J

if f.

and

:\.

(j=1,...,n)

J

k

,

(8.2)

}

denote respectively the lower order of

f

and that of

then

}

max

A

(8.3)

?, .

J

J

in particular, if f 1(z) , . . , fn (z) are of regular growth in the sense of Borel, then also the vector function f(z) = (f1(z)....,fn(z)) is of regular .

growth in this sense.

Proof. From Proposition 7.4 we have the inequalities n

T(r,fj) + 0(1)

ET(r,fk) + 0(1)

T(r,f)

<

(8.4)

k=1

for

j=1, ... n max

. The left hand side shows that

pj

(8.5)

p

In the other direction we assume that p

max p.

>

J

J

. From the right

p > p > max p. such that j inequality (8.4) and Proposition 8.3 we conclude Now choose a number

n

E k=1

p

T(t,fk) p+1

_

+ m

to >

0

,

t

t0

which is impossible, since by the same Proposition each integral 55

T(t, fk) k=1,

dt

.

n

to +1 t0 < is convergent. Thus we must have p max pj and observing (8.4) this proves (8.2). J (8.3) follows immediately from the left hand inequality (8. 4) .

Let

f

_ lim

r *+W

be of order

0 < p < +W

. Then from inequality (8.4) we have

_

--

T(r,f.)

'T(r,f)

rp

T(r,fk)

n

lim

<_

lim rp

r-s+W

rP

k=L

(j = 1, ... n) . Here in the extreme left and right hand sides all terms vanish which belong to component functions of order < p . Thus if f , . . . 'f are the component functions of order p then we have the following ine- Jm qualities between types ,

m T

T

(8.6)

T.

Ji

Ji

i=1

In particular, if

i ll...'f.

f

0
is of order

and of minimum type, then

< +W

are of minimum type.

f

is of convergence class, then we see from inequality (8.4) that f. f. are also of convergence class. If f is of divergence class, then J at least o n e function among f. ,f. must be of divergence class. Since according to Nevanlinna theory the characteristic of each f. is invariable up to an additive bounded term if f. is transformed by a linear 1 If

f

,

.

.

.

J

transformation a.f.

+

8.

]J

Yjfj

(a

]

6

j i

S.

- 8ji)

x

(8.7)

0

+

we have

The order Proposition 8.6 meromorphic vector function :

56

, and if

0 < p < +W f (z) = (f 1(z) , ... , fn (z) )

p

the type

T

of a

in the plane rest the

same if some or all component functions f are subjected to linear transforj mations (8. 7). Now, with f(z) we can consider its derivative defined by f'(z)

If and

p

f.

( f i ( z ) ,

p'

, ,

]

f: ]

and

p.

(j1,..., n)

Pi

,

.

.

.

,f'(z))

f , f' p. denote respectively the orders of , then from Nevanlinna theory it is known that

Pi

Thus, applying Proposition 8.5, we have also p'

mJx p.

=

max pj

=

=

P

This shows

Proposition 8.7 : If

f (z)

(f1( z) , ... , fn (z))

function in the plane, then the order

f'(z)

_

is equal to the order

is a meromorphic vector

p' of the derivative

(fi(z);.... f'(z)) p

of

f (z)

tWhittaker [601 57

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

0

T(r,f)

The generalized Ahlfors - Shimizu characteristic

a e Cn

In Chapter 1, §3 we proved for

,

0<

t
the identity

2n

log Ilf(tel©) - a IIde

1

N(t,f) = V(t,a) + N(t,a,f) + log 1k

+

(a)I1

0

(9.1)

Putting a = 0 = (0, ... , 0) in (9. 1) and replacing ponent vector (1,f) it is clear that N(t,a,(1,f)) N(t,(1,f)) . Using this and the explicit formula

by the

f =

0

n+1

and

com-

N(t,f) _

t r 1

V(t,0)

I

ds

Alog IIf(z)jj dx A dy

s C

0

(z = x+iy)

,

s

we obtain from (9. 1) by differentiation that 1

Alog

2,rt

1 + IIf(z)II2 dx A dy

j Ct 2,r r

d

dt_

I

N(t,f)

+

log V1 + If(teiew dO

27 _

(9.2)

0

By integration from

t = r0 > 0

to

t = r (r0 < r <

R)

we get

r r

Aiog/l +

dt

I

IIf(z),12

dx A dy

27rt

z0

Ct

r

2 it

r

n(t,f)-n(O,f) dt + n(O,f)lo r g r0 t

+

1

log'/1

2n

r0

+

Ilf(re16)II2 dB

0 2 'n

log 1 + 11( rOei0) II 2

(9.3)

dB

0

we have the vectorial Laurent-development

z=0

At

f(z)

zq+1cq+1

zgcq

=

+

+

is the difference between the number of zeq = n (0, 0, f) - n (0 , f) z = 0 , and ros and poles of f at where

Concerning the development of (1,f...... f

at

z=0 i)

)

(column vector)

G)

we distinguish three cases: q> 0

/1 :

ii) q=0 :

)

I

=

(1)

f/

0

(1)

(c1

f

=

0 zq cq

+

z

+

\O1/

+

q+1

+

z2 ('2)

+,\

0

iii) q< 0

:

(f )

zq

(c0

q)J

+

+...

zq+1 q+1

+ 1

z \Q

1/

In these three cases respectively the expression

59

1

log 1 + If(r0eie)II2

r0 + 0

behaves as

log II

like i)

logVI

+

II

f

or

o(r0)or

ii) log

1If(r0eie)II2

+ 1110

1V

iii) -n(0,f)logr0

1I`

+

+

o(r0)

or

+ o(r0)

logllcgil

In the cases i) and ii) the point z = 0 is not a pole of f ; in the case iii) the point z = 0 is a pole of f In case i) we have f (O) = 0 = (0'. ..10) ; in case ii) we have . i) and ii) can be IIc0II = IIf(0)II considered together so that we have .

log 1 + IIf(0)ll2

finite at 1og"1

+

z=0

+ o(r0)

if

f

is

(cases i) and ii))

II f (roeie) II 2

-n(0,f)logr0 + logllcgll + o(r0) z = 0 is a pole of f (case iii) ) Thus, if r

dt (AlogVfl r0

is not a pole of

z=0

+

IIf(z)II2 dx A dy

=

Ct

r

2n

n(t,f)-n(0,f) t dt r

we obtain from (9. 3)

f

0

so that, letting r0

+

1

log 1 +

2n 0

tend to zero,

I

f(reie)I2 de - log V1 + IIf(0) II2 -o(r0)

if

r r

dt

AlogN/1

2 Tr tt

+ IIf(z)II2 dx - dy

J

J

Ct

0

2 Tr

N(r,f)

1

+

2,r

1logVl

+

IIf(reie)II2de - log 1 + I1f(0 )II2

(9.4)

0

If on the other hand r dt

f

, then we get from (9. 3)

1 + If(z)II2 dx A dy

Alog

2,r tt

is a pole of

z=0

0t

r0

2n

r r

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

+

logN/l

2n

+

If(reie)I2 de

0

r0

-log IIcgII - o(r0)

and letting r

r0-r 0

,

r dt

0

1 + IIf(z)I2 dx A dy

Olog

27T t

0t

2n

N(r,f)

+

1°g /1 -L 2n I

+

If(reie)II2 de

-

log IIcgII

(9.5)

1

0

Now the function

61

271

r

1

logV/i + IIf(rei0)II2 d0

2n 0

which appears in (9.4) and (9.5), behaves asymptotically very similar to m(r,f) ; it can serve equally well as m(r,f) as a measure for the mean approximation of f to infinity on circles ac r . We can therefore introduce the following modified proximity function 0

0

m(r,f)

=

m(r,W,f)

with respect to infinity: Definition 9. 1 27T

0

1

m(r,f)

log1 + IIf(Yeie)II2d0 - logV1 + 11f(0)II'

2 it

(9.6)

0

if

z=0

is not a pole of

f

, and

2,r r

0

1

m(r,f)

1 + Iif(rel0)112de - log IIcgII

log

2ir

(9.7)

0

is a pole of f . Here cq is the first non-vanishing coefficient vector in the Laurent development of f (z) at z = 0 if

z=0

.

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

r

0

m(r,f)

+

dt

N(r,f)

elog

2nt 0

1 +

IIf(z)II2dx A dy

Ct

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

62

.

f (z) = f 1(z)

(9.8)

4 If(z)12 nlog(1 + If1(z)12)

_

(1 + Ifl(z)12)2

and that 1

slog /1-1 Ifl(z)12 dx A dy

2

j 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 characteristic function of Ahlfors and Shimizu or the spherical characteristic of fI We now try to find in the present vector valued case the correct geometric meaning of the expression 1

AlogV1

2

+

If(z)II2 dx A dy

Ct

occurring in equation (9. 8). We compute

2 nlog% 1 + IIf(z) I2 dx A dy a

Z.4.

i

aa

2

log

1

+ Ilf(z)I

azaz log/1 + Iif(z)II2

i

2

Z dz A dz

a a log(1

+

Efi fJ

=

dz

a

+ Ekfkfk E_'k1kk

i

(1 + Ekfkfk)( Ekfkfk) - ( Ekfkfk)( Ekfkfk) 1 + Ekfkfk)'

dz A dz

(9.9)

tAhlfors [11, Shimizu [541 63

Here each sum is extended from

is sitting in

Cn

Now

Cnc Pn

k=n as an open set, and we have the inclusion map k=1

Pn

to

.

,wn) -- (1,w1.

(w1,

Pulling back by the inclusion

.wn)

ti

Cn C Pn

we obtain the " F u b i n i

the Fubini-Study metric on Pn -Study metric on Cn " t. It is given

by

1 + Ekwkwk) (Ekdwk QQ dwk ) - (Ekwkdwk ® (Ekwkdwk ) ds2

=

1+

Ekwkwk

)2

(9.10)

Its Kahler form is i w0

Z

(1 + E k w k w k ) ( Ekdwk A dwk )- ( Ekwkdwk ) A (Ekwkdwk (1 + )2 E w w k k k

4 ddclog (1 + Ekwkwk )

If ds2

(9.11)

for example, then P1 is the usual spherical metric on

is the Riemannian 2-sphere

n=1

ds

2

dw

C

,

x( dw 1

(1 +

1

w

1"`1) 2

which is a conformal (Hermitian) metric of constant Gaussian curvature and the associated Kahler form

t

S2, and

dw 1 A dw1 (1 + w1w1)2

4

dul A dv1

(I

+w1w1)2

(wl=u1+iv1)

is the spherical volume form.

We now return to the general case is the pull-back of w0 to CR

tWu [63] 64

n?1

.

(9.11) and (9. 9) show that

Alog

1'V

If(z)

+

1

2dxA dy

(9.12)

This shows that in the general vector valued case

1 + 11f(zo dx A dy

Alog

2

the integral

n?1

f*u

=

(9.13)

J

Ct

Ct

is nothing but the v o I u m e of the image of

z -+

Ct

in

Pn

under the map

(lrfl,...1fn)11,

(If (9. 13) is divided by it we obtain the normalized volume ) . 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 right0 hand side, it is natural to introduce a new modified characteristic function T(r,f) by

T(r,f)

:

0

where

0

0

Definition 9.2

m(r,f)

We will call

.

=

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

is given in Definition 9. 1 0

T(r,f) the"generalized Ahlfors -Shimi-

zu characteristic "orthe "generalized spherical c h a r a c t e r i s t i c ", since it agrees with the characteristic of Ahlfors Shimizu in the scalar case. Summarizing, we can formulate this result: be a vector valued meromorphic function on CR Then denoting by w0 the Kahler form (9.11) of the Fubini-0Study metric on Cn and defining the generalized spherical characteristic T(r,f) by Defi nition 9. 2 , we have the formula : Theorem 9.3

:

Let

f

.

65

r f

0

dt t

1

T(r,f)

2,r

Olog%/1

+ IIf(z)II2 dx A dy

6t

0

r r

1

dt

71

t

(f x const.)

f WO

(9.15)

.

Ct

0

0

This geometric interpretation of T (r , f) gives also a quasi-geometric interpretation of the generalized characteristic T(r,f) of Nevanlinna because of the following

Proposition 9.4

T(r,f)

:

0

T(r,f) as

r-R

T(r,f)

=

0

T(r,f) only by a bounded term:

differs from +

0(1)

(9.16)

,

.

0

Thus, in many investigations T can be used instead of T and vice versa, without any changes of formulas. The estimate (9.16) can be seen as follows. We have 2 Tr

0

1

m(r,f)

log V1 + Ilf(reie)II2 do

27T

-

d

0

where

log'1l

d

if

IIf(0)II2

is not a pole for

z=0

pole for

+

f

.

f

, and

d

=

z=0

if

log IIcg1I

Since logv1 + X11112

log lIfII

log IIf1I

_

+

1ogV_

it follows that

m(r,f)

<

0

m(r,f)

+

d

<

m(r,f)

+

logy

is a

0

m(r,f)

- d

m(r,f)

-

log

5

- d

This shows that the difference 0

m(r,f) - m(r,f) remains bounded for 0 < r < R therefore for the difference

and from (9.14) we see that the same holds

0

T(r,f) - T(r,f) f(z) = (f1(z),...,fn(z)) morphic function. Then the order and type-class of Proposition 9.5

:

Let

be a vector valued merof(z) are the same as

the order and type-class of the integral f C

ai0

r

Proof. Suppose that

C

(r > r0)

Krk

f

r

Then by (9.15)

r

r

K tk dt

0

T(r,f)

<

7T

T

+

0(1)

=

k rk

+

0(1)

r0

This shows at once that the order and type-class of that of the integral in Proposition 9.5. Vice versa if 0

T(r,f) then

<

Krk

(r>r0)

0

T(r,f) cannot exceed

2r r

f

*

109

2

w0

r

1

dt

R

t

Cr

0 <

f*w0

K(2r)k

T(2r,f)

,

(r > r0)

Ct

r

so that the order and type-class of the integral in Proposition 9. 5 cannot ex0 ceed that of T (r, f) This proves Proposition 9. 5. 0 Since the integral representation (9.15) for T (r , f) has the same form as V(r,a) , the same proof as that of Proposition 4.1 shows that .

r

0

lim r+ +m

T(r,f)

*

1n lim

log r

(9.17)

f

w0

r+ +0o C

r

We can apply this to rational vector functions. Using (9. 17), (9. 14) and (4. 8) we conclude If fn (z)) is a vector valued rational f (z) = ( f 1 (( zz)) ,. Proposition 9.6 function, then the normalized volume of the image in Pn of C under f :

is an integer, which is equal to the degree

n(m)

of f

r

* f w0

1 lim

n r++ C

§10

* n(')

=

(9.18)

r

The generalized Riemann sphere

with coordinates can be identified with the real Euclidean space R2n u = (u1,...,u2n) by assigning

The complex Euclidean space

w f---4.

u

Cn

, where

w.

=

1

The real Euclidean space R2n

=

R2n

x

{ 0}

R2n+1

u

2j-1

w = (w 1, ... , wn ) with the coordinates

+ iu 2j

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

u

in

0

=

2n+1

R2n+1 , where the

are denoted by

(u1,...,u2n+1) Let

be the sphere

Stn S

2n

u2

+

u2n

+

.

(u2n+1 -

+

1

0.5)2

0.25

=

It is tangent to R2n at the origin which is its south pole and has north pole p = (0,... , 0,1) . The and (ul,. . . line in R2n+1 through the fixed points p = (0,...,0,1) is parametrized by ,u2n'0) (0, ..., 0, 0. 5)

a(t)

t(u1,...,u2n,0)

=

+

(1-t)(0,...,0,1)

(tut, ... ,tu2n, 1-t)

_

.

0< t 5 1

,

The points of intersection of this line withS2n which satisfy the equation t2(u2

+

+

u2n)

Its only solutions are

(0.5 - t)2

+

t=0

correspond to values t

0.25

and 1

1

=

t

1

,

2 + ... + u2n

+ u2

I

+

lluil2

where we have used the abbreviation Ilull2

=

llwll2

=

uI2

+

2

.

.

.

+ u2n

The first value t = 0 corresponds to the north pole p , and the second F of S2n given by corresponds to the point ' 2n' F 2n+1) .

(

U. Ej

(j=1, ... , 2n)

3

I

1....

(10.1)

Ilull2

+

1lu112 E

2n+1

(10.2)

I

+

Ilull2 69

Asusualwedefine stereographic projection to be this Stn - {p} , which sends a point u E R2n to the point s(u) = (E 1, ...' F 2n+1) E Stn - {p} on the line segment joining u to p . From (10.2) we compute map

s

to

R2n

from

C

1

2n+1

-

1

C

Ilull2

+

1

2n+1

-

1

2n+1

so that from (10.1) uj

=

-

(10.3)

2n+1

Thus stereographic projection which is given by (10.3) .

has an inverse

s(u)

=

E

Now let u = (ul, ... .u2n) in R2n , and respectively E

and

v

u

(vl, ... I V

=

and

(E

1, ....E 2n+1) Stn - {p } . Denote by

their images on

Ti 2n+1)

(j1 , ... , 2n)

,

1

1

s-1(F, )

=

be two points n

(n

=

1

,

1

[u,v]

:

2

((E1- n1)

_

the Euclidean distance of

,

n

+ .

,

,

+

Stn - {p}.

E

n21)

(E 2n+1

2)2

. Using (10.1), (10.2) we

compute (1+I1ull2)2(1+11v1I2)2

u,v 12 [

2

u1

2

(1+ 11 u l12) 2(1+11V11

vl

)2

11112+1

Ilull2+1

2

v2n

( u2n

Ilull2+1

2

1112

Ilull2 C Hull

0+11vI12)u1-(1+I1ull2)v1)2

t+1

+

11v112+1

IIvII 2+1

+...+ ((1+11v112)u2n-(1+llull2)v2n)2 + (Hull

+

2+1)) 2 2(11v112+1)-11112(

Hull

=

(1.11-112)2Hull 2

+

(1+llull2)21V112 -

2+1) Ilul14( Ilull2+1)2 + livll4( Ilull2+1)2 - 21lull21lvl12( Ilull2 +1) ( Ilu 11

(1+IIvll2)2dull 2(l+Ilull2)

=

-

+

(l+Ilull2)21Iv112(1+llvll2)

211ull211v112( llvll2+1)( Ilull2+1)

=

(1+Ilull2)(1+llvll2) 1 (1+11vll2) Ilull2 + (1+Ilull2) llvll2 - 2u.v

=

(l+Ilull2 )(1+llvll2) [Ilull2

=

(1+IluII2) (1+llvll2)

IIu-v

+

-211u11211v112

llvll2

II2

Thus we have obtained the formula Ilu-vll

[ u,v ]

(10.4)

+Ilull2 1

1V

ilvll`

Remark. We conclude from (10.4) that the Euclidean metric 2n+1

ds2

=

F, duj O duj j=1

of

R2n+1

induces on

Stn 2n

ds2

E j=1

the metric

duj 0 duj

n

dw ® dw

(1+IIu112>2

j=1

(1+IIwII22

by the inclusion S2n c R2n+1 t Using the identification Cn = R2n mentioned at the beginning of this section, we now define for any two points a , b E Cn the expression

tSpivak [33] 71

by putting

[ a,b]

IIa-b II

[ a,b ]

(10.5)

V'11 Jlaj` V 1+IIb1`

[ b,a ] is the Euclidean distance of the images of a,b on the sphere Stn under stereographic projection; since the diameter of Stn is 1 , we always have [ a,b ]

We have shown that

0

=

[ a,b I

The number

<

[ a,b ]

_

1

will be called the " g e n e r a l i z e d

chordal

distance "orthe"generalized spherical distance" since for n = 1 it agrees with the ordinary chordal distance of points on the Gaussian plane C 1 = C We recall now that the symbol denotes the ideal element of the Alexandroff one-point compactification Cn U {°°} . We will say that a E Cn tends to °° , if the expression of

a

and

b

,

a

tends to a unit vector as

. Then clearly 1

[a,b]

lim b-+w

1V

+IIaII`

and it is thus natural to define the generalized chordal distance [ a, -I of n any finite point a EC to the "point at infinity" °° to be the number 1

(10.6)

IlaIIL

1+Y

Further, since obviously lim

[ a,-]

we define

[ -,-I

ai

[-,-] 72

:

0

=

by putting =

0

(10.7)

In this manner, the generalized chordal distance [ a,b ] fined for any two points a,b E Cn a {m} and satisfies

[a,b]

0

1

<

It is now natural to call

sphere "

has been de

Stn

the"generalized Riemann

.

The spherical normal form of the generalized first main theorem

§11

In Chapter 1, §3 we proved for

a E Cn

identity (3. 9)

2,r

f 1

alld

log Ilf(re

2 Tr

=

V(r,a) + N(r,a) - N(r,f) + log Ilcq(a)II (11.1)

6

here

log Ilcq(a)II

Now, if

=

z = rein

log Ilf(0) - all

f(0) x a,

if

is not a pole for

w = f(z)

(10.5), (10.6)

, we have from formulas

--

all

[f (rely),a]

_

1IIIf

112

1V

1

so that 1+lIall2

all

[f(rely),°°]

Using this the left hand side of (11.1) can be written 1

T7

2n

2n

r

r 1

log

log

1+

IIaII2 +

log

Z1

[f (rely ) 0

73

and (11.1) can be given the form 2Tr

1

1

d

log

2n

N(r,f)

+

[f(rely),°°]

J

0 27r

1

1

0

log

2 7r

V(r,a) + N(r,a) + log Ilcq(a)II

+

[f (rely ) , a ]

J

0

- log

a e Cn

for

We now introduce a function Definition 11.1

:

If

0

m(r,a)

a E Cn U {w}

by the following formulas:

a x f(0)

,

(11.2)

1+IIaII2

,

f(0) E Cn U

we put

27T

0

d-

1

1

m(r,a)

log

2n

1

log

(11.3)

[f(0),a] 0

In the other cases we put 2Tr

r

0

1

1

m(r,a)

log

2n

d

-

k

(11.4)

if (rely ) , a ] 0

where lv

+11i(0)II2 V1, IIaII

if

log

a = f(0)

Ilcg(a)II k

=

(11.5)

log IIcg1I

if

a = f(0) _ 0

0

As in Nevanlinna theory, we sometimes write m(r,f) for m(r, W) . Definition 11.1 is then consistent with and more general than Definition 9. 1 . The 0 function m(r,a) will be called the"generalized spherical

proximity f unction "orthe"generalized Ahlfors '

A

S h i m i z u p r o x i m i t y function ". Using this function, equation (11.2) can be written as 0

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

0

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

_

(11.6)

(11.6) even holds for a = - , since we defined: V(r,,-) = 0 Note that the sum on the left hand side is exactly the generalized ShimizuAhlfors or generalized spherical characteristic .

0

0

T(r,f)

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 Spherical Formulation):

(fI(z) , ... ,fn(z)) identity 0

T(r,f) holds for

is a non-constant meromorphic function in 0

=

0
m(r,a) R

and

N(r,a)

+

+

V(r,a)

If

w = f(z)= CR , then the

(11.7)

a E Cn U j_I

In applications of the first main theorem sometimes the spherical formulation is of advantage and sometimes the original form is better suited. §12

0

T(r,f)

The mean value representation of

By means of stereog-raphic projection,the right hand side of (11.7) for fixed r (0 < r < R) can be regarded as a function on Stn . We denote by dS2n the 2n - dimensional measure of 5211 . The total volume IS2nl of Stn is

1

(1)'n

n+2 2

r (n+1) 2 0

We consider the integral of m over 5211 . According to Definition 11. 1, 0 for f(0) E Cn U {m} the function m is given by a E Cn , a x f(0)

75

2ir r

0

1

1

m(r,a)

log

2ir

1

-

do

log

[f(reio),a]

(12.1)

If (0),a]

0

Now, 2n

27T

r

1

dS2n

1

log

2II

do

If (re S2n

1

io

2ir

),a]

0

log -

do

1

dS2n

,

[f(reio),a] 0

Stn

and for reasons of spherical symmetry,the inner integral in the last expression is independent of the point f(reio) . This point can be replaced e.g. by the point f(0) , so that the inner integral is equal to 1

log

dS2n

[f(0),a] S 2n

which does not depend on

. From this and (12. 1) we conclude that

0

0

m(r,a) dS2n

0

S 2n

Thus, assuming that (11.7), we obtain

is non-constant and by taking the spherical mean of

f

r

0

1

T(r,f)

(N(r,a) + V(r,a)) dS2n

Is2nI S2n For

a x f(0)

we have r r

N(r,a)

n(t,a) dt t

n

(12.2)

so that

r dt

N(r,a) dS2n

n(t,a) dS2n

t

S 2n

S2n

0

The function n(t,a) as function of the variable a E Cn , which do not belong to the image of f ; if

has value zero for all n > 1 therefore, it vanishes up to a set of 2n - dimensional measure zero on the sphere S2n a

Further, r r

r

dt

V(r,a) dS2n

v(t,a) dS2n

t

S2n

S2n

0

and we know that this vanishes identically for (12.2) can be written 1

Is

n=1

V(r,a) dS2n

2n

Using these facts,

.

if

n>1

,

S 2n

0

T(r,f)

(12.3) r

1

N(r,a) dS2

n

n=1

if

.

S2

If we introduce the abbreviation

A(t,f)

by putting

v(t,a) dS2n

if

n>1

,

A(t,f)

n(t,a) dS2

n

if

n=1

S2 Ti

then (12.3) can be written

r I.

0

T(r,f)

A(t,f) dt t

0

in the case n = 1 this is a well-known formula of Nevanlinna theory. We summarize this result in 0

Theorem 12.1: The generalized spherical characteristic T(r,f) of a nonconstant vector valued meromorphic function in CR can be represented by the integral mean 1

V(r,a) dS2n

if

0

T(r,f)

n>1

,

(12.4)

r 1

N(r,a) dS2

R

n=1

if

S2

Equivalent to this is the formula

r A(t,f)

0

T(r,f)

dt

(12.5)

,

t 0

where

A(t,f)

denotes the spherical mean given by 1

v(t,a) dS2n if n > 1

S2nI

Stn

A(t,f)

=

1

is 2n. I

(n(t,a)+v(t,a))dS2n

(12.6)

Stn

n(t,a) dS2 if n = 1

n

j S2 78

Since by the properties of V(t,a) or n(t,a) the function A(t,f) is 0 positive for 0 < t < R , we conclude from this result that T(r,f) van ishes for r = 0 and is a positive, increasing and strictly convex function of log r for 0 < 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, Chapter 2 By comparison of (12.5) with (9. 15) we obtain the following geometric interpretation of the spherical mean A (t , f) .

.

Corollary 12.2

:

A(t,f)

Alog

f(-0dx

1+V

A dy

1 11

Ct

Ct

(12.7)

We conclude this section by giving an application of the representation (12.4). In the case n > 1 we have r

1

V(r,a)

IS2nI S

0

T(r,f)

dS2n

2n

so that r

dS

2n

(12.8)

0

S 2n

In the first main theorem in spherical formulation 0

T(r,f) we have for

0

=

a X f (O)

m(r,a)

+

N(r,a)

+

V(r,a)

2,r r 1

0

m(r,a)

log

_

1

[f(rea]

do

-

log

[f(0),a)

0

N(r,a)

and

=

0

,

so that 0 m(r,a)

0

T(r,f)

V(r,a)

-

>

V(r,a)

1

1

1+ 0

log

o1

T(r,f)

-

T(r,f)

[f(0),a]

1

1

log

0

+

0

T(r,f)

[f(0),a]

2 7r

1

1

1

0

log

27r

T(r,f)

do

0

[f(reio ),a] 0

Thus, if 0

lim

r+R

T(r,f)

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

0

lim

a

-

1

V(r,a) dS

0

-

2n

lim

T(r,f)

Stn

r-*R

1

-

Stn

V(r,a) dS

0

2n =

T(r,f)

so that

dS

lim

2n

=

0

0

i

T(r,f)

r->R

S 2n

Since the integrand here is non-negative, we conclude that almost everym on Stn lim

r>R an

1

-

V(r,a)

V(r,a) =

T(r,f)

1

- lim r-rR T(r,f)

=

0

i.e., we have

Corollary 12.3

:

V(r,a) lim

(12.9)

0 r-R T(r,f)

holds almost everywhere, unless 0

T(r,f)

<

or

n = 1

Remark. In the Nevanlinna case n = 1 relation (12.9) holds a.e. with the volume function V(r,a) replaced by 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 z1, z2, ... be a finite or infinite sequence of non-vanishing complex numbers. If -+ the sequence is infinite, we assume that z as n --r +W Let q n be an integer > 0 such that the series .

q+1

E

is convergent. Let

E(u,p)

_

(1-u) e

u+ u2 2

+

UP

p

E(u,0) = 1-u

denote the prime factor of Weierstrass. If the sequence ite,it can be shown that the infinite product

(13.1)

z1, z2, ...

is infin-

TTE(zz q) V

converges uniformly in each bounded region of the plane C and represents If the thus an integral function which vanishes exactly in the points zv sequence is infinite and the series .

E

1

z

q

v

is divergent, then q >_ 0 is called the g e n u s of this infinite product. If the sequence is finite, the genus of the corresponding finite product with q=0, 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 zll(m),z21(-),... respectively denote the zeros and poles of fI(z) outside the origin z = 0 . Let q >_ 0 be an integer such that the integral

T(r,f1) (13.2)

dr

r

q+2

is convergent. Then the series q+1

1

2:

E

and z)j 1(0)

both converge. Let

zV 1(c0)

be the smallest integer

k1

q+1

I

such that the series

>0

k1+1

1

zu 1(0) I

converges, and let k2 be the smallest integer ? 0 such that the analogue series for the poles z\; 1(') converges. Then clearly k1 < q , k2 < q Nevanlinnat proved the following fundamental representation of f 1(z) . Under the above conditions the meromorphic function has the representation Theorem 13.1

:

z

z

Pm e

1

(z)

,kl)

1

TTE

r1

f1(z)

zU 1(0) (13.3) z

TTE (

where

r1 > 0

is an integer and

PM (z)

,k2

zV 1(-)

is a polynomial of degree m1
1

In Nevanlinna theory the g e n u s gl of a not id. zero meromorphic function f1(z) of finite order is defined to be the integer TNevanlinna (27] ,p.40 83

max{ ml,kl,k2 }

g1

which is readily obtained from the representation (13.3). If p 1 denotes the order of a not id. zero meromorphic function of finite order, then the following results of Nevanlinna are valid: (i)

gl

(ii)

if

(iii)

Let

a)

g1 + 1

P1

=

p

is not an integer, then

1

f I(z)

g

fl be of integer order

is the largest integer

1

pl >

1


; then the following holds:

If the characteristic function satisfies

T(r,f1) 0

+W

1im

<

r >+

then

f1

r

(13.4)

P

has genus

91 - P 1

b) If

T(r,fI) lim

(13.5)

0

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

T(r,f1) P

r

l +1

dr

(r

0

>

0)

(13.6)

r0

is convergent or divergent. If this integral is convergent, the genus is If this integral is divergent, the genus is g1 = pl 1 , provided that the series gl 11 - 1 .

84

PI

p1

1

1

E zp 1(0)

Z zvl(x)

and

both converge; and the genus is series is divergent.

gl

p1

if at least one of these

Thus the genus g 1 of f 1 is determined uniquely by the order p 1 alone, with the exception of the case where T(r,f1) belongs to the minimum type of the integer order p 1 > 1 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

,.

be a meromorphic vector function of finite order in the plane C ; we assume that f 1, ... Ifn are all zero . Let gj (j=1,...,n) denote the genus of fj(z) . Then the " g e n u s g o f f(z) " is defined to be the integer Definition 13.2

f (z) = ( f 1(z) (z)

Lpt

:

fn (z))

max g

g

1

From result (i) above we have the inequalities g.

g. + 1

P.

Since the order of

f (z)

is

, p

(j=1,...,n) max P. J

13.2

we obtain from Definition

I

Proposition 13.3 : The order p and the genus g of a meromorphic vec tor function f (z) (f,, ... I fn Ft zero ) of finite order in the plane satisfy the inequality g

p

<

g+ 1

From this we conclude in particular

If the order p of a meromorphic vector function f(z) (f 11 ... , fn zero) of finite order in the plane is not an integer, then its genus g is the largest integer < p Corollary 13.4: Eit

Now let

be an integer

p

. Then from Proposition 13.3 we have

?1

either g= p or g= p- 1 In view of the above results of Nevanlinna,assume first that mean or maximum type of an integer order p ' 1 . Then

f(z)

is of

T(r,f) 0

lim

<

<

(13.7)

+00

rP

r-* +W

In Chapter 2, §7 we proved the inequalities n

T(r,fj)

+

T(r,f)

0(1)

Z T(r,fk) + 0(1)

5

(13.8)

,

I

Now, by Proposition 8. 5,

_

T(r,f)

lim r}+00

= max pk

p

T(r,fk)

n

rP

k=1

, and from (13.8) we see that

lim r++00

_

T(r,fk)

r-*+0

rp

lim

rp

PCP

since

T(r,fk) if

0

rP

P

>

P

k

From this and (13. 7) we see that there exists an index

such that

j0

=p

p1 .

0

and

T(r,f. 0

<

10

film-

r}+m

rP

is thus of mean or maximum type of the order result iii)a) of Nevanlinna it follows that f,

=

P1

p. 10

=

p

. By the above

P

0

Since by Proposition 13. 3 always g < p , it follows that the genus g of f(z) is = p g = max g. in this case. Next let f beJof an integer order p 1 , of minimum type and of >_

convergence class. Then 86

+m

T(r,f) lim

T(r,f)

and

0

=

r P+1

dr

(r0 >

+W

<

0)

r0

Using inequality ( 13.8) we obtain

T(r,fj)

r->rp lim

T(r,f.)

p+i dr r

and

0

(j1 , ... ,n)

r0

(13.9)

+m

<

.

By Proposition 8.5 there is an integer j0 such that f. is of order p.=p. By From (13.9) we see that f. is of minimum type and of convergence result iii)b) of Nevanlinna iPfollows that the genus g. of f, is p1. - 1 = p - 1 . Assume that there is an index jl such that fj is of genus gj 1 1 > p . Then by result i) of Nevanlinna we would have p1.1

gi.t

P

L!

so that by Proposition 8. 5 necessarily p. = p . By (13.9) f. is of minimum type of order p and of convergence class. By result iii)lb) of Nevanlinna we obtain g. = p - 1 , which contradicts the assumption g. '- p . This shows thatl g = p - 1 in this case. 11 Finally, let f be of integer order p 1 , of minimum type and of divergence class. We have >>

T(r,f)

T(r,f) lim r->+m

=

and

0

rp

r P+1

dr

=

r0

Applying inequality (13.8) we get

T(r,f.) lim r->+m

0

rp

and there is an integer

j0

for which

(j=1,.. ,n)

p.

=

0

p

and

+00

T(r,fj

)

(13.10)

dr

0

r0

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

I

E

=

p. JO

if at

p =

p

1

E zv.

and

zuj0 (0)

g. JO

(oo) 0

zuj (0) and z vj (co) respectively denote the zeros diverges, where and poles of f ].0 outside the origin. In Othis case we conclude from the general inequality g

max g

=

that

p

g

p

J

and (13. 10) is Now assume that for all indices jo , for which p. = p J satisfied, both series (13.11) converge. Then by result iii)b) of Nevanlinna we have gJ . = p - 1 for these indices j0 . For indices j l , for (13.10) is not satisfied for jo replaced by j 1 , we have either p is of convergence class. In the or we have p. = p and f. J1 J1 case, since f, is of minimum type, we p lf iii)b) of Nevan linna. In the former case have <

gJ 1

< p-1 is an integer, it follows that g. is not an integer, fthen also g. p - 1 by Corollary 13.4 p

J,

:

Let

f = (f1 , ... , fn)

( f 1, ... , fn it 0)

vector function in the plane, of integer order a) If the characteristic function satisfies

T(r,f) 0

<

lim

5

r++m

then AR

f

rP

has genus

if

p J.

J1

Thus we can summarize: Theorem 13.5

;

p

p

=

1

be a meromorphic

1

If

b)

T(r,f) lim

0

rP

r-*+°

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

T(r,f)

r

(r0

dr

+1

>

0)

(13.12)

r0

is convergent or divergent. If (13.12) converges, the genus g = p -1. If (13.12) diverges the genus g is p , provided that there is an index j0 such that (13.12) diverges for f = f. and at least one J 0 of the two series

L

P

I

Z

and z

ujo

(0)

diverges, where zeros and poles of

zVj0 (W)

respectively denote the ; otherwise the0genus g = p - 1 . and

zP j (0) fJ. 0

P

1

zvj (W)

O

§14

Some relations between

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

In this section we assume for simplicity that the given vector function f(z) is meromorphic in the entire finite plane C We have seen that the generalized proximity function m(r,a) can be used to measure the asymptotic convergence of the vector function f to the point a E Cn . 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 expression is the generalized maximum modulus M(r,a) defined by .

M(r,a) for finite

a

max

=

, and for

M(r,f) = M(r,m)

Izl=r

I

IIf(z) - all

a =

max

IIf(z)II

Izi=r 89

We obviously have

m(r,a)

a

.

logM(r,a)

a E Cn U (W)

Here, if a is finite, the right expression has the disadvantage of becoming infinite if on DC r lies a solution of the equation f(z)

a

=

the same inconvenience holds in the case on

if there are poles of

a

f

DC

r Let us first consider the case where f (z) is an i n t e g r a 1 vector N(r,f) = 0 function; then this difficulty can not arise for a = . From T(r,f) = m(r,f) we obtain 2n r

T(r,f)

=

log IIf(reie)IId©

1 227

log M(r,f)

5

(14.1)

J

0

Now, for 0
and

z = re'

the generalized formula of Poisson

2n

log II f (rely) II

=

21T

s

1If(sei())II

log

2

s

2

2

do r +r 2 - 2srcos(6-f)

0

-

s2-z.(0)z 1

E log s(z-z.(0))

Izi(0)1< s

s 2-Cz 2n

log

s(z-O

clog

+

Z Izi (m)I<s

s2-z1(-) log

z

s(z-z.(-))

da

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

2 r

s

1

logllf (rely ) II

2

s

2

2

2

r

+r 2 - 2srCos(El-p)

d©

4

0

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

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

log M(r,f)

(14.2)

We summarize (14.1) and (14.2) in the fundamental Theorem 1 4 . 1

isfies for

:

0

T(r,f)

An integral vector function f (z) = ( f 1(z) ( z ) ,. r <s the fundamental inequality <

If in (14.3) we put

T(r,f)

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

logM(r,f)

= kr (k > 1)

s

sat-

(14.3)

we obtain

k + 1 T(kr,f)

logM(r,f)

fn (z))

(14 '`N

From this we conclude

is an integral vector function, then the order of logM(r,f) is equal to the order of T(r,f) . If is of positive finite order p , then logM(r,f) belong to and T(r,f) the same type or convergence or divergence class. Theorem 14.2

:

If

f (z)

_

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

f

Remark. In connection with Theorem 14. 1 we obtain exactly as in Nevanlinna theory tthe following results for non-constant vector valued integral functions:

T(r,f)

log M(r,f) (i)

lim r-* +w

if

0 < p < +m

<

rp

e(2p + 1)lim

r- +-

rp

and if the right hand side is finite.

tHayman [161,p.19-20 91

log M(r,f)

(ii)

lim

0

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

for any

K>1

The last inequality in (14.4) is apparently valid only if f is integral. Similarly to Nevanlinna theory it is interesting however that (14.4) remains essentially valid in the general case of a m e r o m o r p h i c vector function if (O,r) , the latter logM(r,f) is replaced by its mean value in the interval being a quantity, which is finite for each finite r . We shall provet Theorem 14.3 inequality

:

A meromorphic vector function satisfies for each

r > 0 the

r

1

logM(t,f) dt

r

C(k) T(kr,f)

<

(14.5)

0

where the constant C depends on k

can have any value only. k

, and where the expression

>1

Again the proof rests on the generalized formula of Poisson-Jensen-Nevanz = tell linna. Similarly to the previous reasoning we have for 27T F

t2 d6 2 + t - 2st cos(6-4) 2

logllf(Seie)II

2n

2

s 0

Iz

E .(-)I< s

log

J

Here for

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

the integral on the right hand side is less than

,

and in the sum

tsee Nevanlinna [27] ,p.25 92

Is2

- z1 7;)zI

so that for

s2

+ Izi(-)It

<

2s2

0
r

s

f

1lgM(t,f)

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

dt

E

+

log It - 2s

jz.(oo)I<s j 0

dt

Izl(-)I I

0

(14.6)

The integral s f

J(A )

2s

log

dt

0

A < s

It - A I J

0

needed in (14.6) is equal to s

JsJiogIri -A

s

log2s dt -

log It - A I dt = slog 2s -

dT

J

0

0

=

s log2s - JiogiTi

dT

-

log T dr J

0 (A

s log2s

J

log d-

log T dT

0

0

=

J

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

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

J( z.() =

rs

dn(A,f)

=

J(s)n(s,f) - Jn(A.f)dJ(A)

A=0

A =0

Now dJ(A)

log s A A dA

is positive for

0
Z

and negative

s

for

< A < s

2

,

so that

s -

s

n(a,f)dJ

I

n(s,f)(J(z) - J(s))

n(A,f)dJ

J

s

1 A=O

2

and consequently

L:

n(s,f)J(Z) =-sn(s,f)log(4e)

J( 1zj(o)1)

z.(-)I<s Now put

s

=

is an arbitrary number. Then

k>1

where

r ,f -k

rk

rk r

r

n(s,f)logrk = n(r/k,f)

dt

<

t

-

n(t,f)

dt < N(kr,f)

t d

rrk

r vl

so that V-k log(4e)

L.

J(Iz.(m)I)

rT(kr,f)

,

log

IzJ.(°°)I<s

and

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

<

k+1 _

k- 1

r T(/k r,f)

Introducing these bounds into (14.6) gives r

r

1 r logM(t,f) dt

C(k) T(kr,f)

0

with C(k)

+1

V-k log(4e)

-1

logfk

This proves the Theorem. Again as in Nevanlinna theory we have 94

1k + 1

/-k - 1

r T(kr,f)

; T(kr,f)

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 0 < p < +we have further +00 +00 Proposition 14.4

r

For

:

a E Cn u {_}

dt

t p+l

p

a)

'j (

r

n(t,a)

p

1

N(t,a)

2

=

p

tJ+1

J

dt

(14.7)

J

1z.(a)I> 0

0

0

in the sense that these three expressions are all infinite or are all finite and equal.

Proof. Suppose that

n(r,a)

' Cr'

<

(r > r0)

,

Then

r n(t,a)-

N(r,a)

dt + 0(1)

C

<

rp + 0(1)

,

t

(r > r0)

r0

N(r,a)

so that the order, type,. convergence or divergence class of exceed that of n(r,a) . Vice versa, if N(r,a)

C r1

<

(r > r0)

can not

,

then the inverse statement follows from 2r

n(t,a)

n(r,a)log2

=

t

N(2r,a) +0(1) 5 C(2r) p + 0(1) ,(r >r0).

dt

r Further, we have

r.

r ,

E Iz.(a)i> 0

1

z. J

(a)

p

n(t,a)dt

n(r,a)

dn(t,a) =

rp

p

tp+1

tp

j 0

0

r

r r

n(r,a)

f

n(r,a)

dN(t,a) =

+

_

ru

+

ru

to

N(t,a)dt

pN(r,a) +

u

to+1

ru 0

0

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 u and so is at most of minimal type of order u . Thus N(r,a) ru

-0

r-> +W

as

and thus, from what we proved above, also

n(r,a) r > +W

as

0

r

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 < u < +m we have further v(t,a)

u+1 dt

Y 11

t

V(t,a) u+1 dt

(14.8)

t

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 r 1 r logM(t,f) dt

0

is at most of order

p

Now for

a

=

( a l , ... an) E Cn

we have the inequalities

1

M(r,a)

=

max

Izl=r

1

max

Izl=r

Ilf(z) - all

,

(j=1, ...,n)

Ifj(z) - ajI

Putting 1

M(r,a .)

max

J

Izl=r

Ifj(z) - ajI

we can write

M(r,a)

A5M(r,aj)

(j=1, ...,n)

In the following we assume for simplicity that each fj , j1,.. . ,n non - constant . If we apply inequality (14.5) in the case n = 1 to the function

(14.9)

,

is

1

fj(z) - aj

(14.10)

we see that the expression r

r

1

log M(t,aj) dt

r

0

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 fj(z) , has the same order as f. (z) . We conclude that the expression (14. 11) has at most order p . From this and (14.9) we see that the expression .

J

r

r 1

log M(t,a)dt

r J

0

91

is at most of order

n(r,a)

Using the above results, the meaning of

and the generalized first

main theorem we can formulate

f(z) = (f1(z),...,fn(z)), (fl,..-,fn non-constant) Proposition 14.6 Let be a meromorphic vector function of order p . Put p * = min (p 1, ... , pn} Then the following holds: (i) The expressions :

:

V(r,a), v(r,a), N(r,f), n(r,f), m(r,f)

acrd

r f

1 r logM(t,f) dt

0

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

For

a

=

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

p

f

n

the expressions

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

and

r 1

r log M(t,a) dt 0

are at most of order

p*

;

the series

P*+E

1

(14.12)

z. (a) z)(a)> 0

1

converges for every

E>0

Now choose

c>0

M(r,f)

er

p*

<

+W

and assume that for all

p+ e

holds. Then it follows that 98

if

r

> r

0

>0

the inequality

r

>+F+ P+E

1

r logM(t,f) dt

+

1

o(1)

0

so that the left integral would be of order with Proposition 14.6 (i) we conclude

p+E

. From this contradiction

Corollary 14.7 : Let f(z) be meromorphic of finite order be given. Then for infinitely many arbitrarily large values of

p

and let e > 0

r we have

r p+ E M(r,f)

e

<

In the same way we obtain Corollary 14.8

Let

:

he finite. Let large values of r

be meromorphic, (f 1, ... , f n non-constant) and

f (z)

be given. Then for infinitely many arbitrarily

E>0

P*

we have P* + e

M(r,a)

er

<

a E Cn

,

We now put

r

r 1

a(r)

logM(t,a) dt

r

,

(a (.- Cn u {m})

(14.13)

0

Then

r

r

1log M(t,a) tµ+1

d(ta(t)) dt

r0

t= r0

r a(r)

a(t)dt

a(r0)

r

rµ0

+ (µ+1)

tµ+1 r0

Thus, if the integral 99

r r

log M(t,a) (14.14)

dt

p+1 t

r0

is convergent, then so is the integral r r a(t) dt p+1 t

r0

Vice versa, if the last integral is convergent, then for sufficiently large r , since ra(r) is increasing, +m I

E

E>0

and for all

+m r

a(t) dt

r

a(r)

dt

ra(r)

>

t

p+2

(p+1)rp

r

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

f(z) _ (fl(z),...,fn(z)) p

>0

be meromorphic. Then for

the integrals +m

log M(t,a)

II

1

r

t r

r

r

dt

dt

and r0

0

logM(s,a)ds

tp+2 0

are simultaneously convergent or divergent. So the order of

r 1 r log M(t,a)dt 0

is equal to the limit inferior of the numbers

100

p

,

for which the integral

a

log M(t,a)

to+I

dt

r0

is convergent. Let p* be finite and denote by the inequality P*+E M(r,a) < er

is not valid. Then for log M(r,a) r

P+1

A E

the set of r - values such that

(a)

(a E Cn,

E > 0)

P*

log M(r,a)

dr>.

r

u+I

r

dr

P*+E - u -

1

1

dr

drn n

j

A

A

(a) E

and

n=p*+E-u

where

(a)

0
(a)

A E

E

; by Lemma 14.9 and Proposition

14.6 (ii) the integral on the left side is finite. If a = m , the analogue identity holds if we replace

p*

by

p

. Thus

we can formulate

f (z) _ (f 1(z), ... , fn (z)) Proposition 14.10 Let function ( f 1, ... , fn non-constant) . (i)

If

p*

is finite, let

A

be a meromorphic vector

denote the set of r - values

(a) (E > 0)

such that p*+E

M(r,a)

>

(a E Cn)

er

Then the total variation of rn 0 < n < E

such that

(ii)

If

p

in the set

(a)

A

rests finite for all

Ti

.

is finite, let

A

(E > 0)

denote the set of r - values such

that p+E M(r,f)

>

er

Then the total variation of that 0
rn

in the set

A

E

rests finite for all

n

such

101

Proposition 14. 11

Let

f (z) = (f I (z) , ... , fn (z) ) be a non-constant meromorphic vector function. If for some u >0 the integral

r

:

u+I dr

r0 >

0

0

is convergent, then the same holds for the integrals V(r,a)

logM(r,a)

ru+I dr

r u+i

,

dr

(14.15)

r0

r0

and the series 1

,

(a E Cn U (m)) Izi (a) I> 0

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

N(r,a) r

I dr

r0

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 c Cn by combining the inequalities (14.9), (7.19) and the footnote on p.97. This proves Proposition 14.11. Proposition 14.12

:

Let

f(z) _ (fI(z),...,fn(z))

morphic vector function. If for given

102

u

>0

be a non-constant mero-

the integrals

logM(r,a) u+1

r

V(r,a) v+1 dr r

dr

r0

,

(r0 > 0)

(14.16)

r0

and the series 1

E

(14.17)

1z.(a)I u

1zj(a)1> 0

are convergent for some value a E Cn U {o) , then they stay convergent for every a E Cn U {co} , as does the integral

T(r,f) r

u+1

dr

(14.18)

r0

Proof. Since

m(r,a)

is majorized by

log M(r,a)

,

the integral

m(r,a)

ru+1 dr

r0

is convergent for the given value of gent, also the integral

a

. Since the series (14. 17) is conver-

N(r,a) ru+1 dr r0

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 14.11. Combining Lemma 14.9, Proposition 14.11 and Proposition 14. 12 we have 103

Theorem 14.13:

f(z) = (f1(z),...,f(z))

Let

phic vector function. Let

be a non-constant meromor-

denote the order of the mean value

o1(a)

r 1 r log M(t,a) dt 0

Let

denote the limit inferior of the exponents

a2(a)

a>

0

,

for which

the series 1

E Iz

Iz .(a),a

.(a)I> 0

J

J

converges. Let

max{ a1(a), a2(a), a3(a)

a(a)

v(r,a) (or V(r,a)) . Then

denote the order of

a3(a)

a E Cr U {-}

rests invariable for all

}

and is equal to the order

p

of f (z).

From the generalized first main theorem

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

=

T(r,f) + 0(1)

we obtain V(r,a)

N(r,a)

m(r,a)

lim r->+w

+

T(r,f)

(14.19)

+

T(r,f)

T(r,f)

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

T(r,f)

V(r,a)

N(r,a)

T(r,f)

and

T(r,f)

lies in the closed interval [0,11 . We also remark the following: If p = max{p1,...,pn} is the order of f = (fl,...Ifn) (fl,.. f n non-constant) and if p * = min {p 1, .... pn } , then for a E Cn < p* m(r,a) and N(r,a) are of order and V(r,a) is of order 5 If p* < p , then by the first main theorem V(r,a) must be of order since T(r,f) is of order p , or a=m .

104

,

p.

p,

sI(r)

Lemma 14.14: Let the functions increasing for 0 < r0 5 r < +m der of sk(r) . Then s1(r)

lim

provided that

be real, non-negative and k = 1, 2 let ok denote the or-

. For

.

(14.20)

0

s2(r)

r-* +m

s2(r)

,

aI < a2

Proof. Assume first that c < a2 . Then s1(r)

<

r

s2(r)

>

r

a2

<

. Let

+m

be given such that

e>0

a1+ E r > r0

for

a2-e

for a sequence

r = ra + +m

Therefore s1(ro)

aI-a2+ 2e

ra

<

s2(rc)

(ra > r0)

such that 2c < a2-al , then we have (14.20) . It is clear that (14. 20) remains valid in the case a2 = +m . Since m(r,a) , N(r,a) and m(r,a) + N(r,a) are at most of order and if we choose

p*

for

a E Cn

c>0

, we deduce from Lemma 14.14

f (z) = (f 1(z) , ... , fn (z)) Proposition 14.15 Let function ( f 1, ... , fn non-constant) such that p*

T(r,f)

lim

0

r- +m

V(r,a) 1

- lim

r-r+cT(r,f)

<

. Then for all a E Cn

p

N(r,a)

m(r,a) lim r >+m

be a meromorphic vector

T(r,f)

=

0

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

lim r->+m

=

0

T (r,f)

If the vector function f(z) has a non-constant integral component function fR of order p < , then by a theorem of Wiman f (z) tends 2 2 I z I = r. uniformly to infinity on a sequence of circles with unboundedly 105

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

m(r,a) vanishes in this case . This proves

Proposition 14.16: If the meromorphic vector function f(z) has a non-constant integral component function f1 of order p < , then k Z

m(r,a) lira r-*+00

T(r,f)

a E Cn

for all

-

1

V(r,a) + N(r,a) r-++W T(r,f) lim

=

0

.

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 (Theorem 11.2) shows in the case R = +00 that for a non-constant meromorphic vector function f (z) for fixed a

V(r,a)

+

0

N(r,a)

T(r,f)

<

From this we see that for a sequence

v(rk,a) + n(rk,a)

<

+

0(1)

rk - +0

A(rk,f)

+

,

(r < +00)

,

o(1)

A(t,f)

is given by (12.6) or (12.7). Hence for each fixed a the rk + +00 sum v(r,a) + n(r,a) is for a sequence not much larger than the average of this sum over the sphere Stn . This sequence however in general will depend on a . Thus putting Here

v(r,f)

=

sup {v(r,a) + n(r,a) } a E Cnu{-}

(14.21)

it is reasonable to ask if necessarily lim r-. +w

{v(r,f) - A(r,f) }

<

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

tHayman [16] ,p. 14 106

is a non-constant vector valued meromorphic function, then with the abbreviation (14.21) we have Theorem 14.17

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

If

:

v(r,f) 1

<

lim

e

5

r >+m A(r, f)

(14.22)

Proof. We need the following Lemma 14.18':

is a positive strictly increasing and con. Then given Q > 1 there exists a sefor x ' x0 such that if h(x) is any other positive increasing and x such that h(x) < g(x) for x>x0 , we have

Suppose that

vex function of xj -r +quence convex function of

x

h'(xj)

Here

h'(x)

derivative of

g(x)

eQ g'(xj)

<

(j=1,Z,...)

,

denotes the right derivative of g (x)

h(x)

g'(x)

and

the left

.

From the generalized first main theorem 0

T(r,f)

0

V(r,a)

=

we obtain for

r

0

0

r0 >

+

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

(14.23)

0 0

0

T(r,f) - T(r0,f) = V(r,a)+N(r,a)-V(r0,a)-N(r0,a)+m(r,a)-m(r0,a) or

0

0

0

0

V(r,a)+N(r) a)-V(r0,a)-N(r0,a)=T(r,f)-m(r,a)+m(r0,a)-T(r0,f) . We now choose

r0

for a suitable

6

such that

[f(w),f(0)] If

on

[f( 0) , a]

>

< 2

0

f(w)

z

on

f(0) jwj

jwj = r0

for

(14.24)

; then

= r0

it follows that

[f(w),a]

[f(w),f(0)] - [f(0),al

jwj = r0

so that by definition of

S-

B

=

2S

2

0

m (Definition 11.1),

Hayman [16] p. 15 107

27T r

0

0

1

1

- m(r,a) + m(r0,a)

log

27r

do

ie

<

log

2

[f(r0e ),a]

0

Thus in this case we have using (14.24)

r

r

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

0

2

dt = V(r,a)+N(r,a)-V(r0,a)-N(r0,a) < T(r,f)+logQ .

t

(14.25)

r0

On the other hand, if 1

log

log

<

then

[f(0),a] > 2 5

[f(0),a]

2

a

and, using (14.23) and again the definition of r v(t,a) + n(t,a)

0 m

0

dt

V(r,a) + N(r,a) < T(r,f) + log 2-5

<

t j

r0

Thus (14.25) holds for r > r0 Now the two functions

and all

V(r, a) +N(r, a)-V (r0, a) -N (r0, a)

a

and

0

2

T(r,f) + log S

are positive, increasing and convex functions of log r for log r > log r0 and the second function is strictly increasing. Thus by Lemma 14.18 we can 0 r. -+ +co depending on T(r,f) but not on a , such find a sequence that for r = r. and all a j

0

r Td [V(r,a)+N(r,a)]

eQr dr [T(r,f) +log ]

i.e.

v(r.,a) + n(r.,a) for

108

j

= 1, 2....

and all

eQ A(rj,f) a

. We conclude that

v (r) lim

r->+-

A(r,f)

'

Q

e

tend to 1 gives the right hand side of (14.22). The left hand side of (14.22) is correct, since A(r,f) is the mean value of v(r,a) + n(r,a) and as such not larger than v(r) for all r Letting

Q>1

.

109

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

The generalized second main theorem

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

be a meromorphic vector function in was shown in §9, Proposition 9.4 that the characteristic functions

Let

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

and

CR . It

0

T(r,f)

_

Ct

differ by a bounded term only; here c,b denotes the Kahler form of the Fubi0 ni - Study metric on Cn . So T(r,f) as well as T(r,f) can be thought of as measuring the volume of the image of the disc C r under the mapping f (z) in Cr' , equipped with the Fubini - Study metric. Associated with the curve f(z) is its Gauss map ti

f'

CR

P n-1

which is a holomorphic curve in

(15.1)

Pn-1

and is defined by projecting the de-

rived curve f'

=

(f...... fn

.CR

Cn

Pn 1 by natural projection, and by extending the result holomorphically into the poles of f and into the common zeros of f...... fn . In the case of the original Nevanlinna theory n = I of course,the Gauss map is 0 useless, since P is a point. The Gauss map associates in particular with each tangent plane at each point f(z) E Cn of the complex curve f the point, which this plane defines in Pn-I ; this notion of Gauss map generalizes the corresponding notion of differential geometry in R3 , where to each tangent plane of a surface,is associated the point, which the normal defines on into

the unit sphere. According to Chapter 1 (formulas (3.8) and (2.4)) the function

110

r

v(t,0,£)

V(r,0,f)

dt t

with r 1

v(t,0.f)

1

f w

n

Ct

at

where Pn-1

1

w

.77

Alog IIf(C)11 do A dT

2 7T

is the curvature form of the hyperplane section bundle H over of the image of the disc

measures the volume 7rv(t,0,f) in Pn-1 ti Ct under the mapping f So, if we define the function G(r) = G(r,f) r r

G(r)

=

V(r,O,f')

dt

=

0

dt

if*w

t

r

r

r

r

by putting

Clog 11f'(E )1I do A dT

2-Wt

IT

Ct

Ct

0

(15.2)

pn-1 G(r) measures the volume of the image of the disc Cr in ti under the Gauss map f' , if n a 2 , and vanishes if n = 1 ;(this volume corresponds in differential geometry of R3 to the area of the spherical image under the Gauss map). A second geometric interpretation of the function G(r) can be obtained

then

as follows.

D CC

In general, if on a domain h where

=

g dw(2) dw

=

g

(du2+ dv2)

is a positive C - function on

g =

a Hermitian metric assumes the form (w = u + iv) D

, then its volume form is

g duAdv

and its Kahler form is n h

=

gdwndw

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

h

is defined by 111

K

g Alog g

where A

a2

a2

au2

3v2

_

a2

4

aw aw

is the usual Laplacian. Hence _

a 2log g

Ki

- 2 ddclog g

dw A dw

i

aw aw

We now define the Ricci formt Ric i Rich

s log g

a

27T

of the volume form

t by putting

4 ddclog g

=

(15.3)

Thus the formula Ric

K

1

_

(15.4)

2

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

Cn

'CR

the flat metric ds2

of

Cn

=

induces on

dsf

=

dz1 Q dz1 + Co

CR

.

.

.

+ dzn Q dzn

the pseudohermitian metrictt

IIf'(w)II2 dw®dw

,

A

which is Hermitian on

OCR

. According to the above its volume form dsf =dAf

is

dAf

on

2

If'(w)II2 dw A dw

OCR

tCarlson and Griffiths [151 112

By (15.3) , (15.4) the Ricci form of this volume form is Ric dA f

2-' K dAf

2r ddclog IIf'(w)I

(15.5)

where K

- IIf'(w)II-2Alog IIf'(w)II

=

is the Gaussian curvature of the metric dsf2 . Since log IIf'(w)II is sub0 as harmonic on OCR we remark that on OCR , K <_ 0 , and Ric dAf a positive differential form. Since f

r

v(t, 0,f')

=

f'w

n

Ct

2n

slog IIf'( )II da A dT

Ct

=

ddclog IIf'(w)II

2n

I

Ct

the function (15.2) can now also be written as

r r

G(r)

Alog llf'(E )Ilda A dT

_

Ct

0

1

dt

2n

t 0

- KdAf Ct

r dt

Ric dA f

t 6

(15.6)

Ct

From this interpretation we see that curvature

G(r)

measures the growth of the total

f

K dAf Ct of

f(z) . In view of the above the function

G(r)

=

G(r,f)

willbecalledthe "curvature function" or "Ricci func 113

t i o n" 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 f (z) _ ( f 1 ((zz )) ,(z)) be a non-constant meromorphic vector function in CR . Let ak E Cnu(W} (k = 1,...,q) be q s 3 distinct finite or infinite points.Then q

m(r,ak)

+

2 T(r,f) - N1(r) + S(r)

G(r)

(15.7)

k=1

or, in view of the generalized first main theorem, q

(q-2) T(r,f) + G(r)

k [V(r,ak) + N(r,ak) 1 - N1(r) + S(r)

<

k=1

(15.8)

G(r)

Here

N1(r)

=

=

is the curvature function introduced above; and

2 N(r,f) - N(r,f') + N(r,0,f') r f n1(t) - n1(0) n 1(0) log r

dt

(15.9)

t

is the generalized counting function of all multiple finite or infinite points, where n1(t) denotes the number of multiple finite or infinite points of f (z) in t , each such point counted with its multiplicity diminished Iz I by 1 . then the term S(r) satisfies If R = + S(r)

O{log T(r,f) }

=

+

O(logr)

(15.10)

as r - +w without exception if f (z) has finite order and otherwise as r - +W outside a set J of exceptional intervals of finite measure: I dr

<

J If 114

0
then the estimate

S(r) holds as

=

OfiogT(r,f)}

O{log Rlr}

+

without exception if

r -* R

(15.12)

has finite order

f

log T(r,f) lim

=

F

r-*R log

outside a set

of exceptional intervals such

J

+w

<

1

r

r-R

and otherwise as that d

1

R-r

(15.13)

J

In all cases the exceptional set J is independent of the choice of the finite points ak E'Cn and of their number. R = +w and if f is of finite order, then the right hand (Remark: If side of (15. 10) is O(log r) by definition of the order; if 0 < R < +w and if f is of finite order, then the right hand side of (15. 12) is

O{log Rlr}

by definition of the order in this case) .

Cn

Proof. We choose p > 2 (v = 1,...,p) and put

distinct finite points

av

=

(av,...,an)

E

1

F(z)

_

pp

L. IIf(z) - av v=1

From the inequality log (x1x2) we get for

log x1

+

1IgF(re')d j 0

(15.14)

0 < r < R

2n

2,r

(xl,x2 ? 0)

log x2

2 Tr

m(r,O,f') +

log{F(re'o) Ijf'(re1')jj1d

r

.

(15.15)

j

0 115

Put =

6

min I I al - a1 1

1

ixJ

p E (1,2 , ... ,p}

Let for the moment where

IIf(z) - a"II

<

26

be fixed. Then we get in every point

(< 4 since p >- 2)

(15.16)

the inequality IIf(z) - av II

='

6

Ila" - av II - IIf(z) - a" II

>

6

- 2p

36

4

(since p> 2)

. Therefore the set of points on ac r which is determined by (15.16) is either empty or any two such sets for different " have empty intersection. In any case for

vxu

2n

P 1

1ogF(rei$)do

2n

Zn

E u =1

1og F (rel$) d$ J

0

I l f -a"

< 2p

P >

1

1

log

2n

Ilf(rel$ ) - au II IIf-a

P I<

d$

6

2p

Because of 1

log

27r

1

d$ = m(r,a ") -

IIf-a" II IIf-a" 11

IzI =

IzI =

r

m(r,a") -

1

ll f-a" II IIf-a" II' Zp

` 2p

=

116

log

1

2

log

r

d$

it follows that 2 Tr

p 1

rn(r,aµ)

log F (rely) do

27r

-

Plog

u =1 0

so that by (15.15) 2 Tr

P m(r,a}')

m(r,0,f')

-

log

2n

u =1 0

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) with a = 0 , and gives f (z)

Since

T(r,f')

G(r) + N(r,0,f') + m(r,0,f')

=

+

log c'

opment

f'(z)

zc

=

zR,+1cR +

1

+ .

.

,

.

(ci, x

0)

Using this we have P

T(r,f')

G(r) + N(r,0,f') +

?

m(r,aµ) u =1

2Tr r

1

plog d +

2 Tr

0

On the other hand, using (15.14),

T(r,f')

=

m(r,f')

+

N(r,f') 27 r

m(r,f')

+

N(r,f')

12Tr

log

+

IIf'(rel9)II

do

I

0

117

We now introduce the function N1(r)

N(r,O,f')

=

+

2N(r,f)

N(r,f')

-

(15.17)

consists of two components; the first, N(r,0,f') , characterizes the distribution of zeros of f'(z) , i.e. the multiple points in which the function f(z) assumes a finite vector; the second term, 2N(r,f) - N(r,f') refers to the multiple poles of the function f (z) . N 1(r) will be called the "g e n -

N1(r)

eralized counting function of multiple points" of

since it can be written in the form-

f(z)

r

F

n1(0)logr

N1(r)

n1(t) - n1(0) dt

+

,

(15.18)

t

0

is obviously the numn1(t) = n(t,0,f') + Zn(t,f) - n(t,f') I z I s t , each a ber of all multiple finite or infinite a - points of f in point counted with its multiplicity reduced by one. Introducing N1 we get, where

T(r,f')

omitting

,

p

G(r)

+ N1(r) - 2N(r,f)

m(r,ap)

+

+

N(r,f')

p =1 27r r

1

d4

2,r

logllc',11 - plog?P

+

0

2n r

m(r,f)

N(r,f')

+

+

+

1

IIf'(rei1)II

log

2,r

d4,

II f (rein) II 0

We now put we obtain

ap+1

=

m

.

Observing that

N(r,f)

p+1

G(r)

+

I

k=1

118

m(r,ak)

+

N1(r) - 2T(r,f)

=

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

,

2 71

r

1

log Ile', 1

2ir

- p log 2P

0 2Tr r

llf'(reio)II

+

1

106

2n 0

and hence the inequality p+1

G(r)

I m(r,ak)

+

+

2T(r,f) + S1(r)

N1(r)

(15.19)

,

k=1 where 27r

S 1(r)

=

i log Ilf'(re

2

IIf(re

27T

d+

)II

lg{F(re)Ilf'(re)II)d

271

)II

0

0

+ plog

6

-

(15.20)

Using the generalized first main theorem (§3, Theorem 3.2)

T(r,f) for

=

V(r,ak)

k = 1 , ... ,p

,

+

N(r,ak)

+

m(r,ak) + log IIc.,(ak)II + c(r,ak)

m(r, -) = T(r,f) - N(r,m)

and using that

,

V(r,m) =0,

inequality (15. 19) can alternatively be written as p+l

(p-1)T(r,f) + G(r) + N1(r)

_

=

[V(r,ak) + N(r,ak)) + S1(r) + 0(1)

k=1 (15. 21) with

p

0(1)

(log IIcg.(ak)II +

_

e(r,ak))

k=1

We will now try to find an estimate of the function

S1(r)

given in (15.20). 119

Note first that an application of the inequality n

n

log X

x.

log x

log n

+

.

(x.>0)

,

(15.22)

]

J

j=1

j=1

gives 2,r

2-T

r 1

p

log{F(rel") IIf -(rel') II)do

2n

5

log

2,r

k=1

j

j

i IIf(re

)

- a

kII

do + log p

0

0 so that

2,r

2 7r

+

S1(r)

log

5

II f' (rein ) II

IIf'(reio)I) log

+

If(reil)I!

k=1

I 0

0

+ log p

+ p log 6 -

log IIci.

do II f (re 10)

ak

(15.23)

I

We now need the following important Theorem of Nevanlinna theory.

Suppose that the complex scalar valued function 4(z) is is finite and not zero, then for all r c0 = 4(0) meromorphic in CR . If and the inequality s (0 < r < s < R) Theorem 15.2t:

m(r,

)

24 + 3log

<

1

CO

+

2log

r1

+

1

4log s + 31og s-r + 4logT(s, (15.24)

is valid. If 4(0) = 0 the form

, then

or

C

4(Z)

=

K z

+

K

tNevanlinna (27] p.61 120

4(z)

has around the origin a development of

CK+1zK+1 + ...

,

x

( c

K

0)

.

(15.25)

In this case (15.24) is to be replaced by 34

<

5logl K I

+

1

c

7log r1 + 41og s

+

K

1 s-r + 41og T(s, c)

3109

+

31og

+

(15.26)

Since (15.26) is also valid in the first case, where (15.25) holds with K = 0 we will not apply (15.24) but only (15.26), which holds in the most general case. Substituting

(z)

f.(z) - a.

=

(a. E C)

in (15.26) and using the inequality

T(s,f -a .) i

T(s,f .) + logla .I + log 2

_

]

3

1

we obtain

f.(z) - ai

Lemma 15.3: Assume that

admits around the origin the devel-

opment

f.(z) - a. 3

r and

-

f: m(r, f a

L

z

3

+

3K.

3

then for all

K.+1

K.

c.

=

34

z

+

.

.

,

the inequality

(0 < r < s < R) + 510glK .I

31ogs-r1 +

0)

(c. 3K3

3log

+

3

+

.

3K3+I

s

<

c.

1

+

C.

3K.

41ogJa.

+

7logr1 + 4logs

4log 2 + 41ogT(s,f) 3

1

(15.27)

is valid.

From inequality (15.22) we conclude that 2,r 1

log

2n

27T

IIf" (reel) I I f (rely

0

)

-al

d

I

1

log

4n

l

1-11

...

1-ni

fl all2+...+Ifn-anl2

d

0

121

2n r

f.

f.

-aJ .

I

J

f:

1

2,r

f. - a. J

d

2 log n

+

J

0

I

m(r,f a) J

where which

1logn

+

J

denotes summation over those indices j E {1 , ... ,n) , for const.. Using ( 15.27) we obtain for a = (al, ... an) E Cn and

f.

0
r 1

If"(re

log

2n

)II

d

all 0

n {34 + 7log r + 4log s

<

{51ogI

+

K

.I

+

3log sl

3log

J

Using the inequalities

+

+

c 3K.

T(r,f.)

r

+

4log 2 }

+

2log n

4lo g!a.I + 41ogT(s,f J

J

T(r,f) + 0(1)

we obtain

2n f

1

If'(rei ) II

log

2n

II f

(rein) -aII

d

0

<

1

const. + n{71og r1 + 4log s + 3log s-r

+

4logT(s,f)}

,

where "const." is a number, which depends only on the development of 122

(15.28)

f(z)

a = (a,, ... ail) . We now consider the case R = - and distinguish two suhcases.

at the origin and on

First let f (z) and have for large r

be of finite order

(i)

T(s,f)

s

<

p + E:

p

.

In this case we put

s = 2r

rp+£

2p+c

and with (15.28) 2 -n

r

If"(reio)II

+

1

log

2 it

ail 0

0(1) + '4nlogr

<

+

4n(f) + c)Iogr

=

O(log r)

(15.29)

f(z) 4s of infinite order we can use the Lemma of Boreli accord ing to which the increasing function T(r) = T(r,f) satisfies the inequality If

(ii)

I

T(r +

log T(r)

)

(T(r))k

<

(k > 1)

(15.30)

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 valid. We will apply the Lemma of Borel with s

=

k=2

and

1

r

log T(r)

then we have on the ordinary segments log T(s)

and, if

r>1

2log T(r)

<

,

T(r) > e, then 1

log s

log r

logs-r1 =

log log T(r)

+

log( I +

_<-

rlogT(r)

)

<

log r + log 2

logT(r)

and hence finally rNevanlinna [46] gives a proof on p.57. 123

2,r r +

1

lob

2r

,

all

(15.31)

0

0(1)

<

4nlogr

+

11nlogT(r,f)

+

=

O{logT(r,f)} + O(logr)

excluding the extraordinary intervals. Note that in this analysis the exceptional intervals do not depend on a e Cn . This proves part a) of Lemma 1 5 . 4

:

be a non-constant meromorphic

CR

R = + , then

If

(a)

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

Let

vector function in 27 r

IIf'(reio)II

+

1

log

2,r

II

f(re i

d4 )

O{logT(r,f)} + O(logr)

=

,

(a E C n )

- all (15.32)

0

in the case of infinite order, with perhaps,the exception of a sequence intervals of finite total length

J

of

The exceptional intervals are independent of a . If the order of f(z) is finite, then relation (15.32) is valid without restriction, so that in this case 2 Tr P

If' (rein) II

1

log

2n

IIf(re i. ) - all

d

=

0(logr)

(15.33)

0

without restriction. (b)

If

0
O{logT(r,f)} + O{log Rlr}

in the case of infinite order,with perhaps,the exception of a sequence intervals such that 124

(15.34)

J

of

d

I R-r

J

The exceptional intervals are independent of If the order

a

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

p

of

f (z)

1

log R-r

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

be written O {log

(15.35)

R-r -1

In order to prove part b) of Lemma 15.4 we distinguish again two subcases. CR (0 < R < +co) . Then (i) First let f(z) be of finite order p in we have

T(s)

(R - s)-P- c

<

(c >

0)

for any s (0 < s < R) , which is sufficiently near to 2s = R +r so that put

s-r

.

In this case we

R-r

R-s

=

R

2

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

Rlr

}

which proves the second part of b) . (ii) If f(z) is of infinite order in mine s by the condition R2

R2

_

R-s

CR (0 < R < +') , then we deter-

1

log T r,f

RR-r

An application of Borel's inequality (15.30) gives

T(s,f)

2

<

R T(R-s,f)

2

1 T(R-r +logT(r,f)

<

(T(r,f))k 125

log T(s,f)

<

klog T(r,f)

perhaps up to a sequence

J(k)

r

(k > 1)

,

of intervals such that

r

d

dr (R-r) 2

1

R -r

J(k)

J(k)

Further, from

-s-r =

R2

1

R.`

(R-s)(R-r)

log T(r,f)

_

(R-r)2.

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

we see that 1

log - = s-r

1

2log - + 21ogR + log(logT(r,f) + R - r} R-r

O{log Rlr} + 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 (15.19) or (15.21) .

Remark. Using the meaning of N1(r) as an important Corollary the inequality

we conclude from inequality ( 15.8)

q

(q-2) T(r,f) + G(r)

I (V(r,ak) + N(r,ak) 1

<

+

S(r)

(15.36)

k=1

with

r

r

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

n (O,a)logr

N(r,a)

dt

+

(15.37)

,

t 0

where in

n(t,a) I z 15 t

denotes the number of solutions of the equation , each solution counted only once.

f(z) = a

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

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

(15.38)

can not grow faster than T(r,f) + 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 a the sum of expressions ( 15.38) for any 3 different points a E Cn U j_} will grow at least as rapidly as

T(r,f) + G(r,f) + N1(r) except perhaps of a certain exceptional set J of r - intervals which is independent 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 f(z) _ (f1(z),...,fn(z)) is a non-constant meromorphic vector function in C , then we will call "g a n -

eralize d Pic- ard e xception al Cn U { W }

value" any point

aE

for which

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

=

0(logr)

or equivalently for which v(+m,a) .+ n(+co,a) Now assume that f(z) has 3 different generalized Picard exceptional values aI, a2, a3 E Cn u {co} . Then the generalized second main theorem q = 3

with

gives

T(r,f) + G(r,f)

<

O(logr) - N1(r)

+

S(r)

Thus

T(r,f) or dividing by 1

<

O(log r)

<

+

O{logT(r,f))

T (r , f)

O(logr)

O[log T(r,f)}

T(r,f)

T(r,f)

outside the exceptional intervals. Here on the right side the second term tends r -+ +m ; and if f(z) is transcendent, then also the first term to zero as tends to zero. This contradiction proves Corollary 15.5 (Generalized Theorem of Picard) : Let f (z) = (f 1(z) , ... , fn (z)) be a transcendent meromorphic vector function in the plane C . Then 127

there are at most

different points

2

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

a E Cn U {m}

for which

O(log r)

or equivalently for which + n(+W,a)

v(+°°,a)

<

+m

f(z) is entire or has only finitely many poles, then there is at most such point a E Cn .

If 1

Another application of the generalized second fundamental theorem is Corollary 15.6 : Let the non-constant meromorphic vector function f (z) = (f1(z),...,fn(z)) be of finitet order in the plane C . If, for a positive u, the integral

fV(r,a)

+ N(r,a)

r

dr

u +1

(r0 >

(15.39)

0)

r0

converges for

3

different

a E Cn u {W}

,

then the integral

T(r,f) + G(r,f) dr

r u +1 0

converges, so that f(z) is at most of convergence class of order (15.39) converges for every a E Cn u {m}

u

, and

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

r P +l

dr

<

+_

r0

From the last Corollary we obtain the very important

t Nevanlinna [27] ,p.72 128

Corollary 15.7 (Generalized Theorem of Picard-Borel) Let the non-constant meromorphic vector function f (z) = (f 1(z) , ... , fn(z)) be of finite order p in the plane C . Then there are at most 2 different points a E Cn U {W) for which the sum :

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

(or equivalently the sum

v(r,a) + n(r,a))

is of order < p , if f(z) is entire or has only finitely many poles, then there is at most one such point a E Cn The last considerations concerned meromorphic vector functions in the plane. For meromorphic vector functions in a finite disc CR (0 < R < +W) similar corollaries can be proved provided that

T(r,f) log

R)

1

RR-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 a at least one m(r,a), V(r,a), and N(r,a) has the same order of the expressions (type, class) as T(r,f) . By the generalized theorem of Picard-Borel there exist at most 2 different points, for which the sum V(r,a) + N(r,a) is of smaller order (type, class). For any such "generalized Borel exceptional value" a the function m(r,a) must then have the same order as T(r,f). So far we can not say much with respect to the other points a , 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)

T(r,f)

V(r,a)

T(r,f)

N(r,a)

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

T(r,f)

,

T(r,f)

G(r,f)

, and

T(r,f)

In this entire section we assume that 129

T(r,f)

lim

r+R

_

(16.1)

+W

this is no restriction since in this case (16. 1) only is non-constant. (16.1) and the generalized first main the-

R = +-

In the case

means that f (z) orem show that V(r,a)

+ m(r,a)

+ N(r,a)

urn

T(r,f)

r-*R

from this we get _m V(r,a) + N(r,a) li T(r,f) r->R

m(r,a) 1

<

lim

+

r+R T(r,f)

<

1

so that the sum in the middle must be equal to 1 . For any 6 (a) = 6 (a,!) we define the number by putting

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

m(r,a) 6 (a)

6 (a, f)

=

=

lira

=

1

a E Cn U {W}

- lim

r-'R

r7->_R T(r,f)

T(r,f) (16.2)

Then we have 0

a

6(a)

0

1

further,

m(r,f) 6 (co)

lira_

lira

=

=

r-' R T(r,f)

1

-

N(r,f) (16.3)

T(r,f)

The quotient (16.2) has already been considered in the Propositions 14.15 and 14.16.

As on p. 126 we denote by n(t,a) = n(t,a,f) the number of distinct a - points of f (z) in I z I <_ t and again we use the counting function r

N(r,a)

N(r,a,f)

=

=

We put further 130

N(r,-)

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

_

t 0

N(r,f)

r

dt + n(0,a)logr

,

(16.4)

8(a)

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

6(a,f)

=

lim

=

T(r,f)

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

(16.5)

T(r,f)

and

N(r,a) - N(r,a) 0(a)

0(a,f)

-

(16.6)

lim

=

r-R

T(r,f)

then we have in particular

N(r,f) lim

(16.7)

,

r-j,R T(r,f)

N(r,f) - N(r,f) (16.8)

0

=

Given to

R

8(a)

<

1

0

,

<

6(a)

1

.

e > 0 , we have from (16.6) and (16.2) for

r

sufficiently close

,

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

>

(0(a) - c)T(r,f)

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

<

(1 - 6(a) + c)T(r,f)

so that

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

<

(1 - 6 (a) - 0(a) + 2c)T(r,f)

and consequently 8(a)

>

6(a)

+

(16.9)

0(a)

As in Nevanlinna theory (i.e. n=1, Q z) = f 1(z)) by definition (16.2) the quantity 6 (a) can be positve only if the asymptotic mean approximation to the point a of the values of f on circles about zero with increasing radii is relatively strong. Equivalently, 6 (a) is positive only if the growth of the V(r,a) + N(r,a) is deficient in the sense that it is relatively slow in sum Since points a with comparison with the growth of T(r,f) 6 (a) > 0 are clearly exceptional, the number 6 (a) will be called the "d e f i c i e n .

131

y'

of the quotient V(r,a) + N(r,a)

T(r,f) or simply of the point a , quite analogous to Nevanlinna theory. Points 6 (a) > 0 with will be called " d e f i c i e n t ". The quantity 6(a)

will be called the "index of multiplicity "of

a

since

a

6(a)

is positive only if there are relatively many multiple a - points of f ; here by a multiple a - point we understand a point such that the system of equations f(z) = a has multiple roots; these roots are zeros or poles of f'(z) and are thus countable in number. 6(a) will attain its maximum 1 if the relative density of multiple roots is large, and if their orders of multiplicity are unbounded in the vicinity of z = m . For such a point with 6(a) = 1 the deficiency 6 (a) must vanish since 6 (a)

+

6(a)

6

1

Remark. If we define as in Nevanlinna theory the number setting nI(t,a)

=

n(t,a)

-

n1(t,a)

by

n(t,a)

is the number of multiple solutions in jz I < t of the equation f (z) = a , where a solution of multiplicity v is counted only (v - 1)times. We can then introduce the " counting function of multiple a -points " N1(r,a) by putting r n I(t, a)

r

N1(r,a) = N(r,a) - N(r,a) =

nI(0,a)logr

n1(t,a) - n1(0,a) dt

+ t

0

for fixed

r there are only finitely many

we have N1(r)

Y NI(r,a) a E C n U {m}

so that

132

a

for which

NI(r,a)

0

, and

Nr) 1(

Nr,a) 1( lim

lim

r+R T(r,f)

rr--RT(r,f)

aECnU{W}

aECnU{oo}

Now, as an essentially new ingredient,as compared to Nevanlinna theory, we introduce the quantity 6G

=

dG(f)

G(r,f) lim

=

(16.10)

r+R T(r,f)

This non-negative number measures the relative growth as volume of the image of a disc C r under the Gauss map ti

CR >

r+R

of the

Pn-1

compared to the growth of the characteristic T(r,f) . As we have shown before the characteristic T(r,f) itself measures the volume of the image of a disc Cr as r + R under the map f into Cn , where Cn is e quipped with the Fubini-Study metric. 8G will be called the the G a u s s min or Ricci - index" of f(z) . d G 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 if

S(r

)

(16.12)

T(rv,f)

S(r) plays the role of an . It is only then that for a sequence r v + R unimportant error term. Therefore we shall call the function f admissiblet (for the generalized deficiency relation) if (16.12) holds. f is certainly admissible if either R = - , or in case 0 < R < +m , if .

T(r ,f) V

-

+m

(16.13)

lo

holds for a sequence

rv + R

outside the exceptional set of the second

tHayman [16],p.42 133

main theorem. This can be seen as follows: Suppose first that is a rational vector function, then IIf'(z)II 0

->

,

as

0

R = +W

z

.

If

f

> +W

IIf(z) - ak11

I f(z)II

so that in (15.20) S1(r) = 0(1) ; this shows that (16.12) holds trivially in this case. If f is not rational, then

T(r,f) log r

- +w

r -> +-

as

0 < R < +o and so that (16.12) follows in this case at once from (15.10) . If 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 16.1 (Generalized Nevanlinna Deficiency Relation): Let the meromorphic vector function f (z) = (f 1(z) , ... , fn(z)) be admissible. Then the set {a E Cn u {co} , 8(a) > 0} is at most countable and summing over all such points we have 2] [6(a) + 0(a)]

+

6

1 8(a)

<_

G

a

+

6

<

2

G

a (16. 14)

here the quantities 8(a), 8(a), 0(a) and (16.5), (16.6) and (16.10), respectively. Proof. with

q+1

We choose a sequence

different points

{r

v

}

6

G

are defined by (16.2),

satisfying (16.12) and apply (15.8)

al,.. ,aq E Cn ,

aq+1

q

(q-1)T(rv,f) + G(rv)

Y [V(rv,ak) + N(rv,akI + N(rv,f) k=1

- N1(r) + S(r)

134

Observing that N1(r)

N(r,0,f')

=

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

+

and using (16.12) we get (q-l+ o(1))T(rv. f)

q I [V(rv,ak

+ G(r

+ N(r ,ak)] v

k=1

+ N(rv,f) - N(rv,0,f') ak - point of

of multiplicity u is also a zero of order pi-1 of f' and so contributes only 1 to n(t,ak) - n(t,O,f') . Thus we may Now an

f

rewrite the last inequality as + G(r

(q-l+o(1))T(rv,f)

)

q Y (V(rv,ak)

<

v

+

FN (r

v

,ak)I

k=1 + N(r

where

than

Na(r,o,f')

f)

v

- N (r o

v

,o,f')

f' which occur at points other Dividing by T(rv,f) and ignoring

refers to those zeros of

ak - points of

(k = 1,...,q)

f

.

this latter term we deduce that

V(r,ak)

q

Y lim k=1

+

N(r,ak) +

T(r,f)

r-*R

r-*R T(r,f)

q- 1+ lim

lim

r+R T(r,f)

q

1

lim

_ N(r,f)

X [V(r,ak)

I

+

N(r,akN(r,f)

k=1

G(r,f)

r-*R T(r,f)

i.e. q

1 [1 -

e(ak

+

1

-

©(m)

q-

1

+

6

G

k=1 or

135

q

Y O(ak)

+

+

O(co)

6G

2

k=1

This inequality shows that O(a) > for at most 2R - 1 distinct finite points a . Thus the points for which O(a) > 0 may be arranged in a sequence, in order of decreasing O(a) , by taking first those points for O(a) = 1 , then those, if any, for which e(a) > Z , then those of which the remainder for which etc.. If {a(R) } (R = 1, 2.... ) is E) (a) > the resulting sequence and putting 3 a(0) = - , we deduce that q

I O(a(R))

`

6G

+

2

R =0

for any finite clude that

q

, and hence if the sequence

+

6(a(2))

6 G

{a(2.) }

is infinite we con -

2

R=0

This proves Theorem 16.1. Using the definition of e(-) relation (16.14) can be written O(a)

+

1

aECn

-

the right hand inequality of the deficiency

R(r,f) lim

r-+R T(r,f)

+

2

6G

where now the sum is to be taken only over any set of > 0 . This gives with (16.9)

I [6(a) + 0(a)] + 6G aEC

n

<

Y O(a) aEC

n

+

a E Cn

6G

with

- 1 +lim

O(a)

R(r,f) .

r->R T(r,f) (16.15)

We deduce in particular

Corollary 16.2: If under the conditions of Theorem 16.1 the meromorphic vec136

for function

_

has no poles in

f

CR

or more generally, if only

N (r, f)

lira

then the number be replaced by i.e.

(16.16)

0

r-R T(r,f) 2 1

on the right side of the deficiency relation (16.14) can provided that the sums are extended over finite a only,

[6(a) + 6(a)]

6G

+

aECn

e(a)

+

6

(16.17)

G

aECn

For example (16.17) is always satisfied for an entire vector function. From (16.15) we see further Corollary 16. 3 in

CR

If

:

f (z) is an admissible vector valued meromorphic function

, then the index of the Gauss map satisfies the inequality

N(r,f) G

0

<

1

lim

+

(16.18)

r->R T(r,f)

In particular, 0

if

f(z) 0

if

<

f (z)

6G

o

2

is meromorphic; and <

6G

a

1

is holomorp hic .

In the sum

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

+

N(r,a)

the terms m(r,a) and V(r,a) + N(r,a) behave for fixed tically very differently as r + R . For the "normal" points convergence of f(z) to a , which is measured by m(r,a) weak, so that the deficiency d (a) will vanish, i.e.

a

asymptothe mean

,

is relatively

a

137

m(r,a) lira

0

T(r,f)

whereas

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

T(r,f)

r-'R

is positive, then the growth of is relatively V(r,a) + N(r,a) weak; in general the number d (a) is a measure for the growth of the sum If

d (a)

V(r,a) + N(r,a) If in particular for a transcendent meromorphic vector function in C the point a is a generalized Picard exceptional value, then the deficiency d (a) attains its maximum value 1 since logr = o(T(r,f)) . Therefore Theorem 16.1 contains the generalized theorem of Picard (Corollary 15.5), according to which there are at most 2 distinct such values a . On the other hand, for a meromorphic vector function in C the equality 6 (a) = 1 which is equivalent to

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

=

0

T(r,f)

r-++W

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 2 distinct points a for which 2 - 6G d (a)

>

3

and in particular at most 2 points with Corollary 16.2 we see that there is at most d (a)

6 (a) 1

>

;

3

finite point

analogously from

such that

a

1-dG >

2

and in particular at most 1 One can pose the general

138

finite point such that

d (a)

>

1 2

{a Given sequences (a 6 G > 0 , such that and a number

Problem

:

<

0

6k + eP

E Cn U {co})

1 (6k + ek)

I

<

P.

=

6

k

0(a k)

,

=

ek

,

{6

+

6G

,

{89, - 01

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

is there a vector valued meromorphic function 6 (aZ)

R = 01

,

6 (a) = 0(a) = 0

and

for

a¢

with {ak

}

,

6G the Ricci-index of f (z) ? In scalar NevanIinna theory this deep problem, of which only partial solutions have been known earlier, has recently been given a positive answert.

and with

We now introduce an important new concept, which has no significant coun-

terpart in scalar Nevanlinna theory, by setting 6V(a)

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

V(r,a) =

:

1

- lira

r-R T(r,f)

lim r:->-R

.

T(r,f)

(16.19)

In view of the first main theorem we have always 0

V(a)

1

and in particular 6V(a') = 1 In the case of scalar Nevanlinna theory 6V(a) = I for any a E C u {-} so that in(n=1) it is obvious that deed the quantity 6V(a) is of no significance in this case. In general , 6V(a) is positive only if the volume function V(r,a) grows more slowly than the maximum possible growth permitted by the first main theorem. For this reason 6 V(a) will be called the "v o 1 u m e d e f i c i e n c y" of the point a with respect to f , and a point such that 6V(a)

>

0

will be called "volume deficient" V. A first application of this notion will be given in the following. We shall say that a point a9 E Cn U {m} has multiplicity at least mk if all roots of k the equation f (z) = a have multiplicity at least m k . We have

tDrasin [181 139

1

(r,aR)

mR

+ 0(1) 5 1mR[T(r,f) - V(r,aZ)] + 0(1)

N(r,a

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

m [T(r,f) + (m1)V(r,aR)] + 0(1)

5

R

and if

T(r,f)

is unbounded

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

li urn

T(r,f)

r+R

+

1

mR

(1 -

1 ) lim

V(r,a

mR r-R T(r,f)

so that

(1 - m )6(a V R)

19(aR)

R

If

is admissible the deficiency relation (16.14) shows that

f(z)

(1-- )6V(a2)

+

SG

<

2

(16.20)

R

"completely multiple" Let us call a point aR E for f(z) if the equation f(z) = aR has only multiple roots, i.e. aR has multiplicity at least mR = 2 . For such a point we have 1 - mR 1

?

1

2

and we conclude from (16.20) Corollary 16.4 Let f(z) be an admissible meromorphic vector function in CR . Then given p (0 < n < 1) , there are at most :

4-26G (16.21) Ti

distinct points (i)

(ii)

a

a E Cn u {m}

is completely multiple for 6 V(a)

In particular, if

a

f( z) ;

n

6G=2

conditions of Corollary 16.4 . 140

such that the following conditions hold both:

, then there can be no point which satisfies the

f(z) is entire then we see from (16.15) that instead of inequality(16.20) we can write If

(1

V(a

)

G

+

5

1

where now the sum is extended over finite aR only. In Corollary 16.4, if f (z) is entire, the bound (16.21) can be replaced by 2

- 26G n

if only finite completely multiple points are considered; if no such points in this case.

§17

6G=1

,

there are

Further results about deficiencies

In the first part of this section we compute the quantities 6 (a)

,

®(a)

,

8(a)

and

6 V(a)

of §16 for the instructive example of rational vector functions .. .fn(z)) In this case we know from §4 that f (z) has degree *(-)

f(z) = (f1(z),

+ n(+-,m)

and that the following formulas are valid: *

T(r,f)

=

m(r,f) + N(r,f) = n(w) logy + 0(1)

m(r,f)

=

*(m) logr + 0(1)

N(r,f)

=

n(+oo,oo)logr

m(r,a)

_

+

0(1)

*(a) logr + 0(1)

(a E Cn U {m})

From this we compute v(+m,a) + n(+m,a)

*(a)

6(a)

1

-

(17.1)

*

so that in particular

141

6(m

n(co)

a

=

a

6 (a) = 0

the generalized deficiency

We see that in C" U {m} finite or infinite point

up to the

lim f(z)

=

Z-+ w

if, and only if one of the component functions (a = finity) . Since as r -s +oo

f.

has a pole at in-

N(r,a)

n(+cc,a)

N(r, a)

n( +-, a)

T(r,f)

n(oo)

T(r,f)

n(-)

we have

n(+-,a) - n(+`",a) e(a) n(co)

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

m(r,a)

+ V(r,a)

+ 0(1)

T(r,f)

=

can be written

*(a)logr + n(+co,a)logr + v(+co,a)logr

*

=

n(oo)logr + 0(1)

so that we have the relation *(a)

*( )

+ n(+co,a) + v(+m,a)

+ n(+cc,co)

of Proposition 4.2. We compute further, v(+co,a)

6V(a)

=

1-

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

n( )

n(o,)

=

6(a) +

n(+co,a) (17.3)

n(-)

and

e(a)

=

1-

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

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

n(oo)

n(-) =

142

6 (a)

+

6(a)

(17.4)

a*

In particular we have for

a

8(a)

dV(a)

so that for d (a)

- {a} - f(-C)

a E Cn u {00} =

0(a)

d V(a)

=

0(a)

=

8(a)

=

=

0

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

_

=

n(+-,a)

(a) V

_

8(a)

*(00)

1

=

d (a)

What

=

does

*(a)

(17.5)

1

(17.6)

0

=

+ n(+co,a)

(a E Cn) , and of course

V

(m) = 1

;

(17.7)

_

+

0(a)

aEC -f(

0(a)

d

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

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

-

=

We note that for d (a)

=

-

1

so that d

0(W)

,

v(+o,a)

6V(a)

d (W)

,

n(+-,a)

n(+-,a)

0(a)

(a E Cn)

0

d V(a)

=

=

(a E Cn)

0(a)

= m

,

8(W) = 1

(17.8)

C)

e(a)

=

0

the generalized deficiency relation tell us

(6(a)

+

0(a))

+

dG

<

2

a e C n U {«} if

f(z) is a non-constant rational vector function? For rational non-constant vector functions this becomes 143

n(+-,a) - n(+°°,a)

E

6G

+

a

2

,

(17.9)

n(W)

a ECnu{W} n(+-,a)> n(+-,a)

where as above 0 a

E Cu{'} nz-a.

lim f(z)

=

For an entire rational vector function

f(z)

this simplifies to

n(+r,a) - n(+°°,a)

X

6G

+

(17.10)

1

aECn n(+m,a)> n(+o,a) Example 1. As an example we consider the entire rational vector function (z3

f(z)

_

z5)

,

5 3 Here every point a E C2 not of the form z0 cc , is a = (z0 , z0) z0 * 0 not assumed. In every point the function f(z) has an a = 3 5 (z0 , z0) - point of multiplicity 1 , and in z0 = 0 it has an a = (0, 0) point of multiplicity 3 . Let us first examine the deficiency relation (17. 10) We have *(o) = 5 n(r,a) = n(r.a) up to the point a = (0,0) where n(+',a) = 3 n(+m,a) = 1 so that the deficiency relation ,

,

(17.10) is

2

5

+

d

i.e.

1

G

On the other hand we can compute

f'(z)

(3z2

_

N(r,0,f')

=

,

,

m(r,0,f') = o(1)

G(r,f) = V(r,0,f') = 2logr + O(1) On the other hand,

144

=

3 5

directly as follows. From

we obtain using the first main theorem,

T(r,f)

G

T(r,f') = 4logr + O(1)

5z4)

2logr

dG

d

5logr + 0(1)

so that in fact

aG

For completeness let us compute the other equidistribution quantities for this example. We have T(r,f) = m(r,f) = 5logr + 0(1) , so that of d p(-) = 1 . For finite a we distinguish the following course cases. ae

(i) Here

V(r,a)

a 4 (z03

CZ

,

5

zQ)

m(r,a) = o(1) , N(r,a) = = 5logr + 0(1) , so that b V(a)

(ii)

=

a

=

0

=

(z0

,

6(a)

0

d (a)

=

(z0 E C)

,

, and by the first main theorem =

0(a)

z0) * (0.0)

Here m(r,a) = 0(1) , N(r,a) = log r + theorem V(r,a) = 4logr + O(1) so that

V(a) O(a)

(iii) Here

1

=

4

-

4 + 1 =

, and by the first main

5 =

1

6(a)

=

0

0(a)

=

5

a

(0,0)

=

m(r,a)

=

V(a)

1-

=

=

1

n(0,a) = 3 , N(r,a) = 3logr V(r, a) = 2logr + 0(1) , so that

0(1)

first main theorem

5 (a)

=

0(1)

and by the

,

2

3

=

5

©(a)

1-

=

5

2+3

_

5

0

,

0(a)

=

2

1

2

=

5

3-1 5

=

2 5

As compared to scalar Nevanlinna theory, where we have to consider the value distribution quantities 6(a)

,

0(a)

,

©(a)

we have in the vector valued theory the additional quantities 6V(a) and SG. 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. For the point a we have from the definitions and from V(r,°°) = 0 6(W)

=

- lim

1

r-+R

p(er)

- lim

1

=

N(r,f) T(r,f) N(r,f)

r4R T(r,f)

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

6V(-)

e(W)

1

=

lim r-+R

T(r,f)

and f (z) = (f 1(z) , ... , fn (z)) be a meromorphic vector Let Proposition 17.1 function in CR . Then the following conclusions hold.

N(r,f)

N(r,f) 1

lira

0

=

lim

,

G(-)

1

=

e(-)

,

©(°°)

lira

1

=

e(m)

=

r+R T(r,f) 1

lim r-+R

.

r-+R T(r,f) =

0(m)

,

1

T(r,f) =0

lim

,

=

1

N(r,f)

N(r,f) 0

5

0

lim

0

=

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

=

N(r,f)

N(r,f) (ii)

0

=

r-+R T(r,f)

r-+R T(r,f)

lira

,

=1

r-+R T(r,f)

r-+R T(r,f)

N(r,f) (iv)

(v)

146

6(Cc)

e(o°)

0

=

lira

r-+R T(r,f) =

0

_ N(r,f) lim

r-+R T(r,f)

=

1

,

1 (co)

=

0

0.

_ N(r,a)

N (r,a) (vi)

O(W)

lira

0

=

urn

r+R T(r,f)

r+R T(r,f)

We also note

Proposition 17.2 Let f(z) = (f1(z),...,f(z)) be a non-constant meromorphic vector function in CR . Then the following inequalities are valid for a E CnU{-} :

_ N(r,a)

N(r,a) - 6 (a)

S V(a)

lim

rr}R T(r,f)

N(r,a) lim

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

SV(a) - 0(a)

5

r+R T(r,f)

lim

5

lim

(17.12)

r+R T(r,f)

Proof of (17.11). m(r,a)

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

SV(a) - S (a)

=

lim

lira

T(r,f)

r+R

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

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

lim

T(r,f)

r+R

urn

+

r+R

T(r,f)

The right side is

N(r,a) lira

r+R T(r,f)

N(r,a)

and

?

lim

r+R T(r,f)

This shows (17.11) ; (17. 12) is shown analogously.

From Proposition 17.2, from the inequality the definitions we deduce Proposition 17.3:

Let

morphic vector function in

8(a) + S (a)

CR

S (a)

=

1

or from

. Then the following conclusions hold.

V(r,a) (i)

0(a)

be a non-constant mero-

(f1(z),...,fn(z))

f(z)

5

N(r,a)

lim

=

r+R T(r,f) S V(a)

=

1

,

0

lira

,

0(a)

=

0

r+R T(r,f) =

1. ,

©(a)

=

0

147

(ii)

N(r,a) 6 (a)

N(r,a)

lira

0

=

6 V(a)

5

r-*R T(r,f)

<

F1 -m

r-*R T(r,f)

N(r,a) (iii)

lim

=

6V(a)

0

r-*R T(r,f)

6(a)

=

6(a)

=

0(a)

=

V(r,a) (iv)

(3(a)

=

.

N(r,a)

lira

1

0

0

=

lira

,

r->R T(r,f)

_ N(r,a)

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

N(r,a)

` 1-lim r-*R T(r,f)

6(a)

_<<

6V(a)

N(r,a) (v)

6(a)

=

r-*R T(r,f) (vi)

lim

6 V(a)

r+R T(r,f)

1

=

.

N(r,a) 6 V(a)

lim

0

0

=

r-*R T(r,f)

lim

5

r}R T(r,f)

H(a)

=

N(r,a) 0(a)

(vii)

0(a)

=

O(a)

1

lim

=

T(r,f) V(r,a)

=

1

lira

,

=

0

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

=

r--R T(r,f)

(viii)

6V(a)

,

6V (a)

=

0(a)

=

1

0

,

.

V(r,a) =

lira

1

r-R T(r,f) N(r,a) 6 (a)

48

0

=

0

_ N(r,a)

6(a) = 1-lim r-*RT(r,f)

.

(ix)

6 V(a)

=

d (a)

0

0

=

U(a)

,

=

0

N(r,a) (3(a)

=

0

lim

,

T(r,f)

=

0

In §4 we proved for rational vector functions that the limit

V(r,a) lim

lim

=

r++'

log r

r-;+m

v(r,a)

v(+W,a)

exists. The same statement and proof holds for arbitrary meromorphic vector functions the only difference being that the limit v(+W,a) may now be 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 +

N(r,a)

lim n(r,a) r++W

lim r+ +W

log r

:

=

n(+m,a)

Using this remark and the fact that

T(r,f)

-

log r

+W

(as

r + +W)

for any transcendent meromorphic vector function in the plane, we obtain the next 2 propositions. Proposition 17.4

Let

:

f(z) _ (f1(z),...,fn(z))

be a non-constant meroa E Cn U{-) and assume that 1

morphic vector function in the plane. Let of the following 2 conditions is satisfied: (i)

(ii)

V(r,a) lim

0

r++m T(r,f)

is transcendent and

f(z)

v(+m,a)

<

Then it follows that d V(a) = 1

,

d (a)

N(r,a) 1 - lira r++mT(r,f)

,

((a)

N(r,a) =

1 - lim

r++WT(r,f)

Remark. The assumptions of Proposition 17.4 are satisfied in the Nevanlinna 149

a E Cu(}

for all

n=1

case

Proposition 17. 5

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

Let

:

morphic vector function in the plane. Let of the following 2 conditions is satisfied:

aEC

be a non-constant meroand assume that 1

n U {m}

N(r,a) (i)

lim

=

0

r±+W T(r,f)

is transcendent and

f(z)

(ii)

n(+w,a) < +oo

Then it follows that 6 (a)

6 V(a)

=

=

0(a)

8(a)

,

=

0

The last Proposition shows that in many cases the deficiency 6 (a) is equal to the volume deficiency 6V(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(logr) so that the point a is an exceptional point in the generalized sense of Picard; in this case it follows that 6V(a) = 6(a) = 0(a) = 1 ,

We now introduce" lower" and" upper" " c o u n t i n g 6N

'

6N

AN

'

AN

by putting

N(r,a) 6N(a)

_

6(a) N

=

lim

T(r,f)

AN(a)

=

0 N-(a)

=

r-+R T(r,f)

r-*R T(r,f)

Definition 17.6

150

lim

N(r,a) ,

and we introduce the following subsets of

E(6)

_ N(r,a)

N(r,a) lim

=

d e f i c i e n c i e s"

lim

r-R T(r,f)

Cn

:

{a E Cn, 6(a) > 0}

E(e)

_

{a E Cn, 0(a) > 0}

E(6v)

=

{a ECn, 6V(a) > 0)

,

E(c5N)

{a ECn, 6N(a) > 0}

E(AN)

_

{a E Cn, AN(a) > 01

,

E((S N)

{a C Cn, 6N(a) > 0)

{a E Cn, AN(a) > 0}

E(AN)

E(0)

{a E Cn, 0(a) > 0)

_

f(z) is an entire rational vector function, then we see E(6 v) _ E(6) = 0 from the investigation at the beginning of this § that f(_C) = E(6N) = E(6N) = E(AN) = E(AN) If for example

,

For any f in the scalar case n = I From Proposition 17.2 we see that

we have

E( 6 V) = Cn

E(6N) uE(6)

C

E(6V)

C

E(AN) uE(6)

E(5N) u E(O)

c

E(6V)

C

E(AR) U E(())

n>1

If

then

,

E(6N)

,

E(6N)

(17.13)

E(AN)

,

,

E(AR)

,

E(0)

have

2n - dimensional Lebesgue measure zero since these sets are subsets of f( C). If f (z) is admissible and if n 1 , then by the generalized deficiency relation E(6) , E(0) and E(e) can be at most countable. From (17.13) and the deficiency relation we deduce is admissible and if E(6V) is countable. On the other hand, if f(z) is countable, then E(6N) is countable. Proposition 17.7

If

:

f (z)

is countable, then is admissible and if E(6v)

E(A N)

If condition (ii) of Lemma 17.5 is satisfied, then condition (i) is also satisfied and we deduce from Lemma 17.5 and the deficiency relation Let f (z) be a meromorphic vector function in the plane, Proposition 17.8 Assume that for all a C Cn one of the following 2 conditions is satisfied: :

N(r,a) (i)

r++- T(r,f)

the set

6 (a)

=

E(6 v)

0

is transcendent and

f(z)

(ii) Then

-

lim

6 V(a) =

E(C))

=

O(a)

= E(6)

,

n(+m,a) -

+m

holds for all a ECn is countable, and the deficiency rela-

0(a)

=

0

151

tion can be written in any of the following 3 identical formulations:

N(r,f)

(i)

6V

(a)

+

S

<

G

lira

+

1

r-*+xT(r,f)

aECn

N(r,f)

L

(ii)

6 (a)

+

S

<

G

lira

+

1

r->+WT(r,f)

aECn

N(r,f)

(iii)

8( a )

+

<

cS

1

li m

+

r- +W1lr,t)

aECn

Assumption (i) of this proposition means that there are not too 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 . From the first main theorem we obtain the inequality Remark.

-- =

l_im N(r,a) 1

-

r->RT(r,f)

V(r,a)

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

lim

T(r,f)

r'->_R

?

lim

r+RT(r,f)

+

6(a)

It shows that we must have

V(r,a) lira

r->RT(r,f)

if there are sufficiently many a - points so that the expression N(r,a) 6N(a)

lim

assumes the maximum possible value

1

.

In the case R = +ro we assume in the rest of this are non-constant. It follows then that f n

T(r,f.)

--b

152

as

0 < R < +m

In the case

T(r,fj)

+m

-->

+m

r-*+w

,

§

that all

j = 1,...,n

we assume for simplicity always that as

r-+R

,

j = I, ... n

f 1, ... ,

Using the estimate

T(r,f)

T(r,fj)

?

j = 1,...,n

0(1)

+

and the first main theorem of Nevanlinna theory we obtain the following 2 estimates for the same quantity SN(a) for a = (a1....,an) ECn j = 1,...,n ; k=1, ,n . N(r,ak}

N(r,a)

(i)

SN(a)

=

T(r,fk)

< Jim < Jim rr->R T(r,f) r-R

Jim

r7>RT(r,f)

T(r,f) T(r,fk)

lim

<_

(17.14)

rr->RT(r,fj)

N(r,a)

SN(a)

(ii)

=

lim

N(r,ak)

N(r,a) lim

<

r->RT(r,f)

5

lim

r->RT(r,f.)

r-*RT(r,f.)

T(r,fk) lira

(17.15)

Replacing N by N we obtain the analogous estimates for replacing lim by lim we obtain the analogous estimates for for A -(a) For we obtain a=m

N(r,f)

6N(-)

lim

r-;RT(r,f)

= lim

5 lim

(17.16)

AN(m) and

,

0(a)

,

N(r,ak) - N(r,ak) + 0(1)

< lim r+R

T(r,f) N(r,ak) + 0(1)

T(r,f)

0(a)

=

lim r-->R

o lim

T(r,f)

T(r,fk)

T(r,fk) lim

r-R T(r,f) lim

r+R

(17.17)

rr-;RT(r,f.)

N(r,a) <

T(r,f)

AN(m)

(a (=- Cn)

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

(ii)

and

rr;R T(r,f)

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

0(a)

; and

AN(a)

N(r,fk)

` lim

SN(m) and similar estimates for In the same way we obtain for

(i)

6N(a)

(r, a)

T(r,f.). 153

N(r,ak)

N(r,ak) +0(1)

lim

T(r,f.)

r>R

T(r,fk)

N(r,ak) + 0(1) lim r->R

lim

T(r,f.)

J

r->RT(r,f.) J

J

(17.18)

and

N(r,f) - N(r,f) 0(m)

lim rr+R

=

N(r,fk) - N(r,fk)

? lim

T(r,f)

(17.19)

T(r,f)

r-'R

In particular we see from (17. 18) that 0(a)

0(aj,f.)

<

j = 1,...,n

,

(aECn)

,

For the volume deficiency we obtain for ,(a) = lim

S

m(r,ak) + N(r,ak) <

lim r-+R

T(r,f)

r:-+R

T(r,f) T(r,fk)

T(r,fk) <

lim

=

the estimates

a E Cn

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

(i)

(17.20)

lim

T(r,f)

(17.21)

r>R T(r,f.) J

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

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

(ii)

= lim

d V(a)

<

lim

T(r,f)

T(r,fj)

r->R

m(r,ak) + N(r,ak)

T(r,fk)

lim

lim

r-R For the deficiency

d (a)

d (a)

m(r,ak)

lim

=

f.)

we obtain the following estimates for

m(r,a)

(i)

(17.22)

T(r, fj)

r-RT(r,f)

r ->RT(r,f)

lim

T(r,f)

n .

T(r,fk)

T(r,fk) <

lim

aEC

<

lim

r+RT(r,fj) (17.23)

m(r,a) (ii)

d (a)

=

m(r,a) <

lim

lim

r-+RT(r,f)

i

T(r,fk)

m(r,ak) lim

lim

,

rr-+RT(r, fj) (17.24)

N(r,f)

m(r,f) d (co)

=

lira

r-*RT(r,f) 154

=

1 - lim

r-RT(r,f)

a

N(r,f.) 1 r+RT(r,f)

1 - lim

(17.25)

m(r,fk)

m(r,f)

lim

lim

r-*RT(r,f)

r-->R T(r,f)

(17.26)

In particular we see from (17.24) that d (a)

8 (aj,fj)

<

We have further for (i)

O(a)

j

,

(a E Cn)

,

(17.27)

a E Cn

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

1 - lim

z

= 1, ... n

r-R

T(r,f)

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

=

T(r,f)

r-*R

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

lim

<

T(r,fj)

x:->R

N(r,ak) - N(r,ak) + m(r,ak) +0(1) <

lim

r-R

T(r,f.)

= lim

T(r,fk) (17.28)

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

(ii)

O(a)

=

lim

T(r,f)

r-*R

T(r,fk)

N(r,ak) + 0(1) + m(r,ak) <

]im

r-R

=

T(r,f)

lira

T(r,fk) <

lim

(17.29)

r+RT(r,f.) I

N(r,f) O(oo)

=

1 - lim

O(m)

=

1 - lira

r-RT(r,f)

5

1 - lim -

=

lim

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

N(r,f)

r-*RT(r,f)

(17.30)

r- R T(r,f)

r+R

T(r,f)

N(r,fj) - N(r,fj) + m(r,fj) lim r-->R

T(r,f)

(17.31)

In particular we see from (17.28) that 155

0(a)

j = 1,...,n

0(a.,fj)

<

,

(aECn)

(17.32)

These inequalities show that the relative growth of the component functions f (z) and in particular the number J

T(r,fk) min Iim

(17.33)

j,k

has a very strong influence ton 6(a), 0(a)_, 6V(a), 0(a), 6N(a), (aECnk 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 6(a)

=

6V(a)

provided that

P*

6N(a) = H(a)

=

<

=

(aECn),

0

. We formulate

P

Proposition 17.9 : Let f(z) = (f1(z),...,fn(z)) , (fl,...,fn non-constant) be a meromorphic vector function in the plane such that p * < p . Or let such that f(z) be a meromorphic vector function in CR (0 < R < T(r,f .) - +m r - R (j = 1,...,n) and as J

T(r,fk) min lira

=

(17.34)

0

j,k r->RT(r,fj) Then we have for all 0

=

6(a)

=

Cn (n = 2)

aE

0(a)

=

6V(a)

=

0(a)

=

SN(a)

=

6N(a)

(17.35)

In particular, if

f(z) is admissible in addition, then the deficiency relation of the Gauss map: dG reduces to the following estimate of the index lira_

6G(f)

<

1

N(r,f) (17.36)

+

r->RT(r,f)

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 (17.34). So in the case of rational vector functions all component functions f. are of the same order zero, and J (17.9) or example 1 show that the deficiency relation is , in fact, non-trivial in this case; each quantity dV(a), O(a) and 3(a) is positive for certain In order values of a E C2 , and the deficiency d (a) is zero for a * 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). .

f(z)

Example 2.

(ez

_

,

e2z)

Here each component has order 1 . We put e

11f(z)112

2r cos a

so that

+

e

z = re la

and compute

4r cos a

R

2

m(r,f)

=

log(e2rcosa

1

4n

+

e4rcosa)da

o(1)

+

TI'

2 TI

2

r 1

2r cos a da

2 7T

+

o(1)

2r

_

+ o(1)

n

71

2

Since

we have also

N (r , f) = 0 T(r,f)

=

2r

+ o(1)

11

d(m) = 1 . For

so that the order of f(z) is 1 as it should be, and we distinguish three cases i)

a E C2-{(0,0))

with

a * (e z0

,

e

2z0

)

,

aEC2

(zO E C)

Such a point a is not-assumed by f(z) so that N(r,a) = 0 so that by the first main theorem further m(r,a) = 0(1)

. We have

157

V(r,a)

2r + 0(1)

=

Tr

Thus we obtain

dt,(a)

0

=

O(a)

=

a

(e 0

=

0(a)

=

2Z

L

ii)

5(a)

=

,

0)

e

(z0 E C)

,

Here z0 is a root of the equation . All other roots are of th, f (z) = a form z0 + 2k7Ti , where k is an integer. This shows that the number n (t, a) of all roots in jz s t satisfies n(t,a)

tTr + 0(1)

=

so that

N(r,a)

Further,

+

r

O(logr)

m(r,a) = 0(1)

V(r,a)

so that by the first main theorem

- + O(iogr)

_

This gives 6V(a) iii)

=

a

O(a)

2

=

6(a)

=

=

0(a)

=

0

(0,0)

This point is not assumed by

N(r,a) = 0

so that

f(z)

have 2 Tf

1

m(r,a)

=

log

n

da e

4r cos a +e

2r cos a

0 3

1

log

da e

Tr

158

2r cos a

+

e4r cos a

+

o(1)

Further we

3

loge 2r cos a da

o(1)

+

so that, by the first main theorem,

2r

_

+

0(1)

n

V(r,a) = 0(1)

. Thus we have in this

case

6 (a)

=

6 V(a)

=

1

0(a)

=

In particular the quantities maximum possible value 1

6(a)

,

6 (a), 6 V(a)

=

reach in

+ e((010))

a

(0,0)

=

their

,

What does the deficiency relation tell us about U(-)

0

>

+

O(a)

6G

+

?

We have <

2

6G

a E C2-((0,0)} 1

+

1

0

+

+

<

6G

2

so that the index of the Gauss map is zero. We can confirm this latter result by a direct computation:

f (z)

(ez

_

N(r,(0,0),f') so that

G(r,f)

=

2e2z)

,

=

0

T(r,f') = m(r,f')

,

,

m(r,(0,0),f')

V(r,0,f') = O(1)

=

=

2r

+

0(1)

7T

2r n

, and indeed

G(r,f) 6G

lira

=

0

r->+oT(r,f) What can be said about the generalized Nevanlinna deficiencies 6 (a) = 6(a,f) beyond the general deficiency relation? Applying a selection of known results on Nevanlinna deficiencies to the component functions fe(z) of f(z) we will now give a few rather simple conclusions about generalized Nevanlinna deficiencies .

As was shown by R.Nevanlinna a meromorphic function

r(z)

of order

p

159

distinct values ar , b E Cu{-} such that 6(ar,r) = d (br, r,) = 1 only if PC is a positive integer or if p = +m Assume now that for a meromorphic vector function f(z) = (f1(z),...,In (z)) for 2 distinct points a = (al., ... an) E Cn, b = (bl, ... ,b n) E Cn 6(a) = 6(b) = 1 . Then each component we have maximum deficiency function f.(z) must have the same order p = p as f(z) by Propoj d (a) < 6 (a.,f.) 1 , 6 (b) 6 (b., sition 17.9. Because of the inequalities can have

2

<-

we have

f.) < 1 J

6(all fI) =

... = 6(an,fn) = 6(bl,f1)

We deduce that there must exist an index j0 such that 6(aJ.O,fJ.O )

6(b J.O,f J.O )

=

... = 6(bn,In) = 1

distinct complex numbers

2

a. JO

=

,

and

b J0

1

From the above result of Nevanlinna we see that either p = +. This proves ger or

p

is a positive inte-

.

Proposition 17.10

:

f (z) =

Let

function in the plane. Assume that for 6 (a)

d (b)

=

=

1

,

be a meromorphic vector

fn ( z))

( f 1 ( z )) ,

points

2

a

,

b E Cn

(a * b)

f I(z) , ... fn (z) must have the same order p, which is a positive integer or is infinite. Further, under the assumption we Then all component functions have

6G = 0

,

6 (c) = 0

(c E CnU{_} - {a,b})

0(c) = 0

(c E CnU{m})

,

Next we apply the following Theorem 17. 11

0
:

, and

If

c (z)

is a meromorphic function of order

6(a> 0

p

where

p coszrp when = 0 or In particz) . then a is the only deficient value of when 0 , C ular a meromorphic function (z) of order zero can have at most one defi<

p2 r >

tNevanlinna [27] ,p. 51 160

cient valuet. We deduce Let Proposition 17.12 f (z) = ( f I ( z ) , ... , fn(z)) , ( f 1, ... , fn non-constant) be a mcromorphic vector function of order p in the plane, where 0 p < 2' Assume that for some a = (a1, ... ,a) E Cn 1 (a) > 0 when p = 0 or 6 (a) 1 - cos np when p > 0 . Then a is the only finite deficient value of f (z) ; also, a. is the only deficient value of fj(z) for j = 1,. . . ,n , and each f .(z) has order p . In particular a meromorphic vector 3 function f(z) of order zero can have at most one finite deficient point. Under the assumption we have 6 (c) = 0 (c *a,-) , :

N(r,f) cos np

Y 0(c)

+

+ EM r->+WT(r,f)

dG

N(r,f)

CECn

<

1

+

lim

,

r-,+a'T(r,f)

,

(0 < p <

2

(p = 0)

is deficient, each f.(z) has order p by Proposition 5 (a,f) > 0 when 17.9. The assumptions and inequality (17.27) show that p = 0 or 6 (a .,f .) 1 - cos Trp when p > 0 . Theorem 17. 11 shows ] J that for each j = 1 , ... ,n the value a. is the only deficient point of is the only f.(z) . Thus inequality (17.27) shows that a = (a 1, ... , an) finite deficient point of f(z) Proof. Since

a

J

J

>_

J

{ak )R

k = 1, 2, ... be the set of finite deficient points a (a1,...,any ) E Cn for the vector valued meromorphic function f(z) This set is either finite or is countably infinite. For each j = 1, ... ,n and each 2, we have by (17.27) the inequality Now let

0

<

d (a ,fj)

6 (a2')

Thus for each

,

j = 1,. .. , n

we have

tHayman [161,p.114 161

6 (at)

6 (ak ,f))

<

R

`

Y 6 (c,f.) cEC

R

here the last sum is extended over all finite deficient points

6 (a)

<

min J

aECn

of

f .(z)

.

J

We deduce the inequality

1

c

I 6 (c,fi ) cEC

Using the estimates (17. 20) and (17.32) instead of (17. 27) we can do the analogue reasoning for the set of points a such that (3(a) > 0 , or for the set of points such that O(a) > 0 . Summarizing and using Proposition 17.9 we formulate Proposition 17.1.3: Let f(z) = (fl(z),...,fn(z)) , (fl,...,fn non-constant) be a meromorphic vector function in the plane. Then the following inequalities are valid.

X

6 (a)

<

min J

aECn 0(a)

<

J

aECn Y 0(a) aEC

n

min

<

min J

X 6 (c,f.) cEC Y 0(c,f))

cEC I O(c,f.)

cEC

here the left sums can be positive only if all

f 1,

... , fn

have the same order.

Next we apply

Suppose that r(z) is meromorphic and of lower order in the plane, where 0 < A < +m . Then for a > 3 we have Theorem 17. 14

tFuchs [231 , Hayman [161 ,p.90, Weitsman [59] 162

A C

2] S(ag)a where

A (a, ),

A(a,A)

<

depends on

)

a

and

only.

A

We deduce using Proposition 17.13 Proposition 17.15 Let f(z) _ (f I(z),...,fn(z)), be a meromorphic vector function in the plane and let 0 < A. < function of lower order A. such that :

J

non-constant)

(f1....If

f.(z) be a component we Then for a >

J

have

3

Y d(a)a

A(a,Ai ) -

<

cEC

a E Cn

where

Y d(c,fi )a

<

depends on

A (a, A.)

and

a

only.

A.

Next we apply Theorem 17.16 t: If

is an integral function of order

C(z)

p

,

then

C

0

=

c#

(0

1

p

=

2)

(c, C)

(1 <

1 - sin 7rp

<

p

5

C

We deduce using Proposition 17.13 and Proposition 17.9 Let f(z) = (f1(z),...Ifn(z)) , (f1,...,fn non-constant) Proposition 17.17 be a meromorphic vector function in the plane. Assume that f(z) has an integral component function f.(z) of order p (0 5 p 5 1) . Then we have :

J

tEdrei and Fuchs [221 163

(0 < P <

0

=

X 1

(Z ` p

1 - sin-,1 p

acCn

<_

1)

In particular an integral vector valued function of order have any finite deficient value.

p


can not

Remark. Already Proposition 14.16 shows that an integral vector valued function of order p < 1- can not have any finite deficient value. -Nevanlinna Chas shown that if a number p >0 is given, which is not an integer, then there exists a positive number K(p) such that for every meromorphic function i;(z) of order p the inequality

N(r,c, c) + N(r,d, ) lim

K(p)

T(r,

r-,+m

is valid, whatever the complex numbers c,d (c # d) are. Remark. For functions of order less than 1 the best possible value of is

if

K(P) = 1

For

0

:: p

<-

the best value of

p>1

K(p) = slnip

,

2

satisfies the inequalities ttt

K (p)

Isinnp K(P)

2.2p +

<

TI

p

Zlsinrrp

where sin np q + Isin7rp =

IT

P

sq +pl

t Nevanlinna [271,p.51 tt Edrei and Fuchs [22] tttEdrei and Fuchs [21] 164

(q < p

<_ q + 2

,

q ?0 , integer)

I

(q +


0

, integer)

.

K (p )

We now apply t

is an integral function of finite order is not a positive integer, we have Theorem 17. 18

:

If

t.,(z)

Y 6(c,r)

-

1

p

which

K(p

C

Proposition 17.19 : Let f(z) = (f1(z),...,fn(z)) , (f1,...,fn non-constant) be a meromorphic vector function in the plane. Assume that f(z) has an integral component function f.(z) of finite order p which is not a positive

integer. Then we have

Y

6(a)

1

-

K(p)

aECn Remark. Proposition 17. 17 presents a sharp form of Proposition 17.19 for integral vector functions of order less than 1 From Proposition 17. 18 we see in particular that if .

Y

c$ (C,

c*m

for an integral function C(z) of finite order p , then p C is an integer 't is the lower In addition it is known that in this case A = p , where tit s order of and that all the deficiencies are integral multiples of pr-1 ti Further, all the deficient values av of (z) are asymptotic values, i.e. ti t for z «' along suitable paths y v T'( Z) > av Using these results, Proposition 17.19, Proposition 17.13 and the deficiency relation we deduce that if C

A

C,

C,

,

-

Y

6 (a)

-

=

1

aECn t

tt

Hayman [161 ,p. 104

Pfluger [49] -'-ttEdrei and Fuchs [201, [211 165

is an entire component function of finite order p and then p = AA. is a positive integer, which is equal to and all the deficiencies a. of f.(z) are integral J vJ -1 multiples of p Further, all the deficient values a. of f .(z) are J ] asymptotic values of f .(z) and the inequality holds and if f .(z) J of lower order A. J the order of f(z)

.

J

Y

0(a)

aEC

N(r,f)

6,

+

<

lim

r- +°°T(r,f)

n

is valid. If f(z) is entire,it is probable that under the assumptions also all deficiencies of f(z) are integral multiples of p-1 and that the deficient values of f(z) are asymptotic values for the meromorphic vector function f(z) Further, if c(z) is a meromorphic function of finite order p and if

1

2

c E C U{w}

then it is known that the number #(c) of deficient values does not exceed 2p tWe deduce from this and from Proposition 17.9: be a meromorphic vector function of finite Let f (z) = ( f 1 ((zz)) ,. fn (z)) order p and such that Y

d (a)

=

2

aECn Then all component functions f. must have order p , and the deficiency . From the relation shows that d (o) = 0 , 0(a) = 0 (a E Cn U {co}) 6 G = 0 above and from the inequalities

1

2

6 (a)

aECn we see that number number

#(f.) j #(f)

tWeitsman (58] 166

<

1 6 (c,f.)

2

cEC

6( - ,f) = 0 ( j = 1, ... n) J

!5

and that for each j = 1 , ... , n the is 5 2p . This shows that the

of deficient values of f. J of deficient values of f(z)

is

<(2p)

n

.

It should be noted that the last deductions concerning 6 (a) are of less importance than the corresponding results in scalar Nevanlinna theory, since information on 6 (a) does not give full information on the important quantity

_

V(r,a)

lim

1

r-*+-T(r,f)

- 6 V(a)

but only on

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

r-

1

T(r,f)

- 6 (a)

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 [14] Wells [411 , Morrow and Kodaira [26] , Kobayashi and Nomizu [21] , Vol.2, and in the articles Chern [91, Griffiths [261 and [281 , and otherwise scattered throughout additional literature given in the Bibliography. A. Complex manifolds. Let w = (wl,...,wm) be the coordinates w. = u + iv. , (j=1, ... , m) on the complex Euclidean space Cm R2m ] The real basis {du. , dv. } or the complex basis .

.

J

dwj

J

duj + idvj

=

dwj

=

duj - idvj

span the cotangent space to a point. Its dual is the tangent space with the real basis a

a

au.

{

By.

'

]

}

J

and the dual complex basis aW.

a

1

a

a

a

z (au. - 1 u

=

J

'

J

-= awj

a+

1

z (au.

A(w)

as _

=

j

aaw dw j

- dw.

+

j

]

awj

]

or dA

=

as

+

as

after introducing differential operators

168

a

and

a

by

a)

av.

]

J

The total differential of a complex valued function written da

i

on

Cm

can be

as

aA

A

1(wl,

C_ function

.

,

. ,wm)

on an open set

U C Cm

is called holo-

morphic if aA

=

0

if it is holomorphic in each complex variable separately. A complex manifold M is a manifold M provided with a distinguished open covering {U, V.... ) and coordinate charts i. e .

,

O

U --- C m

U

such that

V-- C m

0V

...

is biholomorphic where defined. A func0V-1 ao tion A is on an open set U C M is called holomorphic if OV(U n V) c Cm holomorphic on for all V . A holomorphic coordinate system on U C M is a collection of functions w = (w1,...,wm) such 1 w-1 that w o 0V are holomorphic on w(U n V) and and 0V o OV(U n V) , respectively for each V . A mapping f M -- N of complex 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 point. 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 OU o

OUV

0V-

:

if it is one-one, i.e. , if

f(x) = f(y)

,

x,y E M

implies

x =y

.

Examples of complex manifolds are : 1. Cm with global coordinates w1, .... w m . 2. A 1-dimensional complex manifold is called a Riemann surface . 3. Complex projective space Pm is a complex manifold , the natural projection

Cm+1 11

0 -+ Pm

is compact since there is a continuous surjective map to Pm . Cm can be pictured as from the unit sphere S2m+1 in Cm- Pm given by Pm as an open set by the inclusion sitting in

is holomorphic.

Pm

Cm+1

(w1.... ,wm)

(1,w1, ... Iwm)

ti

denote the hyperplane at infinity with the equation (w0 = 0) . By considering k as the directions in which we can go to infinity in Cm we obtain an identification k = Pm-1 . Thus we can picture Pm as the compactification of Cm obtained by adding on the hyperplane , at infinity.

Let

169

any point, and w = (w1,..., wm) There are 3 different notions of

be a complex manifold, p E M a holomorphic coordinate system around p a tangent space to M at p . Let

1.

M

.

is the usual real tangent space to

TR , p

considered as a real manifold of dimension

at p , where M is . We can write w. = U. + iv. M

2m

J

TR p

span R {

=

a

a

a uJ

a V.

J

J

}

can be viewed as the space of R-linear derivations on the ring of real valued C functions in a neighborhood of p . 2. TC,p = TR p G) R C is called the complexified tangent space to and

M

TR,P

at

p

. We can write a

spanC{

TC,p

u.

a

a

av.

'

-} a

span C{ aw

=

}

]

awj

and TC,p can be viewed as the space of C-linear derivations in the ring of complex valued C functions on M around p

T'

3.

spanC {

p

a w.

}C

TC,

J

p

is called the holomorphic tangent space to M at p . It can be viewed as consisting of derivations which vanish on antiholomorthe subspace of TC P phic functions (i.e. f such that f is holomorphic). a

Tp

=

spanC { - } a w. J

is called the antiholomorphic tangent space to TC,P

being given as the real vector space have the operation of conjugation sending a w.

at

p

, and we have

TP 0 TP TR p

TC ' p

a

M

to -

tensored with

C , we

a

aW

and

T'

p

=

T' p

]

B. Hermitian metrics . An Hermitian metric on a complex manifold 170

M

is

given by a positive definite Hermitian inner product (

)

T'w O T'w -+ C

w

on the holomorphic tangent space at w coordinates w on M the functions

are

C_

.

a

a

hjk(w)

Dw

Tw

the Hermitian metric

(

ds2

j,k

)w

,

w E M such that for local

)w

In terms of the basis

(T'w p Tw )

for each

for

{dw. ( dwk }

® Tw

'

is given by

hjk(w) dw® Q dwk

,

hjk

=

hkj

By applying the Gram-Schmidt process to the basis (dw1....,dwn) of Tw for each w we can locally construct a coframe for the Hermitian metric, i.e. m forms

k

a. j

k = 1, ... , m

lk dw.

1

of type (1, 0) such that ds2

(w), ... , m(w)) is an orthonormal basis for Tw , in This means that terms of the inner product induced on T w by ( , ) w on T'w The real and imaginary parts of an Hermitian inner product on a complex vector space give an ordinary inner product and an alternating quadratic form, respectively, on the underlying real vector space (see Chern [5] p. 10) . Re ds2

.

T R w (2)

T R, w

--- R

is thus a Riemannian metric on 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. Im ds2

.

T

R,w

0 TR,w - R 171

being alternating, represents a real differential form of degree 2; - 2 Im ds2

is called the associated (1, 1)-form or the Kahler form of the metric plicitly we write a.

$

+ J

J

where

a.

,

J

ds2

are real differential forms. Then

(3. ]

((ai + i B)) 0 (a] - i B)))

ds2

X (a O (X . ]

+

s. © R.)

J

J

J

+

i

( - a. 0 S.

5'

J

J

]

J

The induced Riemannian metric is Re ds2

(a. 0 a

=

j.

R. 0 R.)

+

]

J

J

and the associated (1, 1) -form or Kahler form is w

=

-

1

2

Im ds 2

(a 0

1

=

2

a

For example, the Euclidean metric on

is given by

Cm

m

ds2 Writing

=

J X1

w = U.

Re ds2

dw D dwi

, we see that

i v.

j

(du .0 J

du J .

+

is the standard Euclidean metric on ds2

is

dw. A d-w. 3

172

dv. 0 dv. ) J

J

R2m = C m

, and the Kahler foi

For the metric

l

ds2

the volume element associated to a1A

dp

5 1A ...

a. + is. J

1

is by definition

Re ds2 Auum A Sm

On the other hand we have a. n s

_

4)

1

1

so that the mth exterior power is m

MI. a1 A

L)

A

...

The Euclidean volume element of dp

=

u A V I A ... A U I

m

AV

nCL m

Cm

=

m!

du

.

is, for example, dw1 A d w I A . . . A Z dwm Adwm

m

m!

Now let

SCM

be a complex submanifold of dimension 1. The (1, 1) -form associated to the metric induced on S by ds2 is `''IS , and applying what we have just said to the induced metric on S , we have the Wirtinger Theorem for the dimension l :

vol(S)

W

S

This theorem shows that the volume of a complex submanifold S can be expressed as the integral over S 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 H (u(t) , v(t)) in R2 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 differential form in.

R2

173

C. The Fubini-Study metric. The unitary group

ear transformations of

Cm

<W,W >

In

m

(1)

lW11 2

a unitary basis or unitary frame is an ordered set

Cm

F

of

,

consists of the linwhich leave invariant the expression U

{WO,W11 ...1Wm-1}

=

vectors such that

m

< WA.WB >

=

d

0

AB

<_

A,B,C,D < m-1

(2)

The set of all unitary frames for Cm constitutes the frame manifold F(Pm-1) . Choosing a reference frame FO , any frame F is uniquely of the form F

=

T E Um

F0

T

F HT

The correspondence If

W

F - Cm

tesimal displacement) frame vectors

dW

gives an identification F(Pm-1) = U m is a smooth function, the differential (or infinimay be resolved into its components relative to the .

M-1

dW

kI 0 kWk

where the k = < dW, Wk > are 1-forms on F = Um For example, the frame vectors W. themselves may be considered as smooth maps W.

F(Pm-1)

_ Cm

Expanding the differential (or infinitesimal displacement) dW.(F) the basis vectors in the frame F leads to the equations dWA

B 'ABWB

in terms of

(3)

where CAB

< dWA,WB >

are differential forms in U m From the orthogonality relations (2) we get by differentiation 174

(4)

< dWA , WB >

CAB

+

SBA

< WA, dWB >

+

0

=

(5)

0

=

for any fixed T , the give OAB a basis for the left invariant Maurer-Cartan forms in U m (3) and (5) say that under infinitesimal displacement, the frame -F undergoes an infinitesimal transformation with coefficient matrix 0AB . They are the structure equations of a moving frame. Taking the exterior derivative of (4) we get using (3) Since

F)

WA(T

T WA(F)

=

.

dOAB

=

- < dWA,dWB >

C OACWC '

CD

< WC'WD >

D BBD

AC ^ ABC

0AC ^ BBD

'

so that by (5) (6)

X SAC A GCB

C

which are the Maurer-Cartan equations of the unitary group (3) and (2) we get < dW0'WO>

< dW0,dW0 >

U

m

. From (4),

X00

<

B OBWB '

OCWC > C

B,C < OBWB ' OCWC > M-1

_ (7)

C

where the multiplication of differential forms is understood in the sense of ordinary commutative multiplication. From (7) we get m-1

dW0 , dW0 > - < dW0 , W > < WO , dW0 >

C1

0

(8)

175

. The vector

w c Cm-0

Now let WO

- wE Cm-0

=

(9)

IIwII

has length 1 . From (9) we compute IIwII dw - w dIIwII dW0

so that

-

1

< dW0 , W0 >

dllwll

,

IIwII2

< WO , dW0 >

1

dw > -

<w

1

IwII2 1

< dWO , dW0 >

< dw

,

dw> -

1

< wdliwli

<

,

IIwIIdw >

+

IIwII4

w d IIwII

>

1

< wdliwNI

4

,

wd1Iw1I >

IIwII

dIIwI!

1

< dw , dw> -

_

IIwIIdw

IIwII

2 T, -WI?

I

dIIwII

IIwII

IIwII2

dIIwII

< dw , w > -

< w, , dw>

IIwII3

IIwII3

(dIIwHI)2

This gives for (8) M-1 C=1

_

1

2

1

< dw , dw >

< dw , w > < w , dw >

IIwII4

IIwII

(10)

Remark. The last expression could be abbreviated by IIWII2IIdwiI2 IIw1114 {

176

1< w , dw >1 2

}

1 11W

IwII4

A

dwll2

1

2

jwJdwk - wkdw

4

j1k

From this calculation we conclude that we can define a Hermitian metric in Pm-1 by the formula M-1 ds2 C =1

OC'OC

2 3

Lkwkw k)( 'k dwk O dw

'kwkw k)

3

k

Ekwkdw k)

xQ

Ekwkdw

J

(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 1 In terms of the left hand side of (12) the associated Kahler form can be written m-1

m-1

c

C 1 HOC A HOC

1

00C A 0CO

where for the latter expression we have used (5). Now from (6), using 0 , we have 4,00 A 000 - X00 A X00 m-1

C0

m-1 HOC A

C1

`NCO

HOC 11

¢CO

Thus the Kahler form can be written m-1 =

c

2i

1 HOC

1 A

(13)

2i

NCO

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 can be written as i

a

a log jwjl

d do log jjw

=

(14)

Z

In the special case m = 2 (12) is the Fubini-Study metric of Explicitly, we obtain from the right side of (12)

P1

=

S2

177

ds2

2)-Z{(w

(w w +w w

=

1

2

1

1

w +w w 2

1

) (dw

2

1

(D dw

1

+ dw

2

(E dw 2)

- (dw1w1 + dw2w2) ® (w1dw1 + w2dw2) } (w1dw2 - w2dw1) Q (w1dw2 - w2dw1) (wlw1 +

w2w'2)2

We will now express this in the inhomogeneous coordinate w2 (15)

w1

With

w1dw2 - w2dw1

d =

w1dw2 - w2dw1

_ do

2

-2 wI

_

w1

we can write w? wZ 1

.

ds2

dOd

do

(1 + C

(w1w1 + w2w2)Z

2

Thus the Fubini-Study metric on P1 is just the natural metric of the Rie mann sphere S2 of constant curvature 4 Its Kahler form is .

w

2

d4 A d

da A dT

(1 + )2

(1 +

Cr + i T

,

(16)

which is the spherical volume element. This can also more quickly be obtained as follows. is the coordinate on the open set U1 = (w1 x 0) in P1 By using on U1 the lifting w = (1 , d we obtain from (14) a a log(1 + C

w

_

The volume of

i

)

=

4

dAd

2(1+CO is

S2

2n +m

tdtdO

(1+ )Z 0

178

0

=

n

Olog(1 + 4 c) d A d

Differential forms. Let AP(M,R) denote the space of differential forms of degree p on M , and ZP(M,R) the subspace of closed pforms. Since d2 = 0 , d(AP 1(M,R)) C ZP(M,R) . The quotient groups D.

ZP(M,R)

HDR(M,R) dAP-1(M,R)

of closed forms modulo exact ones are called the de Rham cohomology groups of ZP(M) M . Similarly if AP (M) and denote respectively the spaces of complex valued p-forms and of closed complex valued p-forms on M , we have the corresponding quotient ZP(M)

HDR(M)

If

HDR(M,R) 0 C

dAP-1(M)

is a complex- manifold the decomposition

M

TC w(M)

T

of the cotangent space to

w

(M)

+Q

®

=

w

at each point

M

r

A TC,w(M)

T

(

A

wEM

gives a decomposition

Tw (M) 0 A Tw, (M))

p+q=r

Therefore the space of r-forms can be written Ar(M)

Q APq(M)

=

p+q=r where

AP'q(M)

=

{

E Ar(M)

P

is the space of r-forms of type (p, q) . For day

E

q

: (w) E A Tw (M) 0 A T( M) for all w E M} E AP' q(M)

we have

AP+1.q(M) O APq+I(M)

and we can define operators APq(M)

AP'q+I(M)

APq(M)

AP+1'q(M) 179

where

d

a

+

a

=

In terms of local coordinates

.

a form is of type (p,q) if it can be written

(w)

w = (w 1... 1w m )

''Ii (w) dwI A dwJ #Irp

#J =q

where for each multiindex dwl

dwi

=

A

I

...

a

p

and

are given by

a

_

i(w)

a

,

}

P

A dwi

1

The operators

{ il, ... ,i

=

(w)

lJ(w) dwj A dwI A dwJ

I,J,j

awj

I,J,i

aw

IJ(w) dwi A dwl A dwJ i

E. Vector bundles. Let M be a C differentiable manifold. A complex vector bundle E over M consists of a space E and a projection map n E - . M , such that: {U, V, ...1 of M with n 1(U) i) There is an open covering 7 1(U) -+ U X Ck equivalent to U X Ck by a C- map : U ou(n _1(x)) _ (x} x Ck oU must preserve fibers ii) On we require Y n 1(U V) ) :

:

U0 0u (x, V) where

(x

_

,

gUV(x) &V)

U nV -C k - 0

gUV

_

(x

, U)

,

are C functions, called the

transition functions. The transition functions necessarily satisfy the identities gUV(x)

-

gVU(x)

gUV(x)

-

gvW(x)

=

I

gWU(x)

for all =

I

xEUnV for all x c UnVnW

(17) .

(18)

is called a trivialization of E over U . A complex vector bundle on M is called trivial if it is of the form M X Ck . E is called a line bundle if oU

180

k = 1 . E is called holomorphic if M is a complex the fiber dimension manifold and if the transition functions are holomorphic. We give a few examples without details. 1. Let M be a complex manifold, and let T (M) be the complex tanx gent space to M at x . For x e U C M and U-+Cm a OU coordinate chart, we have maps

Tx(M) for each

xEU

spanC {

a ,

3 V.

= C 2m

, hence a map U

U

a

3 U.

UC 2m

Tx(M)

xEU

giving T(M)

U

=

Tx(M)

xEM the structure of a complex vector bundle, called the complex tangent bundle. 2. For each xEM we have a decomposition Tx(M)

T'(M)

=

) T' (M)

T'(M) C T(M) {T'x (M) C Tx(M)} form a subbundle The subspaces called the holomorphic tangent bundle ; it has the structure of a holomorphic vector bundle. 3. If E -o M is a complex vector bundle, then the dual bundle E is the complex vector bundle with fiber Ex = (E x ) ; the M trivialization s EU ------ I- U X C k

U

,

EU = it 1(U)

induce maps

EU --UxCk =

U

*

*

which give

E

has transition functions tion functions lUV

=

t

UxCk

-1 gUV

M the structure* of a manifold. If E then E -. M is given by the transi{gUV } ,

U Ex

t 9VU 181

Similarly, if E -- M , F - M are complex vector bundles of fiber dimension k and Z and with transition functions {gUV } and respectively, then you can define the following bundles: {h UV } , 4.

EO F

,

given by the transition functions E

lUV

E©F

gUV

*

T (M) T

Ck )

+

given by the transition functions

,

lUV 5.

GL(Ck

T(M)

=

(M)

,

r*(P'q)(M)

© hUV E GL(Ck X CQ)

T

*

(M)

the complex cotangent bundle

.

the holomorphic and the antiholomorphic cotangent bundle ;

.

(M) © A T

T

=

A

(M)

Aholomorphic vector bundle with fiber dimension 1 is called a holomorphic line bundle. We will now give an example of this in detail. Pn-1 F. The universal bundle J on . Let w 1 , ... , wn denote Euclidean coordinates on Cn and also the corresponding homogeneous coorPn-1 Pn-1 Pn-1 dinates on , all . Let x Cn be the trivial bundle on fibers being identified with Cn . We will define a holomorphic line bundle Pn-1 Cn J J is a subbundle of Pn-1 Pn-1 ; its fiber Jw over each point wE is the line {Xw}X C Cn represented by w , i.e., 6.

-

{ A(w1.. .wn)

Jw

,

XeC}

We show now that there exists in fact a line bundle with these fibers. Let J denote the disjoint union of all J w . Then any point v E J can be represented (not in a unique manner) in the form v

where it

:

182

=

(Awl,...IXwn)

=

(w1, ...Iwn) E Cn - 0

J - Pn-1

is given by

A(w11....wn) E C n ,

and

AEC

.

Moreover, the projection

7r(a(wl,...,wn)) Putting

Ui

Now if

v

v

A(wl,...,wn) E 7r 1(Ui)

=

1 W.

Aw

.EC

71-1(U

1

1

.

x 01

Pn-1

, we see that

AEC

wi

,

, then we can write

0}

v

.

in the form

1

is uniquely determined by

1

i

w

,

E

W.

ith

1

=

(wl,...,w11)rk,

{v = A(wi,...,wn) E Cn

_

=

and A. 1 mapping

E Pn-1

,w n)

{(wl'

=

1(Ui)

r

n(wi,...,wn)

) -+ U

. 1

. We can define the

v

xC

by setting

Yv)

4i

the fibers of

A(wi,...,wn) E

=

((w1.....wn)"

is bijective and is linear from the fibers of U xC U. . Suppose now that 1

The mapping v

i(A(w1,...,wn))

_

n

1(Ui)

to

1(U. f Uj)

then we have 2 different representations for relationship. We have $i(v)

A.)

((wl, .... wn)"

,

A.)

((w1..... wn)"

,

A.)

v

and we want to compute the

,

where A. 1

=

Aw.

A.

1

J

=

Aw. 1

Therefore A

=

A. 1

=

wi

A.

wj

i.e.

a. 1

=

-

W.

-wja. 1

j

Thus if we put w. 1

gij

w.

183

then it follows that gij gjk ' gki = 1 . We deduce that J given the structure of a line bundle by means of the trivializations the transition functions

can be

*

w.

{oi}

and

U1 .nU. - GL(1,C) =C - 0

1

w.

.

n

of a vector bundle E- M over

Sections. A section s

G.

is a C

UCM

map

U- E

s

s(x) E E x

such that

for all x E U . A frame for E over U CM is a collection s1 , ..., sk of sections of M over U such that is I (x),. .. , sk(x)} is a basis for Ex for all xEU . A frame for E over U is essentially the same thing as a trivialization of E over U : Given E

U

U

------- 0-

UXCk

a trivialization, the sections

Ul(x,ei

si(x)

the canonical basis of

{ei}

si, ... , sk , we can define a

form a frame, and conversely, given a frame trivialization A

=

U(A)

(x

_

in

G A .1 s 1.(x)

Given a trivialization

of

0U

we can represent every section a

=

s

U , it is important to note that over U uniquely as a C vec-

over

of

E

by writing

Ul(x,ei)

0V

ai(x) '

X

E

is a trivialization of E corresponding reoresentation of If

E

(al,...,ak)

ai(x)

Y

for

(A 1, ...' Ak))

,

Ck

over s I V n i7

ai(x)

U1(x,ei)

and

V

a'

(a,, ... , ok)

the

, then '

V1(x,ei)

so

ai(x)

184

ei

=

of (x)

U4 j(x,ei)

,

i.e.

o

gUV a

Thus, in terms of trivializations

EU -+ Ua X Ck }

{a

a

sections of

over

E

{ as

U Ua

correspond to collections

,...,C.. , aka) }a

=

of vector valued C functions such that a

for all {oa}

gas

a

'

as

gas are transition functions of

where the

a

E

relative to

.

A section s of the holomorphic bundle E over U C M is said to be U E holomorphic if s : is a holomorphic map, a frame s = (s1,... is called holomorphic if each s i sk is; in terms of a holomorphic frame {s. } a section 1 ai(x)

s(x)

.

si(x)

is holomorphic if and only if the functions a,i are. H. The hyperplane section bundle H - Pn-1 is the dual H = J of the universal bundle J , i.e., it is the holomorphic line bundle whose fiber over we Pn-1 corresponds to the space of linear functionals on the {Aw}A . It has global sections line given by the linear forms (P n-1 ,H) A(w) + + anwn C . Such a form a w on A(w) determines a 1 I divisor, which is given by the hyperplane A(w) = 0 . In more detail, let al, ... an be constants and n the projection Cn - 0 --- Pn-1 . The linear form A (w) in Cn - 0 has in the local coordinates in 1(U ) i the expression A(w)

wi(a1 ill + ... + I + ... + an

=

inn)

ith

where

w. =

ill

wi

j

,n

Denoting the expression in parentheses, which is essentially the linear form at the left hand side in "non-homogeneous" coordinates in Ui , by ai

=

(a 1 ic

l ++1++a nn ) ith 185

we see that in

n-1(Ui n u.) W.

w. a. 1

so

{o. } 1

--' G. wi J

w. a.

=

1

J

.

J

1

defines a section

s

i

C. J

in the line bundle whose transition functions

are

j

W.

gij

w=

=

1

i

-1

Because of this origin the latter bundle is called the hyperplane section bundle H of Pn 1 It is the dual J of the universal bundle J I. Divisors and Chern class. In the last example the hyperplane A (w) = 0 is defined by ai = 0 in Ui = (wi x 0) , and the transition functions in can be written Ui n u .

.

.

J

More generally, a divisor D on a complex manifold locally finite formal linear combination D

Ia

=

M

is defined to be a

V. 1

1

of irreducible analytic hypersurfaces of M. lection of holomorphic functions

It may be thought of as a colU. - C such that

M. .

1

1

m.

1

gij

mj

are non zero holomorphic functions in Ui n Uj for all i , j . D to be the zeros of the functions mi in U. . In U. in Uj n Uk gij

gjk

'

gki

is defined we have

mk

M.

M.

m

mk mi

It follows that m. 1

gij

M.

are transition functions of a line bundle associated to the divisor D . 186

[Dl

.

It is called the line bundle

In view of the above then we can say that the hyperplane section bundle H Pn-I is the line bundle which is associated to the divisor of a hyperplane in It can be shown that the line bundle (D] associated to a divisor D on M is trivial if and only if D is the divisor of a meromorphic function. If M is compact we have Poincare'duality between H (M, Z) and q H2m q(M, Z) In particular, a divisor D on M carries a fundamental .

homology class {D}

H2(M,Z)

E H2m-2(M,Z)

as an element in the de Rham group HDR(M,R) Then the divisor D is said to be positive if {D} is represented by a closed positive (1, 1) form m . This means that locally We' may consider

{D}

2

i,j

hij dwi

,

dwj

where the Hermitian matrix (hij) is positive definite. It can be shown that collections {g..} and {g'.} of transition funca. E tions define the same line bundle if and only if there exist functions (Ui) satisfying 'i

gij

gij

The transition functions sent a Cech 1-cochain on (18) mean that d ( {gij})

(19)

{ gij E 0(Ui n U.) }

of

E -M repre-

the relations (17), with coefficients in = 0 , i.e. , {gis a Cech cocycle. Moreover, by 'define the same line bundle if and only if and {gij}

(19) two cocycles {gij}

their difference

]

M

{g.. gis a Cech coboundary. Consequently the set of

holomorphic line bundles on The coboundary map

-0

M

is the Cech cohomology group

H (M, VVV

d

HI (M,(0 *)

H2(M,Z)

arising from the cohomology sequence of the exponential sheaf sequence 0

defines the Chern class c($) = 6 ( {gij }) of a line bundle. If the bundle E carries an Hermitian metric in its fibers, with the curvature matrix 0 , Chern has shown the important theorem that c(E) is represented in the de 187

Rham cohomology group

i 27r

188

HD R( M

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Address of author Mathematics Department

University of Siegen D-5900 Siegen West Germany 196

Table of symbols

Cn

1

w

H2(Pn 1 Z)

1

OCR <

12 13

>

1

li

1

f

C

1

v(r,0)

14

CR

1

a

16

f(z) - a

16

z.(a)

17

II

C +,n

1 2

ac

2

z.(0)

8

z(-)

8

Pn-1

9

*

13

J

n(r,a,f) = n(r,a) V(r,a) = V(r,a,f) v(r,a) = v(r,a,f) N(r,a) = N(r,a,f) N(r,f) = N(r,o)

18

18,19 19,20 20

20

u (r, a)

21

log

22

m(r,f) = m(r,f)

22

m(r,a)

22

T(r,f)

22

10

CR

24

F

10

V(r,W)

24

F

10

0

ti

w

9

a

10

a

10

d

10

ac

10

T(r, A)

* n

28 10

*(a)

28

11

n(+oo,co) = n(+or,f)

28

11

n(+W,a)

29

*

C J

25

*

*

J

12

n(a)

29

H

12

v(+-,a)

30

c1(H)

12

v(r,m)

30

197

m(r,c,f

45

Ric j N1(r)

114,118

45

nl(r)

114,118

45

N(r , a)

126 130

n(r,a)

126, 130

45

n(r,f)) = n(r,oo,fi)

N(r,-,f T(r, f.) J

m(r,ai) = m(r,aj,fi)

46

n(r,a.) = n(r,a.,f.)

46

J

7

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

46

3

d (a)

112

,

(a, f)

d(co) = d(c",f) N(r , f) = N(r , w) O(a)

O(a,f)

130

130 130 131

51,54 0(a) = 0(a,f) 52

131

(r,a)

132

N1(r,a)

132

n

1

54 57

m(r,f) =

m0

(r,-,f)

62

64

65 68 69

70, 71, 72 72

74

S 2n

75

A(r,f)

77

E(u,p)

82

g

85

M(r,a)

89

M(r,f) = M(r,-)

89

M(r,a.)

97

J

98

A

198

= 6 G (f)

133

6V(a) = 6V(a,f)

139

6G

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 point, 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

exceptional value, see deficient 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 Cn , 64 function, meromorphic vector valued, 1 Gaussian curvature, 111 Gauss map, 110, 111, 133 Gauss map, index of the, 133 generalized Ahlfors-Shimizu characteristic, 65

generalized Ahlfors-Shimizu proximity function, 74, 75 generalized Borel exceptional value, 129 generalized chordal distance, 72 generalized counting function of multiple a -points, 131 generalized counting function of multiple points, 118 generalized first main theorem, 23, 26, 75 generalized genus, 85 generalized Nevanlinna characteristic function, 22 generalized Nevanlinna deficiency relation, 134 generalized Nevanlinna proximity function, 22 generalized Picard exceptional value, 127 generalized Poisson-Jensen-Nevanlinna formula, 8 generalized Riemann sphere, 73 generalized second main theorem, 114 generalized spherical characteristic, 65 generalized spherical distance, 72 generalized spherical proximity function, 74 generalized theorem of Picard, 127, 138 generalized theorem of Picard-Borel, 129 genus, 82, 83, 85 Hermitian geometry, 9 Hermitian metric, 11, 64, 111 holomorphic curve, 12 holomorphic line bundle, 11 homogeneous coordinates, 9 hyperplane section bundle, 12

index of multiplicity, 132 index of the Gauss map, 133, 137 index, Ricci, 133 inhomogencous 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,

order of growth, 51, 54 200

65

perfectly regular growth, 54 Picard, generalized theorem of, 127 Picard-Borel, generalized theorem of, 129 plurisubharmonic function, 37 point, 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 stereographic 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