THE SPINOR AND WEIERSTRASS REPRESENTATIONS OF SURFACES IN SPACE IVAN SOLONENKO

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THE SPINOR AND WEIERSTRASS REPRESENTATIONS OF

SURFACES IN SPACE

IVAN SOLONENKO

Abstract.

In this paper, following Sullivan ([10]) and Kusner and Schmitt ([5]), we study

conformal immersions of Riemann surfaces into the three-dimensional Euclidean space.

Regarding such immersions as special bundle maps from the tangent bundle of the surface

to the cotangent bundle of the 2-dimensional sphere, we generalize the classical Weierstrass

representation of minimal surfaces to the case of arbitrary conformal immersions. We

study how such an immersion gives rise to a spin structure on the surface together with a

pair of spinors and how the immersion itself can be studied by means of these spinors.

1. Introduction

The Weierstrass representation of a minimal surface in

is a classical method that allows

one to describe the surface – and its diﬀerential-geometric properties like the mean and

Gaussian curvatures and the Gauss map – by means of a pair of functions on an open

subset of the complex plane, one of which is holomorphic and the other one is meromorphic.

Such a representation is local and depends on the choice of a local holomorphic coordinate

on the surface (also known as isothermal coordinates, especially in earlier literature on the

subject).

In his 1989 paper [10], D. Sullivan observed that the Weierstrass representation can be

rendered coordinate-free and global by replacing the aforementioned pair of functions

with a triple of holomorphic 1-forms on the surface. This triple can be thought of as a

holomorphic bundle map from the tangent bundle of the surface to the cotangent bundle of

the 2-sphere. The unique spin structure of

can be then pulled back along this bundle

map to induce a spin structure on the surface together with a pair of holomorphic sections of

the corresponding line bundle of spinors. All of the original formulas describing the surface

and its geometry can be rewritten neatly in terms of these two spinors. Conversely, given a

Riemann surface with a ﬁxed spin structure and a (nondegenerate) pair of holomorphic

spinors, one can write down an explicit integral formula involving these spinors that gives a

(possibly periodic) minimal immersion of the surface into R3.

Later in their article [5], R. Kusner and N. Schmitt observed that the minimality condition

in Sullivan’s paper can be relinquished. The resulting ’generalized Weierstrass represen-

tation’ works for an arbitrary conformal immersion of a Riemann surface to

. This

way, holomorphic 1-forms and spinors become smooth (1

0)-forms and smooth spinors,

arXiv:2210.15558v1 [math.DG] 27 Oct 2022

2 IVAN SOLONENKO

respectively, and one has an added integrability condition that ensures that an abstract

pair of spinors on Minduces a (possibly periodic) conformal immersion of Mto R3.

Both Sullivan’s and Kusner and Schmitt’s expositions of the subject are rather short on

detail and barely contain any proofs. The present article is an attempt to rectify this

issue. Following and improving on the papers of Sullivan, Kusner, and Schmitt, we give a

detailed and thorough exposition of the generalized Weierstrass and spinor representations

of Riemann surfaces in R3.

The paper is organized as follows. In Section 2we review the classical theory of Weierstrass

representations of minimal surfaces in

. In Section 3we discuss a number of technical

points related to

CP1

, its canonical and tautological line bundles, and its Veronese em-

bedding into

CP2

. In Section 4we deﬁne (generalized) Weierstrass representations of a

Riemann surface and study their relation to conformal immersions of the surface to

We also investigate how the geometric properties of a conformal immersion (for instance,

whether it is minimal) can be expressed via its associated Weierstrass representation and

how the latter can be used to recover the immersion itself. Finally, in Section 5we introduce

the notion of a spinor representation of a Riemann surface and show that it is essentially

equivalent to that of a Weierstrass representation. We then reimagine the salient points of

Section 4in the language of spinor representations. As a side quest, we also give a formal

categorical argument why a spin structure on a principal

(2)-bundle is the same as a

so-called square root of the corresponding Hermitian line bundle.

2. The classical Weierstrass representation

We start with a brief review of the classical theory of Weierstrass representations. Let

M⊂R3

be an oriented minimal surface. By passing to local isothermal coordinates, we

can think of

as the image of a conformal embedding

ϕ:U→R3

, where

is an open

subset of

(we can actually allow

to be an immersion, not necessarily an embedding). If

we write

, the conformality and minimality conditions on

can be rewritten as

•(ϕis a conformal) ||ϕu|| =||ϕv||,hϕu|ϕvi= 0 ⇔ hϕz|ϕzi= 0,

•(ϕis minimal ⇔harmonic) ∆ϕ = 0 ⇔ϕz¯z= 0.

Now assume Uis simply connected. Then there is a smooth map b

ϕ:U→R3satisfying

b

ϕu=−ϕv,

ϕv=ϕu,

and it is unique up to a translation in

. It is also a conformal minimal immersion (

adjoint

). The conditions on

are taken to be as they are precisely so that the immersion

ϕ:U→C3

is holomorphic. It is also isotropic, meaning that

hf0|f0i

= 0.

Conversely, given a holomorphic isotropic immersion of

, its real and imaginary

1Throughout the paper, h−|−i will stand for the standard bilinear form on Rnor Cn.

THE SPINOR AND WEIERSTRASS REPRESENTATIONS OF SURFACES 3

parts are adjoint conformal minimal immersions of

. If we write

= (

Φ1,Φ2,Φ3

we have:

Φ2

1+Φ2

2+Φ2

3= 0.

This condition can be rewritten as

(Φ1+iΦ2)(Φ1−iΦ2) = −Φ2

3.(2.1)

Casting aside the cases when

Φ1

iΦ2

on the entire

(which is deﬁnitely not the case

unless ϕis an immersion into a horizontal plane), we can rewrite this as

Φ1+iΦ2=−Φ2

Φ1−iΦ2

We introduce a pair of functions on U:

µ=Φ1−iΦ2,ν=Φ3

Φ1−iΦ2

Clearly,

is holomorphic and

is meromorphic. Observe that the function

µν2

−Φ1−iΦ2

is holomorphic. What is more, since

is an immersion, all

Φi

’s cannot vanish simultaneously

at any point of

. Together with

(2.1)

, this implies that if

p∈U

is a zero of

, it cannot

also be a zero of

µν2

. We conclude that

and

are related by the following condition:

the set of zeroes of

coincides with the set of poles of

, and given any point

in this set,

ordp(µ) = −2ordp(ν).

We can recover f0using these two functions:

Φ1=µ

2(1 −ν2),Φ2=iµ

2(1 + ν2),Φ3=µν.

The function

, in turn, determines

(and hence

and

) up to translations in

means of integration along paths in U:

f(p) = f(p0) + Zp

f0dz,

where

p0∈U

is ﬁxed. This integral does not depend on the choice of a path because the

1-form

f0dz

is exact. We see that, up to a translation in

, the initial immersion

can be expressed as the real part of an integral of some holomorphic 1-form (or rather a

triple of those) written in terms of µand ν:

ϕ(p) = ϕ(p0) + Re Zp

2(1 −ν2)dz, Zp

iµ

2(1 + ν2)dz, Zp

µνdz.(2.2)

Formula

(2.2)

is called the Weierstrass representation of

. Going in the opposite direction,

one can start with a holomorphic function

and a meromorphic function

that are related

4 IVAN SOLONENKO

by the aforementioned condition

: the zeroes of

are precisely the poles of

and, given

any such zero/pole

p∈U

ordp

(

) =

−

ordp

(

). Using the same integral formula

(2.2)

one obtains a conformal minimal immersion U→R3.

We will render this theory coordinate-free and generalize it to arbitrary conformal immersions

of Riemann surfaces into

. Having done that, we will be able to reinterpret the theory

completely in the language of spinors.

3. The sphere, its canonical bundle, and the Veronese embedding

Before we proceed to main part, we need to discuss a number of technical points regarding

the canonical bundle of the sphere and the Veronese embedding.

Agreement.

Whenever we have a map between smooth manifolds, it is assumed to be

smooth, unless mentioned otherwise. Also, we are going to meet a lot of interplay between

real and complex geometry, so let us agree on the following. Whenever

is a complex

vector space or complex vector bundle,

E∗

will stand for its real dual. Its complex dual will

be denoted by

E∗1,0⊆E∗

. The two are surely isomorphic as complex vector spaces/bundles

by means of taking the real part inside

E∗

, where the complex structure on

E∗

is taken to

be dual to the one on

. We often suppress this isomorphism from the notation. In case we

want to stress whether the functionals being considered are real- or complex-valued, we will

stick to this notation to avoid confusion. The conjugate

E∗0,1

E∗1,0⊆E∗

is the antidual

to E(the space/bundle of C-antilinear functionals).

The ﬁrst technical moment is how one identiﬁes

with

CP1

. There is a bunch of similar

ways to do this, all of them use the stereographic projection, but there is a degree of freedom

in picking a pole of

and an aﬃne chart on

CP1

. We choose the stereographic projection

from the North pole N= (0,0,1):

τ:S2{N} −→

∼C,(x, y, z)7→ x+iy

1−z.

Then we identify

with the aﬃne chart

{[z0:z1]|z06= 0}

CP1{[0 : 1]} ⊂ CP1

and send Nto the missing point [0 : 1]. The resulting map S2−→

∼CP1is given by

(x, y, z)7→ ([1 −z:x+iy],if z6= 1,

[0 : 1],if z= 1,

and it is a biholomorphism when the sphere is oriented by a vector ﬁeld pointing inside

the unit ball (this is the opposite of the boundary orientation of

∂B2

). The inverse

One can actually relax this condition to the requirement that

µν2

be holomorphic, but then one has

to consider more general (so-called “branched”) minimal immersions:

is required to be harmonic,

nonconstant, and conformal outside of its singular points (which will automatically be discrete).

THE SPINOR AND WEIERSTRASS REPRESENTATIONS OF SURFACES 5

mapping looks like this:







[1 : z]7→ 2Rez

|z|2+ 1,2Imz

|z|2+ 1,|z|2−1

|z|2+ 1,

[0 : 1] 7→ N.

The projective line, in turn, can be identiﬁed with the standard conic in

CP2

by means of

the Veronese embedding:

CP1→CP2,[z0:z1]7→ [z2

0:z0z1:z2

1].

The image of this map is cut out be the equation

w0w2

. It is a unique smooth plane

conic up to projective automorphisms of

CP2

. It would be convenient for us to apply such

an automorphism, i.e. linearly change the coordinates on C3:









ew0=w0−w2;

ew1=i(w0+w2);

ew2= 2w1.

In these new coordinates, which we are going to denote

w0, w1, w2

from now on, our conic

becomes [

] =

{[w0:w1:w2]∈CP2|w2

0+w2

1+w2

2= 0}

. Let us denote the aﬃne cone

over it by

Q⊂C3

. It can also be described as the set of isotropic vectors of the standard

nondegenerate quadratic form on

. In these new coordinates, the Veronese embedding is

given by:

θ:CP1−→

∼[Q]⊂CP2,[z0:z1]7→ [z2

0−z2

1:i(z2

0+z2

1):2z0z1].

This is the projectivization of the map

θ:C2→C3,(z0, z1)7→ (z2

0−z2

1, i(z2

0+z2

1),2z0z1).

The image of

, and, when restricted to the set of nonzero vectors,

gives a two-sheeted

holomorphic covering

C2{0}Q{0}

. We are going to use this map to relate the

tautological bundles over CP1and [Q] to their canonical bundles.

First of all, observe that the tautological line bundle

OCP1

(

−

1) over

CP1

is a line subbundle

of the trivial rank-2 bundle

CP1×C2

. If we restrict the projection of the latter onto

OCP1

(

−

1), we obtain the blow-up of

at the origin. Altogether, we have the following

commutative diagram:

OCP1(−1) {0}∼//

_



C2{0}

_





OCP1(−1) //

((((

CP1

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THESPINORANDWEIERSTRASSREPRESENTATIONSOFSURFACESINSPACEIVANSOLONENKOAbstract.Inthispaper,followingSullivan([10])andKusnerandSchmitt([5]),westudyconformalimmersionsofRiemannsurfacesintothethree-dimensionalEuclideanspace.Regardingsuchimmersionsasspecialbundlemapsfromthetangentbundleofthesurfacetotheco...

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