Willmore Surfaces in S^3

SI

10 Willmore Surfaces in

Let <

-,

> be

an

indefinite hermitian inner

product

on

EV. To be specific,

we

choose < Vi

Then the set of

complementary

IT1W2 + IT2W1

1, 1 > 0 defines an S3 C RP1, while the hyperbolic 4-spaces, see Example 4. We have

isotropic lines 4-discs

>:--:::

W

are

<

=

(a db)

(10.1)

C

and the

same

holds for matrix representations with respect to

a

basis

(v, w)

such that

=O=<W'W>'

=l.

10.1 Surfaces in S3 an isotropic line bundle with mean curvature sphere S. We look >. Clearly S* adjoint map M -* Z7p -+ SP* with respect to < 1 L implies stabilizes Lj-, and L

Let L be at the

=

=

(dS*)L

=

S*L'

S*L

=

L'

=

L.

Similarly,

Moreover, if Qt belongs

to

S*,

(dS)*L

1

C L

then

Qt

(S*dS* 4 -

(dSS

-

-

*dS*)

*dS)*

I(SdS + *dS)*

4

-A*.

F. E. Burstall et al.: LNM 1772, pp. 61 - 66, 2002 © Springer-Verlag Berlin Heidelberg 2002

L.

10 Willmore Surfaces in

62

kerQt

Therefore We

proceed

=

=

(image(Qt)*)'

=

(imageA)-L

sphere,

curvature

mean

L'

D

to show that S and S* coincide

uniqueness of the S*

S3

on

L and

HIL. By

the

Theorem 2, it then follows that

see

S.

Let

0

E

1'(L),

and write

SO

=

OA,

S*O

0,60

>=

Op

and <

Note that <

0, 0

>= 0

0, JO yields

> makes sense, because of <

+C0

W

From 0 =<

0, So

>

0 =<

we

=

0, L

>= 0.

Differentiation of

0.

obtain

JO, SO>+< 0,(dS)O %-

>+ <

0,SJO

>

I.,

=0

=<

JO, SO

=<

JO,O >A+ P

=

Now

we

apply

*

S*O'JO

> + <

<

>

O,JO

>

CDA + pw.

using *W

0, JSO

=<

>=

WA,

(10.2)

and obtain 0

We conclude A Now

assume

i.e.

S I HIL

=

AOA

SIL =

=

Ip

0 =< =<

+

pwA

S* IL

and

(p

-

)CDA.

-

S* I HIL

50' so

JO'O

I u. Then

=

> + <

S*JO' 0

=& <

=

S*0,50

>

0, SJO

>

> + < > +


> P

&6j + WP.

But

*(.,) =<

=<

0, JSO

>=<

S*O' 90

>=

0, S60 XW.

>= WP

10.2

Comparison

with

(10.2)

shows p

0 It follows

a

A

=

=

p, i.e.

=

& W

A, and

=

WA

+

SIHIL

we

(

=

S*IHIL.

=

Hyperbolic

2-Planes

63

get

&)WA.

-

This

completes the assumptions

of Theorem 2, and S* = S by uniqueness. OA, then Conversely, if S* = S and So =

A < Now S2

0,0

=

-I

SO,O

>=<

implies A2

0,SO

>=<

>=<

1, and therefore

Proposition 18. An immersed holomorphic S*. i.e. a surface in S', if and only if S

>

A

get

<

0,,0 we

curve

=

A <

0, V)

L in

0,0

>.

> = 0.

HP1

is

isotropic,

=

2-Planes

Hyperbolic

10.2 In the

half-space

Poincar6 model of the

hyperbolic space, geodesics are orthogonally intersect the boundary. We consider the models of hyperbolic 4-space in HP', and want to identify their totally geodesic hyperbolic 2-planes, i.e. those 2-spheres in RP' that orthogonally intersect the separating isotropic S3. Using the affine coordinates, from ExS3 This ample 4, we consider the reflexion H -+ H, x -+ -.t at Im H preserves either of the metrics given in the examples of section 3.2. In particular, it induces an isometry of the standard Riemannian metric of RP' which fixes S3. Given a 2-sphere S E End(EV), S2 -I, that intersects S3 in a point 1, we use affine coordinates, as in Example 4, with 1 =:F vH and w or

euclidean circles that

=

.

=

such that < V,

v

>=< W,

W

>=

0,

< V,

W

Then N -H

S with N 2

=

R2

1, NH

=

=

(0

-R

HR, and S'

C Ifff is the locus of

Nx+xR=H. If S' is invariant under the reflexion at

iR

=

H

S3,

then it also is the locus of -Nd

-

or

Rx + xN

=

T1.

According to section 3.4, the triple (H, N, R)

is

unique

up to

sign. This implies

either

(H, N, R)

=

(ft, R, N)

or

(H, N, R)

-R, -N).

By (10.1) either S* S, and the 2-sphere lies within the 3-sphere, intersects orthogonally, and S* -S. We summarize: =

=

or

it

10 Willmore Surfaces in

64

S3

Proposition 19. A 2-sphere S E Z intersects the hyperbolic 4-spaces determined by an indefinite inner product in hyperbolic 2-planes if and only if S*

=

-S.

10.3 Willmore Surfaces in S' and Minimal Surfaces. in

Hyperbolic 4-Space Let L be set of

a

an

connected Willmore surface in S3 C

indefinite hermitian form

HP',

where S3 is the

H. Then its

on

mean

isotropic sphere

curvature

satisfies

S* Let

us assume

that A

$ 0,

and let

S.

L

ker A and

L

L.

image Q

be the

2-step

B.icklund transforms of L. Lemma 13.

Proof.

First

we

have I

Q*=

4

(SdS

-

*dS)*

4

(dSS

-

*dS)

I

4

Now

imap Q

Therefore < L,

L L

is

-

*dS)

=

-A.

(10-3)

S and So S-stable, and S* OA imply and dense on a 0, open subset of M =

>=

=

(-SdS

L'

=

(image Q)'

=

=

kerQ*

=

<

0, 0

> = 0.

ker A

Lemma 14.

-S

for

the

mean

curvature

sphere 9 of L.

L is obviously (-S)-stable. It is trivially invariant under Proof. First L and Q and, therefore, under d(-S) 2(*A *Q). Finally, the Q of (-S) =

=

-

A is

I

4

and this vanishes

sphere by

on

these three

((-S)d(-S)

-

*d(-S))

=

A,

L. The unique characterization -S. properties implies 9 =

of the

mean

curvature

10.3 Willmore Surfaces in

We

now

turn to the

S3 and Minimal Surfaces

in

1-step Bdcklund transform of

d(F+F*)=2*A+2*A*

Hyperbolic 4-Space L. If dF

=

2

*

A,

65

then

2*A-2*Q=-dS.

=

(10.3) Because S*

We

=

now use

S,

we can

choose suitable initial conditions for F such that

F+F*

-S.

(10.4)

affine coordinates with L

[,I.

Then the lower left entry g of

I

F is

a

Bdcklund transform of

f,

(7.9)

and g +

9

and

H.

=

We want to compute the mean curvature Bdcklund transforms we know

(9.3), (9.9). Likewise, 1 if

NO

we

obtain

0)

(0 1) ( R) (1-f)=(lf (R (1 (1 f) (1 Hg) ( -j ) (1 -Hg) (1 (1 f) (9 HgfI (1 -f) H

1

0

0 1

0 1

9 -fi

1

0

properties of

(10.5)

From Lemma 14

-R..

=

From the

sphere Sg.

Hg=f-f,

Nq=-R, see

(10.4) imply

-

ft

-f

0

0

1

0

0

-f 1

-

01

This

implies H

In

particular f

on

that set. It follows that

E Im

-

=

(I 1) ( 0

=

g

=

HH,

-

-N +

=

since H

0

=

-R

f

-I

whence

(f

-

on an

f)H. open set would

0 -N +

N and H E R for

f

(f

M

:

-

-4

-g)

I)H) O Im H

=

mean w

1

R3,

(I H) (-N ; (f-f)H g), (Ig-H) (N+(I-f)H 0) (1H-g) (I g) ( 1-f N+(I-f)H) (i -g) g

S*

H,

=

g

Sg and, because, R

N-

and -N

-Rg

1

0

-

01

-

0

1

0

1

f

-S9-

f

f

N

01

H

0

0

N

1

0

0

1

-

1

0

10 Willmore Surfaces in

66

We have

now

mean curvature spheres of g intersect S' orhyperbolic planes. We know that, using affine Euclidean metric, the mean curvature spheres are tangent

shown that the

and therefore

thogonally,

S3

coordinates and

a

to g and have the

under conformal

are

same mean

curvature vector

as

of the ambient metric.

changes

g. This

Therefore,

property remains in the

hyperbolic

g has mean curvature 0, and hence is minimal. If A =- 0, then w = 0, and the "Micklund tr'ansform" is constant, which may be considered as a degenerate minimal surface. In general g will be singular in the (isolated) zeros

metric,

of

dg

=

1w, 2

but minimal elsewhere.

We show the in

converse:

hyperbolic 4-space,

Let L be

i.e. with S*

=

4

(SdS + *dS)*

4

holomorphic

curve, minimal

-S. Then I

1

1

A*

immersed

an =

(dSS

-

*dS)

4

(SdS + *dS)

-A,

and therefore also

(d * A)* From

Proposition 15

we

=

-d

*

A.

have I

(f

A

d

dw

dw

4

-f dw f) -dw f

Therefore

jw-,

dw

f dw

=

dwf,

and hence

dw(f But

f

is not in

sition 12

S1,

and therefore

f)

dw=O,

=

0.

i.e L is Willmore.

=

-A

implies

w

=

=

(W

-77D. Rom S*

-S

1w

the backward Bdcklund transform h with dh

conditions is in Im H To

Similarly, Propo-

yields *A

and A*

+

=

we -

know TI

=

-H,

and

dH and suitable initial

R.

summarize

Theorem 9

(Richter [11]).

Let <

.,.

> be

an

indefinite

hermitian

product

IV. Then the isotropic lines form an S' C HP', while the two complemen-. tary discs inherit complete hyperbolic metrics. Let L be a Willmore surface in S' C HP'. Then a suitable forward Bdcklund transform of L is hyperon

bolic minimal.

Conversely, an immersed holomorphic curve that is hyperbolic Willmore, and a suitable backward Bdcklund transformation is a Willmore surface in S'. (In both cases the Bdcklund transforms may have

minimal is

singularities.)