Doc Math J DMV

Quaternionic Analysis

on Riemann Surfaces and Differential Geometry

1 2

Franz Pedit and Ulrich Pinkall

Abstract We present a new approach to the dierential geometry of

3 4

surfaces in R and R that treats this theory as a quaternionied ver

sion of the complex analysis and algebraic geometry of Riemann surfaces

Mathematics Sub ject Classication C H

Introduction

Meromorphic functions

Let M b e a Riemann surface Thus M is a twodimensional dierentiable manifold

equipp ed with an almost complex structure J ie on each tangent space T M we

p

2

have an endomorphism J satisfying J making T M into a onedimensional

p

complex vector space J induces an op eration on forms dened as

X JX

A map f M C is called holomorphic if

df i df

1

A map f M C fg is called meromorphic if at each p oint either f or f is

holomorphic Geometrically a meromorphic function on M is just an orientation

2

preserving p ossibly branched conformal immersion into the plane C R or

1 2

rather the sphere CP S

4

Now consider C as emb edded in the quaternions H R Every immersed

4 4

surface in R can b e descib ed by a conformal immersion f M R where M is

a suitable Riemann surface In Section we will show that conformality can again

b e expressed by an equation like the CauchyRiemann equations

df N df

2 3

where now N M S in R Im H is a map into the purely imaginary

quaternions of norm In the imp ortant sp ecial case where f takes values in

1

Research supp orted by NSF grants DMS DMS and SFB at TUBerlin

2

Research supp orted by SFB at TUBerlin

Documenta Mathematica Extra Volume ICM II

Franz Pedit and Ulrich Pinkall

3

R Im H N is just the unit normal vector for the surface f In the dierential

4

geometry of surfaces in R N is called the left normal vector of f Meromorphic

4 4 1

functions f M R fg S HP are dened as in the complex case

To summarize from the quaternionic viewp oint i is just one sp ecial imag

inary quaternion of norm one The transition from complex analysis to surface

theory is done by

i leaving the Riemann surface as it is

ii allowing the whole of H fg as the target space of meromorphic functions

iii writing the Cauchy Riemann equations with a variable i

Line bundles

A classical metho d to construct meromorphic functions on a Riemann surface M

is to take the quotient of two holomorphic sections of a holomorphic line bundle

n

over M For example if M is realized as an algebraic curve in CP then the

n

ane co ordinate functions on CP are quotients of holomorphic sections of the

inverse of the tautological bundle over M Another common way to construct

meromorphic functions is to take quotients of theta functions which also can b e

viewed as sections of certain holomorphic line bundles over M

In Section we intro duce the notion of a holomorphic quaternionic line bundle

L over M Quotients of holomorphic sections of such bundles are meromorphic

conformal maps into H and every conformal map can b e obtained as such a quotient

in a unique way

Every complex holomorphic line bundle E gives rise to a certain holomor

phic quaternionic bundle L E E The deviation of a general holomorphic

quaternionic bundle L from just b eing a doubled complex bundle can b e globally

measured by a quantity

Z

2

W jQj

called the Willmore functional of L Here Q is a certain tensor eld the Hopf

eld

On compact surfaces W is up to a constant the Willmore functional in the

4

usual sense of surface theory of f M H R where f is the quotient of any

two holomorphic sections of L

Abelian differentials

A second classical metho d to construct meromorphic functions on a Riemann

surface M is to use Ab elian dierentials ie integrals of meromorphic forms

In the quaternionic theory there is no go o d analog of the canonical bundle K On

the other hand also in the complex case forms often arise as pro ducts of sections

1

of two line bundles E and KE Notably this is the case in situations where

the RiemannRo ch theorem is applied This setup carries over p erfectly to the

quaternionic case including the RiemannRo ch theorem itself

Documenta Mathematica Extra Volume ICM II

Quaternionic Analysis on Riemann Surfaces

We show that for each holomorphic quaternionic line bundle L there exists

1

a certain holomorphic quaternionic line bundle KL such that any holomorphic

1

section of L can b e multiplied with any holomorphic section of KL the

pro duct b eing a closed H valued form that lo cally integrates to a confor

mal map f into H

df

1

In the case where KL is isomorphic to L itself we call L a spin bundle If

is a nowhere vanishing holomorphic section of a spin bundle then

df

3

denes a conformal immersion into R This construction is in fact a more intrinsic

version of the Weierstrassrepresentation for general surfaces in space that has

received much attention in the recent literature just as when expressed in

4

co ordinates gives a representation for surfaces in R The Hopf eld Q mentioned

ab ove can b e identied as the Diracp otential or mean curvature halfdensity

of the surface f

H jdf j Q

Here H is the mean curvature and jdf j is the square ro ot of the induced metric

Applications

The only geometric application discussed in some detail in this pap er is a rigidity

2 3

theorem for spheres if f g S R are two conformal immersions which are

not congruent up to scale but have the same mean curvature halfdensity then

Z

2

H

This inequality is sharp

Many other applications to b e discussed in a more elab orate future pap er

will concern the geometry of Willmore surfaces critical p oints of the Willmore

3 4

functional b oth in R and R Moreover rudiments of an algebraic geometry of

n

holomorphic curves in quaternionic pro jective space HP can b e develop ed

Conformal surfaces the standard example

3

Let M b e a Riemann surface and f M R a smo oth map The map f is a

conformal immersion if

i df v is p erp endicular to df J v for any tangent vector v where J is the

complex structure on M and

ii jdf v j jdf J v j for v

Documenta Mathematica Extra Volume ICM II

Franz Pedit and Ulrich Pinkall

2

If N M S is the oriented unit normal to f then the conformality condition

can b e rephrased as

df J v N df v

To see the similarity with complex function theory we rewrite this condition using

3

quaternions H R Im H We will always think of R Im H as the imaginary

3

quaternions If x y R then

xy x y x y

With the notation the conformality condition for f b ecomes For the rest

of the article we will take this to b e the dening equation for conformality also in

4

the case of maps not necessarily immersions into R

4

Definition A map f M R H is conformal if there exists a map

2

N M H such that N and

df N df

At immersed p oints this is equivalent to the usual notion of conformality and

3

f determines N uniquely If f is R valued then N is the oriented unit normal

but otherwise N is not normal to f We will call N the left normal to f Moreover

1 1

if f is conformal so is its Mo ebius inversion f with left normal f N f Thus

the ab ove denition is Mo ebius invariant and hence denes conformality of maps

4 1

f M S HP H fg

Holomorphic quaternionic line bundles

A quaternionic line bund le L over a base manifold is a smo oth rank real vector

bundle whose b ers have the structure of dimensional quaternionic right vector

spaces varying smo othly over the base Two quaternionic line bundles L and L

1 2

are isomorphic if there exists a smo oth bundle isomorphism A L L that is

1 2

quaternionic linear on each b er We adopt the usual notation Hom L L and

H 1 2

End L Hom L L etc for the spaces of quaternionic linear maps

H H

The zero section of a quaternionic line bundle over an oriented surface has

co dimension so that transverse sections have no zeros Thus any quaternionic

line bundle over a Riemann surface M is smo othly isomorphic to M H

Complex quaternionic line bundles

Example Given a conformal map f M H with left normal N the quater

nionic line bundle L M H also has a complex structure J L L given by

J N for L

We make this additional complex structure part of our theory

Definition A complex quaternionic line bund le over a base manifold is a pair

L J where L is a quaternionic line bundle and J End L is a quaternionic

H

2

linear endomorphism such that J

Documenta Mathematica Extra Volume ICM II

Quaternionic Analysis on Riemann Surfaces

Put dierently a complex quaternionic line bundle is a rank two left complex

vector bundle whose complex structure is compatible with the right quaternionic

structure Two complex quaternionic line bundles are isomorphic if the quater

nionic linear isomorphism is also left complex linear

The dual of a quaternionic line bundle L

1

L f L H quaternionic linearg

has a natural structure of a left quaternionic line bundle via

1 1

for H L and L Using conjugation we can regard L as a

right quaternionic line bundle If L has a complex structure then the

1

complex structure on L is given by

J J

1

so that L is also complex quaternionic

Any complex quaternionic line bundle L can b e tensored on the left by a

complex line bundle E yielding the complex quaternionic line bundle EL On a

Riemann surface M we have the canonical and anticanonical bundles K and K

It is easy to see that

KL f TM L J g

and

KL f TM L J g

In this way we have split the quaternionic rank bundle Hom T M L which

R

has a left complex structure given by as a direct sum KL KL of two complex

quaternionic line bundles

If E is a complex line bundle then L E E b ecomes a complex quater

E

nionic line bundle with J i i and right quaternionic structure

1 2 1 2

given by

i i i j

1 2 1 2 1 2 2 1

Conversely for a given complex quaternionic line bundle L J we let E f

L

L J ig b e the i eigenspace of J Then E L is a complex line subbundle

and E E is isomorphic to L This leads to the following

L L

Theorem The above correspondences

E L L E

E L

give a bijection between isomorphism classes of complex line bund les and isomor

phism classes of complex quaternionic line bund les This bijection is equivariant

with respect to left tensoring by complex line bund les and respects dualization

Definition The degree of a complex quaternioninc line bundle L over a

compact Riemann surface is the degree of the underlying complex line bundle E

L

ie deg L deg E

L

Documenta Mathematica Extra Volume ICM II

Franz Pedit and Ulrich Pinkall

On a compact Riemann surface isomorphism classes of complex line bundles

are characterized by their degrees Thus complex quaternionic line bundles also

are characterized by their degrees Given a trivializing section of L we have

2 2

J N for some N M S Im H N and one easily checks that

deg L deg N

Holomorphic quaternionic line bundles

Definition Let L J b e a complex quaternionic line bundle over a Riemann

surface M and let L denote the smo oth sections of L A holomorphic structure

on L is given by a quaternionic linear map

D L KL

satisfying

d J d D D

for M H

The quaternionic linear subspace kerD L is called the space of holomor

0

phic sections and is denoted by H L

One can check that the KLpart of a quaternionic connection on L gives a

holomorphic structure D which may b e used as motivation for the ab ove formula

Any complex holomorphic structure on the underlying complex line bundle

E is an example of a holomorphic structure D These holomorphic

L

structures on L are characterized by the condition that D and J commute The

failure to commute is measured by

Q D JDJ



which is a section of T M End L Now

H

End L End L End L

H +

splits into linear maps commuting and anticommuting with J The former is a

trivial complex bundle with global sections I d and J The latter is a nontrivial

1

E E Since Q anticommutes with J complex line bundle isomorphic to

L

L

and satises Q J Q we see that Q is a section of the complex line bundle

K End L We call Q the Hopf eld of the holomorphic quaternionic line bundle

L Thus any holomorphic structure D is uniquely decomp osed into

D Q

with commuting with J Vanishing of the Hopf eld Q characterizes the usual

complex holomorphic structures Two quaternionic holomorphic line bundles are

isomorphic if there is an isomorphism of complex quaternionic line bundles which

intertwines the resp ective holomorphic structures On a compact Riemann surface

this implies that the underlying complex holomorphic structures are isomorphic

Documenta Mathematica Extra Volume ICM II

Quaternionic Analysis on Riemann Surfaces

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and that the Hopf elds are related up to a constant phase u S Thus the

mo duli space of quaternionic holomorphic structures b ers over the Picard group

1

of M The b er K End L S over E is given by the Hopf elds

E

A global invariant of the quaternionic holomorphic line bundle L is obtained

2

by integrating the length of the Hopf eld Q we dene the density jQj by

2

Q Q jQj v I d v TM

v v

M

where we identify J invariant quadratic forms with forms on M The Wil lmore

2

functional of D is the L norm of the Hopf eld

Z

2 2

jjQjj jQj

The vanishing of jjQjj characterizes the complex holomorphic theory

Example We have already seen that a conformal map f M H with left

2

normal N M S induces the complex quaternionic bundle L M H with

J N We dene the canonical holomorphic structure D on L to b e the one for

which the constant sections are holomorphic ie D is characterized by D

Any other section of L is of the form for some M H and implies

that is holomorphic i

d N d

Thus the holomorphic sections of L are precisely the conformal maps with the

0

same left normal as f In particular dimH L

1

The Mo ebius invariance of the holomorhic structure follows since f induces

an isomorphic holomorphic structure on M H Thus we have assigned to each

1 4

conformal map into HP S a quaternionic holomorphic line bundle with at

least two holomorphic sections

1

The Hopf eld for this holomorphic structure is Q N dN dN and

4

2 ? 2 2

jH j K K jdf j jQj

?

where H is the mean curvature vector of f and K is the curvature of the normal

2

bundle We see that jQj is a Mo ebius invariant density which is consistent with

the Mo ebius invariance of our setup Thus the Willmore energy of our holomor

R

2 2

phic structure jjQjj jQj W f is up to top ological constants just the

Willmore energy of f

So far we have seen how a conformal map induces a holomorphic line bundle

with at least two holomorphic sections As in the classical complex theory we

4

have the converse construction ie all conformal maps into S arise as quotients

of holomorphic sections

Example Let L M b e a quaternionic holomorphic line bundle and assume

0

that dimH L with holomorphic sections such that has no zeros

2

Then J N for some N M S We dene f M H by

f

Documenta Mathematica Extra Volume ICM II

Franz Pedit and Ulrich Pinkall

then implies that df N df ie f is conformal with left normal N

An interesting sp ecial case comes from conformal maps with Q It can

b e shown that they are sup erconformal in the sense that their curvature ellipse is

a circle Since Q sup erconformal maps are critical for the Willmore energy

4 3

and thus Willmore surfaces in S In case f is R valued Q simply means

that f is a conformal map into the sphere The sup erconformal maps all arise

3 4

as pro jections from holomorphic maps into the twistor space CP over S and

have b een studied by various authors In our theory these maps arise as

quaternionic quotients of holomorphic sections of doubled complex holomorphic

line bundles

In the ab ove construction the structure of zeros of quaternionic holomorphic

sections b ecomes imp ortant Applying a result of Aronsza jn we can show

Theorem Let be a nontrivial holomorphic section of a quaternionic holo

morphic line bund le L over a Riemann surface M Then the zeros of are isolated

and if z is a centered local coordinate near a zero p M

k k +1

z O jz j

where is a local nowhere vanishing section of L The integer k and the value

p L are wel ldened independent of choices We dene the order of the zero

p

p of by ord k

p

We conclude this section with a degree formula

Theorem Let be a nontrivial section of a quaternionic holomorphic line

bund le L over a compact Riemann surface M Then

X

2

deg L jjQjj ord

p

p2M

In contrast to the complex holomorphic case where negative degree bundles

do not have holomorphic sections we see that in the quaternionic theory the

Willmore energy of the bundle comp ensates for this failure and we still can have

holomorphic sections

1

Equality in is attained by holomorphic bundles L where L E

E is a doubled complex holomorphic bundle E and L has a nowhere vanishing

1

meromorphic section The holomorphic structure on L then is obtained by

1

to b e holomorphic dening

We conjecture the following lower b ound for the Willmore energy on holo

0

morphic line bundles over the sphere let n dimH L and d deg L then

2 2

jjQjj n nd

Examples are known where equality holds Using the degree formula we can prove

under certain nondegeneracy assumptions For d the case of spin

bundles see the next section this estimate has b een conjectured by Taimanov

Documenta Mathematica Extra Volume ICM II

Quaternionic Analysis on Riemann Surfaces

Abelian differentials

Definition A pairing b etween two complex quaternionic line bundles L and



L over M is a nowhere vanishing real bilinear bundle map L L T M H

satisfying

J J

for all H L L

A pairing b etween L and L is actually the same as an isomorphism of complex

1

quaternionic line bundles L KL given as where

X

X

If is a form on M with values in L and is a section of L then we dene an

Lvalued form as

X Y X Y Y X

Similarly for L and a form with values in L we set

X Y X Y Y X

Lemma For each K E nd L there is a unique K E nd L such that

for al l L L The map is complex antilinear

J J

Theorem If two complex quaternionic line bund les L and L are paired then

for any holomorphic structure D on L there is a unique holomorphic structure D

on L such that for each L L we have

d D D

The Hopf elds Q and Q of D and D are conjugate

Q Q

Thus a holomorphic structure on L determines a unique holomorphic struc

1 1

such that L and KL b ecome paired holomorphic bundles In this ture on KL

situation the RiemannRo ch theorem is true in the familiar form of the theory of

complex line bundles on compact Riemann surfaces of genus g we have

0 0 1

dimH L dimH KL deg L g

4

Theorem suggests a way to construct conformal immersions f M R H

0

If L and L are paired holomorphic bundles and H L are b oth nowhere

vanishing sections then is a closed form that integrates to a conformal

4

immersion into R p ossibly with translational p erio ds In fact this construction

is completely general

Documenta Mathematica Extra Volume ICM II

Franz Pedit and Ulrich Pinkall

Theorem Let f M H be a conformal immersion Then there exist

paired holomorphic quaternionic line bund les L L and nowhere vanishing sections

0 0

H L H L such that

df

L L and are uniquely determined by f up to isomorphism

In the setup of the theorem cho ose lo cally nonvanishing sections L

L satisfying J i J i Then there is a

R Ri valued co ordinate chart z on M satisfying

dz

We can write

j j

1 2 1 2

with R Rivalued functions Expanding we obtain a generalization

3 4

of the Weierstrass representation of surfaces in R to surfaces in R The

equations Q and Q unravel to Dirac equations for and

Definition A holomorphic line bundle over M is called a spin bund le if

there exists a pairing of with itself such that the second holomorphic structure

on provided by Theorem coincides with the original one

As a direct consequence of the denition of a pairing we obtain in the case of

spin bundles the relation

Therefore for any holomorphic section of a spin bundle the equation

df

3

denes a conformal map into R Im H p ossibly with translational p erio ds This

is in fact a co ordinatefree version of the Weierstrass representation for surfaces in

3

R which could b e obtained by a calculation similar to the one given ab ove for

4

R We now show that the Dirac p otential H jdf j featured in this representation

can b e identied with the Hopf eld Q of

For a spin bundle the map K End puts a real structure on

K End and therefore allows us to dene a real line bundle

R ReK End f K End g

12

We now show that R can b e identied with the real line bundle D of half

densities over M A halfdensity U is a function on the tangent bundle TM which

is of the form

q

g X X U X p

p p p

1

where C M and g is a Riemannian metric compatible with the given confor

mal structure For each the function X j X j is a halfdensity

Documenta Mathematica Extra Volume ICM II

Quaternionic Analysis on Riemann Surfaces

On the other hand it can b e checked that for each we can dene a

section of R as

J

12

There is a canonical isomorphism R D which takes to j j for all

Theorem Let be a holomorphic section of a spin bund le over M Then

3

there is a conformal immersion f M R on the universal cover of M with

only translational periods such that

df

Identifying the halfdensity jdf j as explained above with a section of K E nd

the mean curvature of f is given in terms of the Hopf eld Q of as

H jdf j Q

We conclude by indicating a pro of of the rigidity theorem for spheres stated

in Section The hyp otheses imply that in the situation of the theorem ab ove

has a dimensional space of holomorphic sections Since in the case at hand the

conjecture has b een proven we take n and d deg and obtain

Z Z

2 2 2

H jdf j jQj

References

D Ferus F Pedit U Pinkall Quaternionic line bund les and dierential

geometry of surfaces In preparation

4

T Friedrich The geometry of tholomorphic surfaces in S Math Nachr

S Montiel Spherical Wil lmore surfaces in the foursphere Preprint Granada

G Kamb erov F Pedit U Pinkall Bonnet pairs and isothermic surfaces

Duke Math J Vol No

B G Konop olchenko Induced surfaces and their integrable dynamics Stud

Appl Math Vol

3

I A Taimanov The Weierstrass representation of spheres in R the Wil l

more numbers and soliton spheres Preprint No SFB TU Berlin

Documenta Mathematica Extra Volume ICM II

Franz Pedit and Ulrich Pinkall

Ulrich Pinkall Franz Pedit

Fachb ereich Mathematik Department of Mathematics

TU Berlin University of Massachusetts

Strasse des Juni Amherst MA

Berlin USA

Germany franzgangumassedu

pinkallmathtub erlinde

Documenta Mathematica Extra Volume ICM II