The Spinor Representation of

Minimal Surfaces

Rob Kusner Mathematics Department

University of Massachusetts at Amherst

Nick Schmitt

Center for Geometry, Analysis, Numerics and Graphics University of Massachusetts at Amherst

Contents

Introduction 1

I Spinors, Regular Homotopy Classes and the Arf Invariant 4

1 The spinor representation 4 2 Spin structures and spin manifolds 4 3 The quadratic form associated to a spin structure 5 4 The spin structure on the Riemann sphere 8 5 The spinor representation of a surface 9 6 Regular homotopy classes and spin structures 11 7 Theta characteristics and spin structures 11 8 Spin structures on hyperelliptic Riemann surfaces 12 9 Group action on spinors 17 10 Periods 19 11 Spin structures and nonorientable surfaces 19

II Minimal Immersions with Embedded Planar Ends 22

12 Embedded planar ends 22 13 Algebraic characterization of embedded planar ends 22 14 Embedded planar ends and spinors 23 15 F and K as vector spaces 25 16 A bilinear form Ω which kills K 27

III Classification and Examples 30

17 Genus zero 30 18 Existence and non-existence of genus zero surfaces 30 19 Moduli spaces of genus zero minimal surfaces 32 20 Ω on the Riemann sphere 33 21 Genus zero surfaces with four embedded planar ends 34 22 Genus zero surfaces with six embedded planar ends 35 23 Projective planes with three embedded planar ends 37 24 Genus one 42 25 Ω on the twisted torus 42 26 Ω on the untwisted tori 43 27 Non-existence of tori with three planar ends 45 28 Minimal tori with four embedded planar ends 48 29 Klein bottles: conformal type, spin structure and periods 51 30 Minimal Klein bottles with embedded planar ends 53 Appendix 58 A The Pfaffian 58 B Elliptic functions 59 References 61 Graphics 63

Abstract

The spinor representation is developed and used to investigate minimal surfaces in ℝ^{3} with embedded planar ends. The moduli spaces of planar-ended minimal spheres and real projective planes are determined, and new families of minimal tori and Klein bottles are given. These surfaces compactify in S^{3} to yield surfaces critical for the Möbius invariant squared mean curvature functional W. On the other hand, all Wcritical spheres and real projective planes arise this way. Thus we determine at the same time the moduli spaces of W-critical spheres and real projective planes via the spinor representation.

Introduction

In this paper we investigate the interplay between spin structures on a Riemann surface M and immersions of M into three-space. Here, a spin structure is a complex line bundle S over M such that S ⊗ S is the holomorphic (co)tangent bundle T (M) of M. Thus we may view a section of a S as a “square root” of a holomorphic 1-form on M. Using this notion of spin structure, in the first part of this paper we develop the notion of the spinor representation of a surface in space, based on an observation of Dennis Sullivan [27]. The classical Weierstrass representation is

(g, η) ⟶ Re ∫ (1 − g^{2}, i(1 + g^{2}), 2g)η,

where g and η are respectively a meromorphic function and one-form on the underlying compact Riemann surface. The spinor representation (Theorem 5) is

(s_{1}, s_{2}) ⟶ Re ∫ (s_{1}^{2} − s_{2}^{2}, i(s_{1}^{2} + s_{2}^{2}), 2s_{1}s_{2}),

where s_{1} and s_{2} are meromorphic sections of a spin structure S. Either representation gives a (weakly) conformal harmonic map M → ℝ^{3}, which therefore parametrizes a (branched) minimal surface.

One feature of the spinor representation is that fundamental topological information, such as the regular homotopy class of the immersion, can be read off directly from the analytic data (Theorem 6). In fact, for the special case where the Riemann surface M is hyperelliptic, we are able to give an explicit calculation of the Arf invariant for the immersion (Theorem 8); the Arf invariant distinguishes whether or not an immersion of an orientable surface is regularly homotopic to an embedding. We also consider in Part I the spinor representation for nonorientable minimal surfaces in terms of a lifting to the orientation double cover (Theorem 11). This is sufficient for constructing examples later in the paper, but is less satisfying theoretically. In a future paper, we plan to consider the general case from the perspective of “pin” structures, and also give a more direct differential geometric treatment of the Arf invariant.

The second part of this paper focuses on general properties of minimal surfaces with embedded planar ends from the viewpoint of the spinor representation. It is wellknown (see [2], [13], [14]) that such surfaces conformally compactify to give extrema for the squared mean curvature integral W = ∫ H^{2}dA popularized by Willmore. Conversely, for genus zero, all W-critical surfaces arise this way [2].

Using the spinor representation to study these special minimal surfaces has the computational advantage of converting certain quadratic conditions to linear ones. This is carried out in Part II, where we refine the tools we need. In fact, associated to a spin structure S on a closed orientable Riemann surface M is a vector space K of sections of S such that pairs of independent sections (s_{1}, s_{2}) from K form the spinor representations of all the minimal immersions of M with embedded planar ends (Theorem 13). Thus the problem of finding all these immersions is reduced to an algebraic problem (Theorem 15). In order to better understand K, a skewsymmetric bilinear form Ω is defined from whose kernel K is computable (Theorem 17).

The third (and final) part of this paper is devoted to the construction of examples and to classification results. Specifically, for a given finite topological type of surface, we want to explore the moduli space M of immersed minimal surfaces (up to similarity) of this type with embedded planar ends: the dimension and topology of M, convergence to degenerate cases (that is, the natural closure of M), and examples with special symmetry (which correspond to singular points of M). The tools mentioned above permit the broad outline of a solution, but require ingenuity to apply in particular cases. For example, the form Ω allows the moduli space to be expressed as a determinantal variety which determines how the location of the ends can vary along the Riemann surface M. However, this determinantal variety is only computable when the number of ends is small. Furthermore, the basic tools, being algebraic geometrical, ignore the real analytic problems of removing periods and branch points. The latter require much subtler and often ad hoc methods.

Previously known results concerning genus zero minimal surfaces with embedded planar ends include the following:

• examples have been found for 4, 6, and every n ≥ 8 ends [2], [14], [23];

• there are no immersed examples with 3, 5, and 7 ends [3];

• the moduli spaces for immersed spheres with 4 and 6 ends, and projective planes

with 3 ends have been determined [3]. In Part III our new theorems include the following:

• a new proof of the non-existence of examples with 3, 5 and 7 ends is given using the skew-symmetric form Ω (Theorem 18);

• the moduli space for 2p ends (2 ≤ p ≤ 7) is shown to be 4(p − 1)-dimensional (Theorem 21);

• the point which compactifies the moduli space of projective planes with 3 ends is proved to be a Möbius strip, and all symmetries of these surfaces are found (Theorem 25). A recent result concerning genus one is the construction in [5] of examples with

four embedded planar ends, assuming a rectangular lattice. We give further results: • there are no three-ended tori (Theorem 26);

• there is a real two-dimensional family of four-ended immersed examples on each conformal type of torus (Theorem 27);

• there exists an immersed Klein bottle with four ends (Theorem 29).

For higher genus, the general methods we have developed here also yield (possibly branched) minimal immersions with embedded planar ends, but it becomes more and more difficult to determine precisely when branch points are absent or periods vanish: we again postpone this case to a future paper.

Most of the theorems presented here were worked out while we visited the Institute for Advanced Study during the 1992 Fall term, and were first recorded in [26]. It is a pleasure to thank the School of Mathematics at the Institute for its hospitality, as well as Sasha Bobenko, Peter Norman and Dennis Sullivan for their comments and interest. In particular, we should mention that Bobenko has recently announced some related results for constant mean curvature surfaces (Surfaces in terms of 2 by 2 matrices: Old and new integrable cases, in: A. Fordy and J. Wood, Harmonic Maps and Integrable Systems: Vieweg, 1994).

Part I

Spinors, Regular Homotopy Classes and the Arf Invariant

1 The spinor representation

The notion of a spin structure is developed and used to describe the spinor representation of a surface in space. Section 3 defines a “quadratic form” which can be used to completely classify the spin structures on a surface, and section 4 computes coordinates for the unique spin structure on the Riemann sphere. In the next two sections, the spinor representation of a surface is explained and related to the regular homotopy class of the surface. Section 7 shows equivalent characterizations of spin structures, the most useful of which will be that of representing spin structures by holomorphic differentials. These differentials are computed on hyperelliptic Riemann surfaces. Section 9 takes up the question of group action on spinors, and computes the group which performs Euclidean similarity transformations. Two surfaces which are transforms of each other under the action of this group are considered to be the same. The final two sections discuss briefly the technicalities of periods and nonorientable surfaces.

2 Spin structures and spin manifolds

A spin structure on an n-dimensional (spin) manifold M is a certain two-sheeted covering map of the SO(n)-frame-bundle on M to a Spin(n)-bundle (see [20], [17]). When n = 2, this notion of spin structure may easily be reduced to the following definition in terms of a quadratic map between complex line bundles, as the figure below depicts:

[figure: M; S; T (M); μ]

Figure 1: Spin structure

\underline{Definition 1.} A spin structure on a Riemann surface M is a complex line bundle S over M together with a smooth surjective fiber-preserving map μ : S ⟶ T (M) to the holomorphic (co)tangent bundle T (M) satisfying

μ(λs) = λ^{2}μ(s) (2.1)

for any section s of S.

Two spin structures (S, μ) and (S′, μ′) on a Riemann surface M are isomorphic if there is a line bundle isomorphism δ : S ⟶ S′ for which μ = μ′δ. Hence two spin structures may be isomorphic as line bundles and yet not be isomorphic as spin structures. The number of nonisomorphic spin structures on a Riemann surface M is equal to the cardinality of H^{1}(M, ℤ_{2}). (This count remains true for spin manifolds in general: see [20].) In particular, if M is a closed Riemann surface of genus g, there are 2^{2g} = #H^{1}(M, ℤ_{2}) such structures on M.

On an annulus A = {r_{1} < z < r_{2}} there are exactly two nonisomorphic spin structures, which can be given explicitly as follows. The tangent bundle T (A) may be identified with A × ℂ by means of the global trivialization

a \frac{∂}{∂z} |_{p} ↦ (p, a).

Let S_{0} = S_{1} = A × ℂ and define maps μ_{k} : S_{k} ⟶ T (A) for k = 0, 1 by

\begin{matrix}μ_{0}(z, w) = (z, w^{2}), \\ μ_{1}(z, w) = (z, zw^{2}). \end{matrix}

Then (S_{k}, μ_{k}) are spin structures on A since μ_{k} satisfies the condition (2.1). Though S_{0} and S_{1} are isomorphic line bundles over A, they are nonisomorphic spin structures. For if S_{0} and S_{1} were isomorphic spin structures with bundle isomorphism δ : S_{0} ⟶ S_{1} satisfying μ_{0} = μ_{1}δ, then δ would be of the form (z, w) ↦ (z, f(z, w)). Then w^{2} = zf^{2}, implying that z has a consistent square root on ℂ^{∗}, which is impossible.

3 The quadratic form associated to a spin structure

In this section, the Riemann surface M, its holomorphic (co)tangent bundle, and the spin structure are replaced with the corresponding real manifold and real vector bundles. In particular, all vector fields in this section are real vector fields.

To each spin structure S on the Riemann surface M we associate a ℤ_{2}-valued quadratic form

q : H_{1}(M, ℤ_{2}) ⟶ ℤ_{2}.

To say that q is quadratic means that for all α_{1}, α_{2} ∈ H_{1}(M, ℤ_{2}) we have

q(c_{1} + c_{2}) = q(c_{1}) + q(c_{2}) + c_{1} · c_{2}.

where c_{1} · c_{2} denotes the mod 2 intersection number of c_{1} with c_{2}.

To define q(c), let α : S^{1} ⟶ M be a smooth embedded representative of c (the existence of such an α follows from results in [19]). Let v be a smooth vector field along α which lifts to a section of the spin structure along α, and let w(α, v) denote the total turning number, mod 2, of the derivative vector α′ against v along α. Define q(c) = w_{v}(c) + 1.

To show that q is quadratic, the following technical lemma is stated without proof. (A Jordan trail is a closed tracing along a curve which tracing does not cross itself. The existence of the Jordan trail is assured in [12].)

\underline{Lemma 2.}

(i) Let α : S^{1} ⟶ M be an immersion, and let a be the number of self-crossing points of α. Let v be a smooth non-zero vector field along α(S^{1}) on M. Let β be a Jordan trail for α. Then

w_{v}(β) = w_{v}(α) + a.

(ii) Let α_{1} and α_{2} : S^{1} ⟶ M be immersions, with a common base point α_{1}(t) = α_{2}(t). Let α_{1} ∗ α_{2} : S^{1} ⟶ M denote the closed curve consisting of α_{1} followed by α_{2}. Let v be a smooth non-zero vector field along α_{1}(S^{1}) ∪ α_{2}(S^{1}). Then

w_{v}(α_{1} ∗ α_{2}) = w_{v}(α_{1}) + w_{v}(α_{2}).

\underline{Lemma 3.} If α : S^{1} ⟶ M is an embedded curve on a spin surface M with spin structure S, and v_{1}, v_{2} are smooth nonzero vector fields along α, then the following are equivalent:

(i) w(α, v_{1}) = w(α, v_{2}) (ii) v_{1} and v_{2} alike lift or do not lift along α to smooth sections of S.

Proof. We may assume M is an annulus containing α(S^{1}) as the unit circle, with spin structure S_{k} (k = 0 or 1) as in section 2. Any vector field S^{1} ⟶ ℂ is of the form t ↦ t^{p}[f(t)]^{2}, where f is smooth and

p = \{ \begin{matrix}k & if v lifts, \\ 1 − k & if v does not lift. \end{matrix}

Then, with w_{α}(h_{1}, h_{2}) defined as the mod 2 winding number of h_{1} against h_{2} (or equivalently, of h_{2}/h_{1}) along α,

w_{α}(v_{1}, v_{2}) = w_{α}(t^{p}[f_{1}(t)]^{2}, t^{q}[f_{2}(t)]^{2}) = w_{α}(t^{p}, t^{q}) ≡ p + q (mod 2)
= \{ \begin{matrix}0 & if v_{1}, v_{2} alike lift or do not lift, \\ 1 & otherwise. \end{matrix}

But w_{α}(v_{1}, v_{2}) = w(α, v_{1}) + w(α, v_{2}), and the result follows. □

\underline{Theorem 4.} The form q : H_{1}(M, ℤ_{2}) ⟶ ℤ_{2} defined above is well-defined, that is, independent of the choice of the vector field v and the choice of embedded representative α. Moreover, q satisfies

q(c_{1} + c_{2}) = q(c_{1}) + q(c_{2}) + c_{1} · c_{2}.

Proof. Let α_{0}, and α_{1} : S^{1} ⟶ M be embedded representatives of c ∈ H_{1}(M, ℤ_{2}). Let v_{0}, v_{1} be smooth nonzero vector fields which lift along α_{0}, α_{1} respectively to sections of the spin structure S. Let α_{t} (t ∈ [0, 1]) be a homotopy of α_{0} and α_{1}. Extend v_{0} to a smooth nonzero vector field in an annulus containing the image of α_{t}.

Then w(α_{t}, v) is a continuous function of t, and an integer, hence it is constant. In particular,

w(α_{0}, v_{0}) = w(α_{1}, v).

But v = v_{0} lifts along α_{0} to a smooth section of S. So v must also lift along α_{1}. But since v_{1} also lifts along α_{1},

w(α_{1}, v) = w(α_{1}, v_{1}).

Thus

w(α_{0}, v_{0}) = w(α_{1}, v_{1}),

showing that q is well-defined.

Now, to show q is quadratic, let α_{1}, α_{2} be embedded representatives of c_{1}, c_{2} ∈ H_{1}(M, ℤ_{2}), and let

a = # of self-crossing points of α_{1} ∗ α_{2} ≡ α_{1} · α_{2} − 1 (mod 2).

Let β be a Jordan trail for α_{1} ∗ α_{2}. Then

w(β, v) = w(α_{1} ∗ α_{2}, v) + a = w(α_{1}, v) + w(α_{2}, v) + α_{1} · α_{2} + 1

by Lemma 2(i). Hence

\begin{aligned}q(c_{1} + c_{2}) & = & w(β, v) + 1 = (w(α_{1}, v) + 1) + (w(α_{2}, v) + 1) + α_{1} · α_{2} \\ & = & q(c_{1}) + q(c_{2}) + c_{1} · c_{2}. \end{aligned}

□

A well-known result (see, for example, [24]) is that the equivalence class of the quadratic form q : H_{1}(M, ℤ_{2}) ⟶ ℤ_{2} under linear changes of bases of H_{1}(M, ℤ_{2}) is determined by its Arf invariant

Arf q = \frac{1}{\sqrt{#H}} ∑_{α∈H}(−1)^{q(α)}, (3.2)

where H = H_{1}(M, ℤ_{2}). The quadraticity of q insures that this invariant has values in {+1, −1}. For a compact surface of genus g, there are 2^{2g−1} + 2^{g−1} spin structures for which the Arf invariant of the corresponding quadratic form is +1, and 2^{2g−1} − 2^{g−1} spin structures for which it is −1 (see section 8).

4 The spin structure on the Riemann sphere

The following description of the unique spin structure on S^{2}, as well as the spinor representation of a surface in the next section, are adapted from [27]. Identify

S^{2} ≅ [Q] = {[z_{1}, z_{2}, z_{3}] ∈ ℂℙ^{2} | z_{1}^{2} + z_{2}^{2} + z_{3}^{2} = 0},

where Q is the null quadric

Q = {(z_{1}, z_{2}, z_{3}) ∈ ℂ^{3} | z_{1}^{2} + z_{2}^{2} + z_{3}^{2} = 0}.

Then T (S^{2}) may be identified with the restriction to [Q] of the tautological line bundle

Taut(ℂℙ^{2}) = {(Λ, x) ∈ ℂℙ^{2} × ℂ^{3} | x ∈ Λ}

(here, ℂℙ^{2} is thought of as the lines in ℂ^{3}), so

(4.3) T (S^{2}) ≅ Taut(ℂℙ^{2})|_{[Q]} = {(Λ, x) ∈ [Q] × Q | x = 0 or π(x) ∈ Λ},

where π : Q ⟶ [Q] is the canonical projection. Given this, the unique spin structure Spin(S^{2}) on S^{2} may then be identified with the tautological line bundle

(4.4) Spin(S^{2}) ≅ Taut(ℂℙ^{1}) ≅ {(Λ, x) ∈ ℂℙ^{1} × ℂ^{2} | x ∈ Λ}, with the associated mapping μ given by

μ([z_{1}, z_{2}], (s_{1}, s_{2})) = ([σ(z_{1}, z_{2})], σ(s_{1}, s_{2})),

where σ : ℂ^{2} ⟶ Q is the map defined by

σ(z_{1}, z_{2}) = (z_{1}^{2} − z_{2}^{2}, i(z_{1}^{2} + z_{2}^{2}), 2z_{1}z_{2}). (4.5)

As may be checked, the map μ satisfies the conditions of Definition 1.

When T (S^{2}) and Spin(S^{2}) are restricted respectively to their nonzero vectors and nonzero spin-vectors, they have single coordinate charts

{nonzero vectors in T (S^{2})} ⟶ Q \ {0}
{nonzero spin-vectors in Spin(S^{2})} ⟶ ℂ^{2} \ {0}

defined by taking the second component in each of (4.3) and (4.4) respectively. In this case, μ may be thought of as the two-to-one covering map σ : ℂ^{2} \ {0} ⟶ Q \ {0}.

5 The spinor representation of a surface

To describe the spinor representation, let M be a Riemann surface with a local complex coordinate z, and X : M ⟶ ℝ^{3} a conformal (but not necessarily minimal) immersion of M into space. Since X is conformal, its z-derivative ∂X = ω can be viewed as a null vector in ℂ^{3}, or via (4.3), as a map into the (co)tangent bundle T (S^{2}); so with the (not necessarily meromorphic) Gauss map g associated to X, we get the bundle map (ω, g) as in Figure 2.

[figure: M; S^{2}; T (M); T (S^{2}); g; ω]

Figure 2: Bundle map

The Weierstrass representation is determined by (g, η) where η is the (not necessarily meromorphic) differential form defined by

ω = (1 − g^{2}, i(1 + g^{2}), 2g) η.

Conversely, given a bundle map (ω, g) of T (M) into T (S^{2}), if ω satisfies the integrability condition Re dω = 0, then

X = Re ∫ ω : M ⟶ ℝ^{3}

is a (possibly periodic) immersion with Gauss map g.

The spinor representation of the immersion, shown in Figure 3, is obtained by lifting ω to the spin structures on M and S^{2}.

[figure: M; S^{2}; T (M); T (S^{2}); S; Spin(S^{2}); g; ω; ω̂; μ; σ]

Figure 3: Spinor representation of a surface

\underline{Theorem 5.} Let M be a connected surface, and (ω, g) a bundle map of T (M) into T (S^{2}). Then

(i) there is a unique spin structure S on M such that ω lifts to a bundle map ω̂ : S ⟶ Spin(S^{2});

(ii) there are exactly two such lifts ω̂, and these differ only by sign.

Proof of (i): Considering Spin(S^{2}) as a ℤ_{2}-bundle on T (S^{2}) when restricted to nonzero spin-vectors and vectors respectively, let S be the (unique) pullback bundle of Spin(S^{2}) under ω, and μ, ω̂ as shown. Extend S, ω̂, and μ to include the zero spin-vectors.

Proof of (ii): If ι : Spin(S^{2}) ⟶ Spin(S^{2}) is the order-two deck transformation for the covering Spin(S^{2}) ⟶ T (S^{2}), then ι ∘ ω̂ is another map which in place of ω̂ makes the diagram commute. Conversely, if ζ : S ⟶ Spin(S^{2}) is such a map, then for x ∈ S, ζ(x) is ω̂(x) or ι ∘ ω̂(x) and continuity implies that ζ = ω̂ or ιω̂. □

The spinor representation is determined by the pair of sections ω̂ = (s_{1}, s_{2}) of S related to ω by the equation

ω = (s_{1}^{2} − s_{2}^{2}, i(s_{1}^{2} + s_{2}^{2}), 2s_{1}s_{2}).

Thus the Weierstrass representation and the spinor representation are related by the equations

η = s_{1}^{2} and g = s_{2}/s_{1}.

The case of a nonorientable M is dealt with in section 11 by the taking of the spin structure on the oriented two-sheeted cover of M.

6 Regular homotopy classes and spin structures

Let X_{1}, X_{2} : M ⟶ ℝ^{3} be two immersions of a surface into space. Recall the distinction between regular homotopy equivalence of the immersions X_{1}, X_{2}, and regular homotopy equivalence of the corresponding immersed surfaces — these immersed surfaces are regularly homotopic if there is a diffeomorphism φ of M such that X_{2} is regularly homotopic to X_{1} ∘ φ — so this latter equivalence relation is coarser.

\underline{Theorem 6.} Let X_{1}, X_{2} : M ⟶ ℝ^{3} be two immersions of a surface into space, let S_{1}, S_{2} the spin structures induced as in Theorem 5, and let q_{1}, q_{2} be the associated quadratic forms as in Theorem 4. Then

(i) X_{1} and X_{2} are regularly homotopic if and only if q_{1} ≡ q_{2}(mod 2). (ii) The surfaces X_{1}(M) and X_{2}(M) are regularly homotopic if and only if Arf q_{1} =

Arf q_{2}. In particular, an immersed surface is regularly homotopic to an embedding if and only if its Arf invariant equals +1.

Sketch of proof. Define q̃(α) as the linking number (mod 2) of the boundary curves of the image of a tubular neighborhood of α in ℝ^{3}. Then

q(α) = 0 ⟺ the Darboux frame along α is nontrivial as an element of π_{1}(SO(3)) ⟺ q̃(α) = 0.

Hence q ≡ q̃(mod 2). But X_{1}, X_{2} are regularly homotopic if and only if q̃_{1} ≡ q̃_{2}(mod 2), and the corresponding immersed surfaces are regularly homotopic if and only if Arf q̃_{1} = Arf q̃_{2} (see [24]). □

7 Theta characteristics and spin structures

Theorem 7 ties the notion of spin structure with other concepts from algebraic geometry. Recall that a theta characteristic on a Riemann surface is a divisor D such that 2D is the canonical divisor.

\underline{Theorem 7.} Given a Riemann surface M, there are natural bijections between the following sets of objects:

(i) the spin structures on M; (ii) the complex line bundles S on M satisfying S ⊗ S ≅ T (M); (iii) the theta characteristics on M; (iv) the classes of non-identically-zero meromorphic differentials on M whose zeros

and poles have even orders, under the equivalence

φ_{1} ∼ φ_{2} ⟺ φ_{1}/φ_{2} = h^{2} for some mero- morphic function h on M.

Proof. (i) ⟺ (ii): Given a line bundle S on M satisfying S ⊗S ≅ T (M), S is a spin structure with mapping μ : S ⟶ S ⊗ S defined by μ(s) = s ⊗ s. Conversely, given a spin structure S on M, the map μ(s) ↦ s ⊗ s is well-defined and a vector-bundle isomorphism, so T (M) is isomorphic to S ⊗ S.

(ii) ⟺ (iii): Via the natural correspondence between the line bundles on M with the divisor classes, this set of line bundles is bijective with with the theta characteristics.

(iii) ⟺ (iv): Again, there is a natural bijection between the meromorphic differentials with zeros and poles of even orders and the theta characteristics. Given such a differential φ, the corresponding theta characteristic is \frac{1}{2}(φ). Moreover, two such differentials correspond to theta characteristics in the same linear equivalence class if and only if their ratio is the square of a meromorphic function on M. For

φ_{1}/φ_{2} = h^{2} ⟺ \frac{1}{2}(φ_{1}) − \frac{1}{2}(φ_{2}) = (h).

□

The spin structures on a compact Riemann surface are also bijective with the various translates ϑ[{a_{0} \atop b_{0}} ] of the theta functions on the surface (see [21] for the definition of ϑ[{a_{0} \atop b_{0}} ]).

8 Spin structures on hyperelliptic Riemann surfaces

In the special case of a hyperelliptic Riemann surface, the spin structures and their corresponding quadratic forms are computed explicitly.

\underline{Theorem 8.} Let

M = {[x_{1}, x_{2}, x_{3}] ∈ ℂℙ^{2} | x_{2}^{2}x_{3}^{2g−1} = ∏_{i=1}^{2g+1}(x_{1} − a_{i}x_{3})}

be a hyperelliptic Riemann surface of genus g, where A = {a_{1}, …, a_{2g+1}} ⊂ ℂ is a set of 2g + 1 distinct points. Let z = x_{1}/x_{3} and w = x_{2}/x_{3}. For each subset B ⊆ A, define

f_{B}(z) = ∏_{a∈B}(z − a) and η_{B} = f_{B}(z)dz/w.

Then (i) Any differential η_{B} represents a spin structure in the sense of Theorem 7. (ii) The set of 2^{2g} meromorphic differentials

{η_{B} | B ⊆ A, #B ≤ g}

represent the 2^{2g} distinct spin structures on M. (iii) With q the quadratic form corresponding to η_{B}, let γ be a curve in M whose

projection to the z-plane is a Jordan curve which avoids ∞ and A, and let C ⊆ A be the set of branch points which lie in the region enclosed by γ (so #C is even). Then

q([γ]) = #(B ∩ C) + \frac{1}{2}#C (mod 2).

(iv) With η_{B} and q as in (iii),

Arf q = \{ \begin{matrix}+1 & if 2g − 2#B + 1 ≡ ±1 (mod 8), \\ −1 & if 2g − 2#B + 1 ≡ ±3 (mod 8). \end{matrix}

Proof of (i). Let P_{i} = P_{a_{i}} = [a_{i}, 0, 1] and P_{∞} = [0, 1, 0] be the branch points of the two-sheeted cover z : M ⟶ ℂℙ^{1}. Then the divisor of η_{B} is

2 ((g − #B − 1)P_{∞} + ∑_{a∈B} P_{a}) .

Since this divisor is even, the differential represents a spin structure by Theorem 7. Proof of (ii). Note that there are \binom{2g+1}{r} differentials in the (r+1)^{th} row, totaling

∑_{r=0}^{g} \binom{2g+1}{r} = 2^{2g}. All but those in the last row are holomorphic.

In order to prove that these differentials represent distinct spin structures, we first compute the relations on the divisors of the form ∑ k_{i}P_{i} + k_{∞}P_{∞}. Two such divisors are equivalent if and only if there is a meromorphic function M whose divisor is their difference. Since the functions w and z − a_{i} have respective divisors

\begin{aligned}(w) & = & P_{1} + ⋯ + P_{2g+1} − (2g + 1)P_{∞}, \\ (z − a_{i}) & = & 2P_{i} − 2P_{∞}, \end{aligned}

we have the independent relations

\begin{aligned}P_{1} + ⋯ + P_{2g+1} & ≡ & (2g + 1)P_{∞}, \\ 2P_{i} & ≡ & 2P_{∞} (i = 1, …, 2g + 1). \end{aligned}

To show that there are no other relations independent of these, let ∑ k_{i}P_{i}+k_{∞}P_{∞} ≡ 0 be a relation. Then ∑ k_{i} = k_{∞}, and by the relations above, we may assume each k_{i} is 0 or 1. Hence the general relation may be assumed to be of the form D − dP_{∞} ≡ 0, where D is a sum of distinct P_{i} ∈ A, and d = #D. Let h be a function with divisor D − dP_{∞}. Since the only pole of h is at P_{∞}, h is a polynomial in z and w, so there are polynomial functions f_{1} and f_{2} of z such that

h(z, w) = f_{1}(z) + wf_{2}(z).

Then

2g + 1 ≥ d = − ord_{P_{∞}} h = − ord_{P_{∞}} (f_{1} + wf_{2}) ≥ − ord_{P_{∞}} wf_{2} = deg f_{2} + 2g + 1.

Thus d = 2g + 1, and D = P_{1} + ⋯ + P_{2g+1}, so no new relation can exist.

We want to show that η_{B_{1}} and η_{B_{2}} represent identical spin structures if and only if B_{1} = B_{2} or B_{1} = B′_{2}, where the prime notation C′ designates the complement A\C in A. If B_{1} = B_{2}, then this is clear; if B_{1} = B′_{2}, then η_{B_{2}}/η_{B_{1}} = (f_{2}/w)^{2} is a square of a meromorphic function on M, and so η_{B_{1}} and η_{B_{2}} represent the same spin structure by Theorem 7.

Conversely, suppose that η_{B_{1}} and η_{B_{2}} represent the same spin structure. Then by Theorem 7, η_{B_{2}}/η_{B_{1}} = h^{2} for some meromorpic function h on M. But

2(h) = (h^{2}) = (η_{B_{2}}/η_{B_{1}}) = 2((d_{2} − d_{1})P_{∞} + D_{2} − D_{1}),

where D_{1} = ∑_{a∈B_{1}} P_{a}, D_{2} = ∑_{a∈B_{2}} P_{a}, d_{1} = #B_{1}, and d_{2} = #B_{2}. So (d_{2} − d_{1})P_{∞} + D_{2} − D_{1} ≡ 0. By the relations (8), this divisor is equivalent to ∑_{a∈B_{1}∘B_{2}} P_{a} − #(B_{1} ∘ B_{2})P_{∞}, where B_{1} ∘ B_{2} is the symmetric difference (B_{1} ∪ B_{2}) \ (B_{1} ∩ B_{2}). Since the relations (8) generate all such relations, it follows that B_{1} ∘ B_{2} is either ∅ or A, that is that B_{1} = B_{2} or B_{1} = B′_{2}.

Proof of (iii). It follows from the definition of q that q([γ]) is the degree (mod 2) of the map f(z)/w thought of as a map from the curve γ on M to ℂ \ {0}. Let h = (f/w)^{2}. Then

deg h = ∑_{h(p)=0} ord_{p} h + ∑_{h(p)=∞} ord_{p} h,

the sums being restricted to points within γ. This computes to

deg h = #(B ∩ C) − #(B ∪ C) = 2(#(B ∩ C) − \frac{1}{2}#C),

which shows that

q([γ]) = #(B ∩ C) + \frac{1}{2}#C (mod 2).

Proof of (iv). In order to compute Arf q, we first compute ∑ q(α), where α ranges over H_{1}(M, ℤ_{2}). Correspondingly, the set of branch points C in the region enclosed by α range over the subsets of A of even cardinality. Hence ∑ q(α) is the number of such subsets for which q(α) = 1, that is, for which

#(B ∩ C) − #(B′ ∩ C) ≡ 2 (mod 4).

The Spinor Representation of Minimal Surfaces 15 The set of such subsets is

{R ∪ S | R ⊆ B, S ⊆ B′, #R − #S ≡ 2 (mod 4)}.

The cardinality of this set is

∑ q(α) = ∑_{i−j≡2} \binom{b}{i} \binom{b′}{j} ,

where b = #B, b′ = #B′, and the sum is over i and j with i − j ≡ 2 (mod 4).

To compute this sum, define

ξ(c, k) = ∑_{i≡k} \binom{c}{i} .

Then

∑ q(α) = ∑_{i} \binom{b}{i} ∑_{j≡i+2} \binom{b′}{j} = ∑_{k} \binom{b}{i} ξ(b′, j + 2)
= ∑_{p=0}^{3} ∑_{n} \binom{b}{4n + p} ξ(b′, p + 2) = ∑_{p=0}^{3} ξ(b, p)ξ(b′, p + 2).

Using a fact about Pascal’s triangle

ξ(c, k) = 2^{(c−2)/2} (2^{(c−2)/2} + cos \frac{π}{4}(c − 2k)) ,

we have

∑ q(α) = 2^{(2g−3)/2} ∑_{p=0}^{3} (2^{(b−2)/2} + cos \frac{π}{4}(b − 2p)) (2^{(b′−2)/2} + cos \frac{π}{4}(b′ − 2p))
= 2^{g−1} (2^{g} − \frac{1}{\sqrt{2}} ∑_{p=0}^{3} cos \frac{π}{4}(b − 2p) cos \frac{π}{4}(b′ − 2p))
= 2^{g−1} (2^{g} − \sqrt{2} cos \frac{π}{4}(2g − 2b + 1))
= \{ \begin{matrix}2^{g−1}(2^{g} − 1) & if & 2g − 2b + 1 ≡ ±1 (mod 8), \\ 2^{g−1}(2^{g} + 1) & if & 2g − 2b + 1 ≡ ±3 (mod 8). \end{matrix}

Since (−1)^{t} = 1 − 2t for t = 0 or 1,

Arf q = \frac{1}{2^{g}} ∑(−1)^{q(α)} = \frac{1}{2^{g}} (2^{2g} − 2 ∑ q(α))

is +1 or −1 according as 2g − 2b + 1 is ±1 or ±3 (mod 8). □

As an example, we compute the values of q for the four spin structures on a Riemann torus T. Let ℂ/{2ω_{1}, 2ω_{3}} = Jac (T) be the Jacobian for T, and let e_{i} = ℘(ω_{i}), ω_{2} = ω_{1} + ω_{3}. Then φ(u) = (℘(u), ℘′(u)) maps the Jacobian to the Riemann surface M defined by w^{2} = 4(z − e_{1})(z − e_{2})(z − e_{3}). The differentials as in (ii) of the above theorem pull back to

\begin{aligned}du & = & φ^{∗}(dz/w), \\ (℘(u) − e_{i})du & = & φ^{∗}((z − e_{i})dz/w). \end{aligned}

With α_{i} the generator of H_{1}(T, ℤ_{2}) defined by

α_{i} : [0, 1] ⟶ Jac(T), α_{i}(t) = 2tω_{i},

the values of q and Arf q are tabulated.

Table 1: Values of q and Arf q for spin structures on tori

\begin{matrix}η & q_{η}(0) & q_{η}(α_{1}) & q_{η}(α_{2}) & q_{η}(α_{3}) & Arf q_{η} \\ du & 0 & 1 & 1 & 1 & −1 \\ (℘(u) − e_{1})du & 0 & 1 & 0 & 0 & +1 \\ (℘(u) − e_{2})du & 0 & 0 & 1 & 0 & +1 \\ (℘(u) − e_{3})du & 0 & 0 & 0 & 1 & +1 \end{matrix}

An immersion corresponding to q for which Arf q = +1 is regularly homotopic to the torus standardly embedded in ℝ^{3}. The value Arf q = −1 corresponds to the twisted torus, which can be realized as the “diagonal” double covering of the standardly embedded torus as shown, but is not regularly homotopic to an embedding.

[figure: standard torus]

Figure 4: The twisted torus

9 Group action on spinors

The largest linear group acting on Q is the “linear conformal group”

ℂ^{∗} × SO(3, ℂ).

The orbit of an immersion into Q under this action is an 8-real-dimensional family of immersions (which action, however, will not respect the vanishing of periods — see section 10). The subgroup

ℝ^{+} × SO(3, ℝ)

is the group of similarity transformations of Euclidean 3-space. Hence the homogeneous space

(9.6) (ℂ^{∗} × SO(3, ℂ)) /(ℝ^{+} × SO(3, ℝ)) ≅ S^{1} × (SO(3, ℂ)/SO(3, ℝ)) .

is the 4-real-dimensional parameter space of non-similar surfaces in the above orbit.

The S^{1} factor gives rise to the well-known “associate family” of minimal surfaces, which are locally isometric and share a common Gauss map. The other factor has a simple (though apparantly less known) geometric interpretation as well. The Gauss map is the ratio of two spinors, so SO(3, ℂ) ≅ PSL(2, ℂ) acts on the Gauss map via post-composition with a fractional linear transformation of S^{2}; indeed, the quotient by SO(3, ℝ) ≅ PSU(2) leaves the hyperbolic three-space H^{3}, so the second factor can be thought of as the non-rigid Möbius deformations of the Gauss map.

The above observations are justified by the following well-known fact (see, for example, [8], [25]).

\underline{Theorem 9.} There is a unique two-fold covering homomorphism

T : GL(2, ℂ) ⟶ ℂ^{∗} × SO(3, ℂ)

such that for any A ∈ GL(2, ℂ),

T (A)σ = σA, (9.7)

where σ : ℂ^{2} ⟶ Q is as in equation (4.5), and A and T (A) act by left multiplication on C^{2} and Q respectively. Moreover, T is a two-fold covering homomorphism when restricted to the following groups:

T : GL(2, ℂ) ⟶ ℂ^{∗} × SO(3, ℂ),
T : SL(2, ℂ) ⟶ SO(3, ℂ),
T : ℝ^{∗} × SU(2) ⟶ ℝ^{+} × SO(3, ℝ),
T : SU(2) ⟶ SO(3, ℝ).

Proof. We define T and omit many of the calculations. ℂ^{3} may be identified with the set Γ of traceless 2 × 2 complex matrices via

(x_{1}, x_{2}, x_{3}) ⟷ ( \begin{matrix}x_{3} & −x_{1} + ix_{2} \\ −x_{1} − ix_{2} & −x_{3} \end{matrix} ) = X,

with the subset ℝ^{3} ⊂ ℂ^{3} identified with Γℝ = \{X ∈ Γ | X = \overline{X}^{t}\} The inner product on ℂ^{3} becomes

⟨X, Y ⟩ = ∑_{1}^{3} x_{i}y_{i} = \frac{1}{2} trXY ,

and

⟨X, X⟩ = \frac{1}{2} trX^{2} = − det X,

so Q ⊂ ℂ^{3} is identified with

Γ_{Q} = {X ∈ Γ | det X = 0}.

Similarly, ℂ^{2} may be identified with the set Δ of matrices of the form

( \begin{matrix}x_{1} & x_{1} \\ x_{2} & x_{2} \end{matrix} ) .

The map σ : ℂ^{2} ⟶ Q becomes under these identifications σ : Δ ⟶ Γ_{Q} given by

σ(X) = XKX′, where K = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ), and X′ denotes the classical adjoint

( \begin{matrix}a & b \\ c & d \end{matrix} )′ = ( \begin{matrix}d & −b \\ −c & a \end{matrix} )

satisfying XX′ = X′X = (det X)I and (XY)′ = Y′X′.

Then in order to satisfy equation (9.7), T must be defined, for X ∈ Γ, by

T (A)X = AXA′.

It follows that T (A) is linear and maps Γ to itself, and that T : GL(2, ℂ) ⟶ GL(3, ℂ) is a homomorphism with kernel {±I}. That T maps into the four groups listed follows from the equation

⟨T (A)X, T (A)Y ⟩ = (det A)^{2}⟨X, Y ⟩

and the fact that T (A)(Γℝ) = Γℝ for A ∈ ℝ^{∗} × SU(2). □

Lifting the group action on Q to an action on ℂ^{2} \ {0} via T, the homogeneous space in equation (9.6) can also be written

(9.8) (GL(2, ℂ)) / (ℝ^{∗} × SU(2)) ≅ S^{1} × (SL(2, ℂ)/SU(2)) ≅ S^{1} × H^{3},

where H^{3} is hyperbolic three-space.

10 Periods

Given an immersion X : M ⟶ ℝ^{3}, the period around a simple closed curve γ ⊂ M is the vector in ℂ^{3}

∫_{γ} ∂X.

If the real part of a period is not (0, 0, 0), the resulting surface is periodic and does not have finite total curvature. It is a considerable problem to “kill the periods” — that is, choose parameters so that the integrals around every simple closed curve in M generates purely imaginary period vectors. Non-zero periods can arise along two kinds of simple closed curves:

• a simple closed curve around an end p ∈ M,

• a non-trivial simple closed curve in H_{1}(M, ℤ). For a simple closed curve γ around an end p ∈ M,

∫_{γ} ∂X = 2πi res_{p} ∂X.

This integral is zero at embedded planar ends.

Using the spinor representation, the condition that a period along a closed curve γ ⊂ M be pure imaginary can be expressed by

∫_{γ} (s_{1}^{2} − s_{2}^{2}, i(s_{1}^{2} + s_{2}^{2}), 2s_{1}s_{2}) ∈ iℝ^{3},

equivalent to

\begin{aligned}∫_{γ} s_{1}^{2} & = & \overline{∫_{γ} s_{2}^{2}} \\ ∫_{γ} s_{1}s_{2} & ∈ & iℝ. \end{aligned} (10.9)

These equations are preserved by the group ℝ^{∗} × SU(2) of homotheties.

11 Spin structures and nonorientable surfaces

To deal with immersions of a nonorientable manifold M into space, we pass to the oriented two-fold cover of M. The following rather technical results are required in Part III. Without proof we state:

\underline{Lemma 10.} Let

A : S^{2} ⟶ S^{2} be the antipodal map,
A_{∗} : T (S^{2}) ⟶ T (S^{2}) the derivative of A,
Â_{∗} : Spin(S^{2}) ⟶ Spin(S^{2}) one of the lifts of A_{∗} to Spin(S^{2}) .

Then in the coordinates of section 5 we have

A_{∗} = Conj,
Â_{∗} = ± ( \begin{matrix}0 & i \\ −i & 0 \end{matrix} ) ∘ Conj.

The lifts of the antipodal maps are shown in Figure 5.

[figure: S^{2}; S^{2}; T (S^{2}); T (S^{2}); Spin(S^{2}); Spin(S^{2}); A; A_{∗}; Â_{∗}; σ]

Figure 5: Lifts of the antipodal map

\underline{Theorem 11.} Let M be a nonorientable Riemann surface, and X : M ⟶ ℝ^{3} a conformal minimal immersion of M into space.

Let π : M̃ ⟶ M be an oriented double cover of M, and X̃ = X ∘ π the lift of X to this cover. Let I : M̃ ⟶ M̃ the order-two deck transformation for the cover. With ω = ∂X̃, and in the notation of Lemma 10, we have

(i) gI = Ag, (ii) ωI_{∗} = A_{∗}ω, (iii) ω̂Î_{∗} = ±Â_{∗}ω̂.

Proof. Since X̃ is the lift of X, we have that X̃ = X̃I. Hence Re ω = Re ∂X̃ = dX̃ = dX̃I_{∗} = Re ωI_{∗}. Since X is a conformal minimal immersion, ω is holomorphic. Hence ω and ωI_{∗} are either equal or conjugate. But I_{∗} is orientation reversing, so they are conjugate and ωI_{∗} = \overline{ω} = A_{∗}ω, proving (ii).

From

gIπ_{1} = gπ_{1}I_{∗} = π_{3}ωI_{∗} = π_{3}A_{∗}ω = Aπ_{3}ω = Agπ_{1}

and the surjectivity of π_{1}, (ii) follows. Similarly, from

σω̂Î_{∗} = ωπ_{2}Î_{∗} = ωI_{∗}π_{2} = A_{∗}ωπ_{2} = A_{∗}σω̂ = σÂ_{∗}ω̂

(iii) follows. □

We remark that the proper treatment of nonorientable surfaces should really be via “pin” structures (Pin(n) being the corresponding two-sheeted covering group of O(n)), and that in this case we should have an analytic formula for the full ℤ_{8}-valued Arf invariant of the associated ℤ_{4}-valued quadratic form on H^{1}(M, ℤ_{2}).

Part II

Minimal Immersions with Embedded Planar Ends

12 Embedded planar ends

The first section of this part of our paper discusses the behavior of a minimal immersion at an embedded planar end. Lemma 12 translates this geometric behavior to a necessary and sufficient algebraic condition on the order and residue of the immersion at the end. Arising naturally from this algebraic condition is a certain vector subspace K of holomorphic spin-vector fields (sections of a spin structure) which generates all minimal surfaces with embedded planar ends (Theorem 15). More precisely, two sections chosen from K form the spinor representation of a minimal surface, and conversely, any such surface must arise this way. However, such a surface is usually periodic, and possibly a branched immersion. In order to compute K explicitly, a skew-symmetric bilinear form Ω is next defined (Definition 16) whose kernel is closely related to the space K. In Part III, this form is used to prove existence and nonexistence theorems for a variety of examples.

13 Algebraic characterization of embedded planar ends

The geometric condition that an end of a minimal immersion be embedded and planar can be translated to algebraic conditions (see for example [4]). Let X : D\{p} ⟶ ℝ^{3} be a conformal minimal immersion of an open disk D punctured at p such that lim_{q→p} |X(q)| = ∞. The image under X of a small neighborhood of p (and by association, p itself) is what we shall refer to as an end. The behavior of the end is determined by the residues and the orders of the poles of ∂X at p as follows.

Let ψ_{1}, ψ_{2}, ψ_{3} be defined by

∂X = (ψ_{1} − ψ_{2}, i(ψ_{1} + ψ_{2}), 2ψ_{3}).

The Gauss map for this immersion (see [22]) is

g = ψ_{3}/ψ_{1} = ψ_{2}/ψ_{3}.

First note that for X to be well-defined, we must have for any closed curve γ, which winds k times around p,

0 = Re ∫_{γ} ∂X = k Re (2πi res_{p} ∂X),

and so res_{p} ∂X must be real. Assume this, and assume initially that the limiting normal to the end is upward (that is g(p) = ∞). In this case,

ord_{p} ψ_{2} < ord_{p} ψ_{3} < ord_{p} ψ_{1},

so the first two coordinates of X(q) grow faster than does the third as q → p. It follows that ord_{p} ψ_{2} cannot be −1, because if it were then

res_{p} ∂X = (− res_{p} ψ_{2}, i res_{p} ψ_{2}, 0)

would not be real. Hence ord_{p} ψ_{2} ≤ −2. The image under X of a small closed curve around p is a large curve which winds around the end | ord_{p} ψ_{2}| − 1 times. The end is embedded precisely when ord_{p} ψ_{2} = −2.

If an end is embedded, its behavior is determined by the vanishing or nonvanishing of the residues of ∂X. For an embedded end, −2 = ord_{p} ψ_{2} < ord_{p} ψ_{3}, so ψ_{3} has either a simple pole or no pole. If ψ_{3} has a simple pole (and hence also a residue), the end grows logarithmically relative to its horizontal radius and is a catenoid end. If ψ_{3} has no pole, the end is asymptotic to a horizontal plane and is called a planar end. Moreover, in this latter case, res_{p} ψ_{2} must vanish (again, if it did not, res_{p} ∂X would not be real), and so res_{p} ∂X = (0, 0, 0).

For an end in general position the same conclusions hold, because a real rotation affects neither ord_{p} ∂X nor the reality or vanishing of res_{p} ∂X. In summary, we have

\underline{Lemma 12.} Let X : D\\{p\} ⟶ ℝ^{3} be a conformal minimal immersion of a punctured disk. Then p is an embedded planar end if and only if

ord_{p} ∂X = −2 and res_{p} ∂X = 0,

where ord_{p} ∂X denotes the minimum order at p of the three coordinates of ∂X.

14 Embedded planar ends and spinors

The conditions in the lemma above can be translated into conditions on the spinor representation of the minimal immersion. This leads to the definition of a space K of spin-vector fields, pairs of which form the spinor representation satisfying the required conditions.

Throughout the rest of Part II, the following notation is fixed:

M is a compact Riemann surface,

S is a spin structure on M,

P = [p_{1}] + … + [p_{n}] is a divisor of n distinct points.

(14.10)

The points p_{1}, …, p_{n} will eventually be the ends of a minimal immersion of M whose spinor representation will be a pair of sections of S.

Let H^{0}(M, O(S)) and H^{0}(M, M(S)) denote respectively the vector spaces of holomorphic and meromorphic sections of S. Define

(14.11)

F = F_{M,S,P} = {s ∈ H^{0}(M, M(S)) | (s) ≥ −P }
H = H_{M,S} = H^{0}(M, O(S))
K = K_{M,S,P} = {s ∈ F | ord_{p} s ≠ 0 and res_{p} s^{2} = 0 for all p ∈ supp P }.

\underline{Theorem 13.} Let X : M ⟶ ℝ^{3} be a minimal immersion with spinor representation (s_{1}, s_{2}). Then p ∈ M is an embedded planar end if and only if s_{1}, s_{2} ∈ K and at least one of s_{1}, s_{2} has a pole at p.

Proof. By Lemma 12, p is an embedded planar end if and only if

ord_{p} ∂X = −2 and res_{p} ∂X = 0.

The first of these equations is equivalent to the condition

s_{1}, s_{2} ∈ F, and at least one of s_{1}, s_{2} has a pole at p.

Given this, the conditions ord_{p} s_{1} ≠ 0, ord_{p} s_{2} ≠ 0 in the definition of K follow because if one were 0, the other would be −1, giving s_{1}s_{2} a nonvanishing residue.

It remains only to show that for s_{1}, s_{2} ∈ F,

res_{p} s_{1}^{2} = 0, res_{p} s_{2}^{2} = 0 ⟹ res_{p} s_{1}s_{2} = 0.

This is an application of the following lemma. □

\underline{Lemma 14.} Let S be a spin structure on a closed Riemann surface M, and let s_{1}, s_{2} be meromorphic sections of S with ord_{p}s_{1} ≥ −1, ord_{p}s_{2} ≥ −1 for some p ∈ M. Then

2 res_{p}s_{1}s_{2} = \{ \begin{matrix}[\frac{s_{2}}{s_{1}} ]_{p} res_{p}s_{1}^{2} + [\frac{s_{1}}{s_{2}} ]_{p} res_{p} s_{2}^{2} & if ord_{p}s_{1} = ord_{p}s_{2}, \\ 0 & if |ord_{p}s_{1} − ord_{p}s_{2}| ≥ 2. \end{matrix}

Proof. With z a complex coordinate near p satisfying z(p) = 0, let φ be a section of S satisfying φ^{2} = dz. Also let

s_{1} = (\frac{a_{−1}}{z} + a_{0} + …) φ,
s_{2} = (\frac{b_{−1}}{z} + b_{0} + …) φ

be the expansions of s_{1} and s_{2}. Then

res_{p} s_{1}^{2} = 2a_{−1}a_{0},
res_{p} s_{2}^{2} = 2b_{−1}b_{0},
res_{p} s_{1}s_{2} = a_{−1}b_{0} + a_{0}b_{−1}.

In case ord_{p} s_{1} = ord_{p} s_{2}, then either a_{−1} ≠ 0, b_{−1} ≠ 0, so that

[\frac{s_{2}}{s_{1}} ]_{p} res_{p} s_{1}^{2} + [\frac{s_{1}}{s_{2}} ]_{p} res_{p} s_{2}^{2} = \frac{b_{−1}}{a_{−1}} (2a_{−1}a_{0}) + \frac{a_{−1}}{b_{−1}} (2b_{−1}b_{0})
= 2(b_{−1}a_{0} + a_{−1}b_{0})
= 2 res_{p} s_{1}s_{2},

or a_{−1} = b_{−1} = 0, and the three residues vanish.

In case | ord_{p} s_{1} − ord_{p} s_{2}| ≥ 2, then a_{−1} = a_{0} = 0 or b_{−1} = b_{0} = 0, so again the three residues vanish. So in each case the formula is verified. □

15 F and K as vector spaces

The following theorem develops some of the properties of the spaces F and K. The most important of these is that K is in fact a vector space. In this section we write K for the holomorphic cotangent bundle (that is, the canonical line bundle) of M.

\underline{Theorem 15.} With M, P, and S as in equation (14.10), and F, H, and K as in equation (14.11), let g = genus (M) and n = deg P. Then

(i) if n ≥ g, then dim F = n; (ii) K and H are subspaces of F; (iii) if n ≥ g, then K ∩ H = 0.

Proof of (i): The dimension of F can be computed by means of the Riemann-Roch theorem (see, for example, [10]) which states

dim H^{0}(M, L) − dim H^{0}(M, K ⊗ L^{∗}) = deg L − g + 1

for an arbitrary line bundle L. Let R be the line bundle corresponding to the divisor P, and let L = S ⊗ R. Then:

• H^{0}(M, L) ≅ F by the isomorphism s ⊗ r ↦ s, where r is a section of R with divisor P;

• H^{0}(M, K ⊗ L^{∗}) = 0, since deg(K ⊗ L^{∗}) = g − 1 − n, which is negative by hypothesis;

• deg L − g + 1 = n; from which it follows that

dim F = dim H^{0}(M, L) = n.

Proof of (ii): H ⊆ F is a subspace by definition. To show that K ⊆ F is a subspace, let s ∈ K and p ∈ P, let z be a conformal coordinate near p with z(p) = 0, and let φ be a section of S which satisfies φ^{2} = dz. Let

s = (\frac{a_{−1}}{z} + a_{0} + …) φ

be the expansion of s. Then

s^{2} = (\frac{a_{−1}^{2}}{z^{2}} + \frac{2a_{−1}a_{0}}{z} + …) dz,

so that

res_{p} s^{2} = 0 ⟺ a_{−1} = 0 or a_{0} = 0

and

ord_{p} s^{2} ≠ 0 ⟺ a_{−1} ≠ 0 or a_{0} = 0.

Together, these two conditions are equivalent to the condition

a_{0} = 0.

Thus (15.11) s ∈ K ⟺ the constant term in the expansion of s van- ishes at each p ∈ P.

K is a vector space because if s_{1}, s_{2} satisfy condition (15.11), then so does any ℂ-linear combination of s_{1} and s_{2}.

Proof of (iii): Let s ∈ K ∩ H be a section which is not identically zero. Since s ∈ K, we have that ord_{p} s ≠ 0 for all p ∈ supp P — that is, at each such p, s has either a pole or a zero. But since s ∈ H, s cannot have a pole at p, and hence has a zero, so (s) ≥ P. Conversely, if (s) ≥ P, then s ∈ K ∩ H, so

K ∩ H = {s ∈ F | (s) ≥ P }.

Thus for s ∈ K ∩ H not identically zero,

n ≤ deg s = g − 1.

Hence if n ≥ g, then K ∩ H = 0. □

16 A bilinear form Ω which kills K

In order to understand the vector space K more explicitly, a skew-symmetric bilinear form Ω is defined whose kernel contains K. This form may then be used in many cases to compute K, and thereby moduli spaces of minimal surfaces with embedded planar ends.

\underline{Definition 16.} With M, P, and S as in equation (14.10) define Ω = Ω_{M,P,S} : F × F ⟶ ℂ by

Ω(s_{1}, s_{2}) = ∑_{p∈P} ξ(p; s_{1}, s_{2}),

where

ξ(p; s_{1}, s_{2}) = \{ \begin{matrix}\frac{1}{2} [\frac{s_{2}}{s_{1}} ]_{p} res_{p}s_{1}^{2} & if \frac{s_{2}}{s_{1}} does not have a pole at p, \\ res_{p}s_{1}s_{2} & if \frac{s_{2}}{s_{1}} has a pole at p. \end{matrix}

The form Ω can be computed as follows: for p ∈ P, let z be a conformal coordinate near p with z(p) = 0, let φ^{2} = dz, and let

s_{1} = (\frac{a_{−1}}{z} + a_{0} + …) φ,
s_{2} = (\frac{b_{−1}}{z} + b_{0} + …) φ

be the expansions of s_{1} and s_{2}. Then

ξ(p; s_{1}, s_{2}) = b_{−1}a_{0}. (16.12)

\underline{Theorem 17.} With H, K as in equation (14.11), Ω satisfies the following: (i) Ω is a skew-symmetric bilinear form on F; (ii) ker Ω ⊇ K + H; (iii) if K ∩ H = 0, then ker Ω = K ⊕ H; (iv) if n = deg P ≥ genus (M), then ker Ω = K ⊕ H.

Proof of (i): For a given s_{1}, s_{2} ∈ F, let

P_{0} = {zeros of s_{2}/s_{1}} ∩ P,
P_{∞} = {poles of s_{2}/s_{1}} ∩ P,
P_{1} = P \ (P_{0} ∪ P_{∞}),

so that P_{0}, P_{∞}, P_{1} are disjoint and their union is P. Then

Ω(s_{1}, s_{2}) = \frac{1}{2} ∑_{p∈P_{1}} [\frac{s_{2}}{s_{1}} ]_{p} res_{p} s_{1}^{2} + ∑_{p∈P_{∞}} res_{p} s_{1}s_{2}, (16.13)

Ω(s_{2}, s_{1}) = \frac{1}{2} ∑_{p∈P_{1}} [\frac{s_{1}}{s_{2}} ]_{p} res_{p} s_{2}^{2} + ∑_{p∈P_{0}} res_{p} s_{1}s_{2}.

Adding the above two equations, and using the lemma in section 14 yields

Ω(s_{1}, s_{2}) + Ω(s_{2}, s_{1}) = 2 ∑_{p∈P} res_{p} s_{1}s_{2},

which vanishes by the residue theorem, since all poles of s_{1}s_{2} are in P.

Proof of (ii): To show that that K ⊆ ker Ω, let s_{1} ∈ K, so that res_{p} s_{1}^{2} = 0 and ord_{p} s_{1} ≠ 0 for all p ∈ P. Let s_{2} ∈ F be arbitrary, so that ord_{p} s_{2} ≥ −1. Referring to equation (16.13), the first sum is zero because res_{p} s_{1}^{2} = 0 at each p ∈ P by hypothesis. To show that each term in the second sum is zero, let p ∈ P_{∞} so that ord_{p} s_{1}/s_{2} ≥ 1. Then

ord_{p} s_{1} = ord_{p} s_{2} + ord_{p} s_{1}/s_{2} ≥ 0.

But ord_{p} s_{1} ≠ 0, so ord_{p} s_{1} ≥ 1. Then

ord_{p} s_{1}s_{2} = ord_{p} s_{1} + ord_{p} s_{2} ≥ 0,

so res_{p} s_{1}s_{2} = 0.

To show that H ⊆ ker Ω, let s_{1} ∈ H, so that s_{1}^{2} has no poles, and let s_{2} ∈ F be arbitrary. Then the first sum in equation (16.13) is zero because s_{1}^{2} has no poles. The second sum is zero by the residue theorem — to show that all poles of s_{1}s_{2} are in P_{∞}, note that

ord_{p} s_{1}s_{2} = ord_{p} s_{1} + ord_{p} s_{2} ≥ − ord_{p} s_{1} + ord_{p} s_{2} = ord_{p} s_{2}/s_{1}.

So if s_{1}s_{2} has a pole at p, then so does s_{2}/s_{1}; this puts p ∈ P_{∞}.

Proof of (iii): Ω can be “factored” as the composition of two maps as follows: near each point in supp P = {p_{1}, …, p_{n}}, choose a conformal coordinate z_{i} with z_{i}(p_{i}) = 0. Let φ be a section of S satisfying φ^{2} = dz, and let

s = (\frac{b_{i}}{z_{i}} + a_{i} + …) φ

be the expansion of s at p_{i}. Define maps A, B : F ⟶ ℂ^{n} by

A(s) = (a_{1}, …, a_{n}),
B(s) = (b_{1}, …, b_{n}).

The Spinor Representation of Minimal Surfaces 29 By the local computation of equation (16.12), Ω = B^{∗}A = −A^{∗}B, or with a choice of basis for F, Ω = B^{t}A = −A^{t}B as matrices. (Note that while Ω is independent of the choice of coordinates, A and B are not.) Moreover, we have

ker A = K and ker B = H;

the first is by equation (15.11), and the second is immediate from the definition of H. (This incidentally provides another proof of (ii).)

Now let Â = A|_{ker B^{∗}A} and note that

ker Â = (ker A ∩ ker B^{∗}A) = ker A

and

image Â = A(ker B^{∗}A) = image A ∩ ker B^{∗}.

Applying the rank-nullity theorem to Â,

dim ker Ω = dim ker B^{∗}A
= dim ker Â + dim image Â
= dim ker A + dim( image A ∩ ker B^{∗})
≤ dim ker A + dim ker B.

So under the assumption that K ∩ H = 0,

dim ker Ω ≤ dim ker A + dim ker B = dim K + dim H = dim K ⊕ H. But ker Ω ⊇ K ⊕ H, so ker Ω = K ⊕ H, proving (iii).

Part (iv) follows directly from (iii) above and Theorem 15(iii). □

Part III

Classification and Examples

17 Genus zero

In the first half of Part III, the skew-symmetric form Ω developed in Part II is used to investigate minimal genus zero surfaces with embedded planar ends. The first two sections demonstrate the non-existence of examples with 2, 3, 5, or 7 ends, and the dimension of the moduli space of examples with 4, 6, 8, 10, 12 and 14 ends is computed. The following three sections compute explicitly the moduli spaces for the families with 4 and 6 ends, and in section 23, the moduli space of the three-ended projective planes is investigated. The remaining sections (following the heading Genus one) of Part III are devoted to constructing minimal tori and Klein bottles. All of these surfaces are found (or shown not to exist) by the following general method: after computing Ω on a simple basis, its pfaffian, which is a function of the ends, is set to zero. The resulting condition on the placement of the ends — that is, the determinantal variety — together with further conditions arising from the demand that the immersion have no periods and no branch points, forms a set of equations whose simultaneous solution (or impossibility of solution) gives the desired result.

18 Existence and non-existence of genus zero surfaces

The non-existence of genus zero minimal unbranched immersions with 3, 5 or 7 embedded planar ends was first proved in a case-by-case manner in [3]. The following is a new proof, using the ideas of Section 12.

\underline{Theorem 18.} There are no complete minimal branched or unbranched immersions of a punctured sphere into space with finite total curvature and 2, 3, 5, or 7 embedded planar ends. There exist unbranched examples with 4, 6, and any n ≥ 8 ends.

Proof. Examples with 2p ends (p ≥ 2) are given in [14], and with 2p + 1 ends (p ≥ 4) in [23]. For the cases n = 3, 5, or 7, by the lemma below, 2 ≤ dim K ≤ [\sqrt{n}] ≤ 2 (here [q] denotes the greatest integer less than or equal to q), so dim K = 2, which contradicts the other statement of the lemma that n − dim K is even. The case n = 2 is proved in [14] (or is proved likewise by the lemma). □

We remark that there is also a simple topological proof of the non-existence of genus zero examples with 3 ends, using ideas in [13] and [15]. The trick is to exploit the SO(3, ℂ)-action discussed in section 9 to deform the Gauss map — on a punctured sphere with planar ends there is no period obstruction to doing this — so that the compactified S^{2} is transversally immersed with a unique triple-point, which is impossible. (By carefully treating the periods introduced by this explicit SO(3, ℂ) deformation of the Gauss map, the same kind of argument should generalize to exclude orientable minimal surfaces of any genus with three embedded planar ends — see section 27 for a proof in the case of tori.)

\underline{Lemma 19.} Let P be a divisor on the Riemann sphere S^{2} as in equation (14.10) with n = deg P ≥ 2, and let K = K_{S^{2},S,P} be as in equation (14.11). Then

(i) n − dim K is even; (ii) If there exists a complete branched or unbranched minimal immersion of S^{2} into

space with finite total curvature and n embedded planar ends in supp (P), then 2 ≤ dim K ≤ \sqrt{n}.

Proof of (i): By Theorem 17, ker Ω = K ⊕ H. But H = 0 because there are no holomorphic differentials on the sphere, so ker Ω = K. Since Ω is skew-symmetric, rank Ω = n − dim K is even (see Appendix A).

Proof of (ii): The sections s_{1} and s_{2} in the spinor representation (s_{1}, s_{2}) of such a surface are independent, showing the inequality 2 ≤ dim K. To show the other inequality, let z be the standard conformal coordinate on S^{2} = ℂ ∪ {∞}, and let P = ∑[a_{i}] (where the a_{i} ∈ ℂ are distinct) be the divisor of the n ends. Let g_{1}η, …, g_{m}η be a basis for K, where η^{2} = dz. Define f : S^{2} = ℂℙ^{1} ⟶ ℂℙ^{m−1} by

f = (g_{1}, …, g_{m}).

Then f is well-defined and holomorphic even at the common zeros and the common poles of g_{1}, …, g_{m}. Let

h(z) = ∏(z − a_{i}).

It follows from

(hg_{i}) = (h) + (η) + (g_{i}η) ≥ (P − n[∞]) + [∞] − P = −(n − 1)[∞]

that

d_{0} = deg f ≤ n − 1.

To show that f ramifies at each a ∈ supp P, let h_{i}(z) = (z − a)g_{i}(z). Then h_{i} does not have a pole at a. Moreover, since by hypothesis there exists a minimal surface with ends at supp P, at least one of the g_{i} has a pole at a, so the h_{i} cannot all be zero at a. Hence the appropriate condition that f ramify at a is

(h_{i}h′_{j} − h′_{i}h_{j})|_{a} = 0 for all i, j.

This is satisfied because of the condition (15.11) defining K: the expansion of g_{i} at a is

g_{i} = \frac{c_{i}}{z − a} + o(z − a),

so the expansion of h_{i} at a is

h_{i} = c_{i} + o(z − a)^{2},

and so h′_{i}(a) = 0 for all i. Since f ramifies at each a ∈ supp P, we have

r_{0} = ramification index of f ≥ n.

Now let f_{k} : ℂℙ^{1} ⟶ ℙ(Λ^{k+1}ℂ^{m}) defined by f_{k} = f ∧ f′ ∧ … ∧ f^{(k)} in ℂ^{m} be the kth associated curve of f, and use the Plücker formulas (an extension of the Riemann-Hurwitz formula — see [9]) which on ℂℙ^{1} are

−d_{k−1} + 2d_{k} − d_{k+1} − 2 = r_{k},

where d_{k} is the degree of f_{k}, and r_{k} is the ramification index of f_{k}. In the table below, multiplying the numbers on the left by the inequalities on the right and adding yields

d_{0} ≥ (m + n)(m − 1)/m.

But n − 1 ≥ d_{0}, so it follows that n ≥ m^{2}. □

Table 2: Values for the Plücker formulas

m − 1  2d_{0} − d_{1} − 2 = r_{0} ≥ n
m − 2 −d_{0} + 2d_{1} − d_{2} − 2 = r_{1} ≥ 0
⋮  ⋮  ⋮  ⋮
2 −d_{m−4} + 2d_{m−3} − d_{m−2} − 2 = r_{m−3} ≥ 0
1 −d_{m−3} + 2d_{m−2}  − 2 = r_{m−2} ≥ 0

19 Moduli spaces of genus zero minimal surfaces

The following two theorems deal with the moduli spaces of genus zero examples.

\underline{Theorem 20.} Let P be a divisor on S^{2} as in equation (14.10) and K = K_{S^{2},S,P} as in equation (14.11) with m = dim K ≥ 2. Then the space of complete minimal branched immersions of S^{2} into ℝ^{3} with finite total curvature and embedded planar ends at supp P is the complex 2(m − 1)-dimensional space Gr_{m,2}(ℂ) × (S^{1} × H^{3}).

Proof. Each point of the Grassmanian Gr_{m,2}(ℂ) represents a two-dimensional subspace of K. Each such subspace generates the space S^{1} × H^{3} of branched immersions (equation (9.8)). □

\underline{Theorem 21.} For each p ≥ 2 there exists a real 4(p−1)-dimensional family of minimal branched immersions of spheres punctured at 2p points with finite total curvature and embedded planar ends. For 2 ≤ p ≤ 7, the moduli space of such immersions is exactly 4(p − 1)-dimensional.

Proof. Let P = ∑[a_{i}] be a divisor of degree 2p on S^{2}, and S the unique spin structure on S^{2}. Let H and K be as in equation (14.11). Then pfaffian Ω = 0 (see Appendix A) if and only if dim K ≥ 2 if and only if there exists a surface with 2p ends at supp P. Counting real dimensions, the space of 2p ends is 4p-dimensional; the Möbius transformations of S^{2} reduce the dimension by 6, and the pfaffian condition on the ends reduce the dimension by another 2, so the space of ends which admit surfaces is (4p − 8)-dimensional. For each admissible choice of ends, by the above theorem there is a real (4 dim K − 4)-dimensional space of surfaces. Altogether, this totals 4p + 4 dim K − 12, which is at least 4p − 4 since dim K ≥ 2.

In the case that 2 ≤ p ≤ 7, by Lemma 19, 2 ≤ dim K ≤ [\sqrt{2p}] ≤ [\sqrt{14}] = 3, so dim K, being even, must be exactly 2. □

20 Ω on the Riemann sphere

For the examples in sections 21–23 we need to compute Ω on the Riemann sphere. Let z be the standard conformal coordinate on S^{2} = ℂ ∪ {∞}, and let φ^{2} = dz represent the unique spin structure on S^{2}. Let P = [a_{1}] + … + [a_{n−1}] + [∞] with the a_{i} ∈ ℂ distinct. We have H = 0 since there are no holomorphic differentials on the sphere. A basis for F is

\{t_{1}, …, t_{n−1}, t_{n}\} = \{\frac{φ}{z − a_{1}} , …, \frac{φ}{z − a_{n−1}} , φ\} .

These sections are in F since

(t_{n}) = −[∞], (t_{i}) = −[a_{i}],

and are independent because they have distinct poles, and so are a basis for F since dim F = n. By the local calculation (16.12) for Ω,

Ω(t_{i}, t_{j}) = \{ \begin{matrix}\frac{1}{a_{j} − a_{i}} & (1 ≤ i ≤ n − 1; 1 ≤ j ≤ n − 1; i ≠ j), \\ −1 & (1 ≤ i ≤ n − 1; j = n), \\ 1 & (i = n; 1 ≤ j ≤ n − 1), \\ 0 & (i = j). \end{matrix}

21 Genus zero surfaces with four embedded planar ends

The family of minimal genus zero surfaces with four embedded planar ends was computed first in [2]. A different computation is included here for completeness.

\underline{Theorem 22.} The space Σ_{4} of complete minimal immersions of spheres punctured at four points into space with finite total curvature and embedded planar ends is S^{1} × H^{3}.

Proof. Let z be the standard conformal coordinate on S^{1} = ℂ ∪ {∞}. By a Möbius transformation of the Riemann sphere S^{2}, the ends can be normalized so that two of the ends are 0 and ∞ and the product of the other two is 1. Naming the normalized ends

{a_{1} = a, a_{2} = 1/a, 0, ∞},

the matrix for Ω in the basis

\{\frac{1}{z − a_{1}} , \frac{1}{z − a_{2}} , \frac{1}{z}, 1\}

is

Ω = ( \begin{matrix}0 & \frac{1}{a_{2}−a_{1}} & −\frac{1}{a_{1}} & −1 \\ \frac{1}{a_{1}−a_{2}} & 0 & −\frac{1}{a_{2}} & −1 \\ \frac{1}{a_{1}} & \frac{1}{a_{2}} & 0 & −1 \\ 1 & 1 & 1 & 0 \end{matrix} )

(see section 20). The pfaffian of Ω (see Appendix A) computes to a nonzero multiple of

(a^{2} − \sqrt{3}a + 1)(a^{2} + \sqrt{3}a + 1).

This pfaffian must be zero in order for ker Ω = K to be at least two-dimensional and hence to generate surfaces. Setting this pfaffian to zero yields interchangeable solutions for a, one of which is

a = (\sqrt{3} + i)/2.

With φ^{2} = dz as usual, a basis for K is

t_{1} = (\frac{\sqrt{3}z − 1}{z(z^{2} − \sqrt{3}z + 1)}) φ and t_{2} = (\frac{z(z − \sqrt{3})}{z^{2} − \sqrt{3}z + 1}) φ,

the family of immersions is then given by X = Re F, where

F = ∫ (s_{1}^{2} − s_{2}^{2}, i(s_{1}^{2} + s_{2}^{2}), 2s_{1}s_{2})

and

( \begin{matrix}s_{1} \\ s_{2} \end{matrix} ) = Q ( \begin{matrix}t_{1} \\ t_{2} \end{matrix} ) ,

where Q ∈ ℂ^{∗} × SL(2, ℂ). The surfaces are identical (up to a rotation or dilation in space) when Q ∈ ℝ^{∗} × SU(2). Thus a parameter space for this family of surfaces is S^{1} × H^{3} (see section 9). That these surfaces are immersed is shown in the next section. □

22 Genus zero surfaces with six embedded planar ends

Herein is computed the family of minimal genus zero surfaces with six embedded planar ends.

\underline{Theorem 23.} The space Σ_{6} of complete minimal immersions of spheres punctured at six points into space with finite total curvature and embedded planar ends is V × (S^{1} × H^{3}), where V is an algebraic subvariety of (ℂℙ^{1})^{3} with codimension 1.

Proof. On the sphere S^{2} = ℂ ∪ {∞} with standard conformal coordinate z, the ends can be normalized so that two of the ends are at 0 and ∞, and the product of the remaining four ends is 1. With this normalization, let the ends be {a_{1}, a_{2}, a_{3}, a_{4}, 0, ∞}. Set

\begin{aligned}σ_{1} & = & a_{1} + a_{2} + a_{3} + a_{4}, \\ σ_{2} & = & −(a_{1}a_{2} + a_{1}a_{3} + a_{1}a_{4} + a_{2}a_{3} + a_{2}a_{4} + a_{3}a_{4}), \\ σ_{3} & = & a_{1}a_{2}a_{3} + a_{1}a_{2}a_{4} + a_{1}a_{3}a_{4} + a_{2}a_{3}a_{4}. \end{aligned}

The matrix for Ω in the basis

\{\frac{1}{z − a_{1}} , \frac{1}{z − a_{2}} , \frac{1}{z − a_{3}} , \frac{1}{z − a_{4}} , \frac{1}{z}, 1\}

is

Ω = ( \begin{matrix}0 & \frac{1}{a_{2}−a_{1}} & \frac{1}{a_{3}−a_{1}} & \frac{1}{a_{4}−a_{1}} & −\frac{1}{a_{1}} & −1 \\ \frac{1}{a_{1}−a_{2}} & 0 & \frac{1}{a_{3}−a_{2}} & \frac{1}{a_{4}−a_{2}} & −\frac{1}{a_{2}} & −1 \\ \frac{1}{a_{1}−a_{3}} & \frac{1}{a_{2}−a_{3}} & 0 & \frac{1}{a_{3}−a_{4}} & −\frac{1}{a_{3}} & −1 \\ \frac{1}{a_{1}−a_{4}} & \frac{1}{a_{2}−a_{4}} & \frac{1}{a_{3}−a_{4}} & 0 & −\frac{1}{a_{4}} & −1 \\ \frac{1}{a_{1}} & \frac{1}{a_{2}} & \frac{1}{a_{3}} & \frac{1}{a_{4}} & 0 & −1 \\ 1 & 1 & 1 & 1 & 1 & 0 \end{matrix} )

(see section 9). The pfaffian of Ω (see Appendix A) is pfaffian Ω = τ_{1}τ_{3} + σ_{1}σ_{3} − 20, (22.14) where

τ_{1} = σ_{1}^{2} + 3σ_{2} and τ_{3} = σ_{3}^{2} + 3σ_{2}.

The condition that the pfaffian be 0 defines the algebraic subvariety

V = {(σ_{1}, σ_{2}, σ_{3}) ⊂ (ℂℙ^{1})^{3} | pfaffian Ω = 0}

of codimension 1. Assuming that the pfaffian is zero, and letting φ^{2} = dz, a basis for the kernel of Ω is {t_{1}, t_{2}}, where

t_{1} = (\frac{b_{3}z^{3} + b_{2}z^{2} + b_{1}z + b_{0}}{z(z^{4} − σ_{1}z^{3} − σ_{2}z^{2} − σ_{3}z + 1)}) φ,
t_{2} = (\frac{z(c_{3}z^{3} + c_{2}z^{2} + c_{1}z + c_{0}}{z^{4} − σ_{1}z^{3} − σ_{2}z^{2} − σ_{3}z + 1}) φ,

and

\begin{matrix}b_{0} = σ_{2}, \\ b_{1} = −σ_{2}σ_{3}, \\ b_{2} = σ_{2}τ_{3} − 2σ_{1}σ_{3} − 10, \\ b_{3} = σ_{1}τ_{3} + 5σ_{3}, \end{matrix} \begin{matrix}c_{0} = σ_{3}τ_{1} + 5σ_{1}, \\ c_{1} = σ_{2}τ_{1} − 2σ_{1}σ_{3} − 10, \\ c_{2} = −σ_{1}σ_{2}, \\ c_{3} = σ_{2} \end{matrix}

(the special case σ_{2} = 0 for which the above sections are linearly dependent is ignored here). The family of immersions is then given by

X = Re ∫ (s_{1}^{2} − s_{2}^{2}, i(s_{1}^{2} + s_{2}^{2}), 2s_{1}s_{2})

where

( \begin{matrix}s_{1} \\ s_{2} \end{matrix} ) = Q ( \begin{matrix}t_{1} \\ t_{2} \end{matrix} ) ,

and Q ∈ ℂ^{∗} × SL(2, ℂ)/ℝ^{∗} × SU(2) ≅ S^{1} × H^{3} as in the previous section. □

That the four- and six-ended families are immersed follows from the lemma below, which in turn follows directly from the definitions of the spaces in equation (14.10).

\underline{Lemma 24.} On the sphere with its unique spin structure S, let P_{1} = ∑[p_{i}] as in equation (14.10), and P_{2} = P_{1} + [a], (a ∉ supp (P_{1})). Let F_{i} = F_{S^{1},P_{i},S} and K_{i} = K_{S^{1},S,P_{i}} (i = 1, 2) as in equation (14.11). Then K_{2} ∩ F_{1} = {s ∈ K_{1} | s(a) = 0}.

Now, to complete the proof that the above examples are immersed, let P_{1} be the divisor of ends of even degree n < 9, and let (s_{1}, s_{2}) be the spinor representation of a minimal branched immersion. Supposing this surface is not immersed, let a be a branch point of the surface, and set P_{2} = P_{1} + [a]. Then s_{1} and s_{2} are independent sections in K_{1} and s_{1}(a) = 0, s_{2}(a) = 0, so by the lemma, s_{1}, s_{2} ∈ K_{2}. Applying Lemma 19 iii), we have that

2 ≤ dim K_{2} ≤ [\sqrt{n}] ≤ 2,

so dim K_{2} = 2. This contradicts the fact that n + 1 − dim K_{2} is even (Lemma 19 i)).

23 Projective planes with three embedded planar ends

It was shown in [14] that any minimal immersion of a punctured real projective plane with embedded ends has only planar ends, and has at least three of them. Hence those which are the subject of the following theorem are the examples of minimal projective planes with the fewest number of embedded ends. One method for determining the moduli space of finite total curvature minimally immersed projective planes punctured at three points was given in [3]. Here we provide another description of this moduli space using the spinor representation. Note that all these surfaces compactify to give surfaces minimizing W = ∫ H^{2}dA among all immersed real projective planes [13], with minimum energy W = 12π.

\underline{Theorem 25.} Let Π_{3} be the moduli space of complete minimal immersions of real projective planes punctured at three points with finite total curvature and embedded planar ends modulo Euclidean similarities. Then

(i) Π_{3} is homeomorphic to a closed disk with one point M_{0} removed from the boundary;

(ii) the point M_{0} represents the Möbius strip with total curvature −6π in the sense that if γ : ℝ^{+} ⟶ Π_{3} is a curve with lim_{t→∞} γ(t) = M_{0}, then there is a oneparameter family of immersions X_{t} parametrizing the surfaces γ(t) such that as t → ∞, X_{t} converges uniformly on compact sets to a parametrization of the Möbius strip;

(iii) the surfaces with non-trivial symmetry groups are represented by the boundary of the disk, which represents a one-parameter family of surfaces which have a line of reflective symmetry; among these, the only surfaces with larger symmetry groups (other than M_{0}) are two surfaces which have, respectively, the symmetry groups ℤ_{2} × ℤ_{2}, and D_{3}, the dihedral group of order 6.

Proof of (i): The two-sheeted covering of the projective plane is the Riemann sphere S^{2} = ℂ∪\{∞\}, with order-two orientation-reversing deck transformation I(z) = −1/\overline{z}. By a motion in PSU(2) the six preimages on the sphere of three points in the projective plane can be normalized as in section 22 to be

{a_{1}, I(a_{1}), a_{2}, I(a_{2}), 0, ∞}

with the product of the first four equal to 1. With this choice, following the notation of section 22, we have

σ_{2} ∈ ℝ; σ_{3} = −\overline{σ}_{1}; τ_{3} = \overline{τ}_{1}.

For each choice of ends satisfying equation (22.14), up to dilations and isometries of space there is a unique minimal immersion of the projective plane, whose spinor representation is given by \sqrt{i}(t_{1}, t_{2}), with t_{1}, t_{2} as in section 22. For if \sqrt{i}(t̂_{1}, t̂_{2}) is the spinor representation of another immersion with the same ends, then a motion in ℂ^{∗} × PSL(2, ℂ) can make t̂_{1} = t_{1}, and the compatibility condition in Theorem 11 forces t̂_{2} = ±t_{1}. Hence the moduli space Π_{3} can be parametrized as a quotient space of

Γ = \{(σ_{1}, σ_{2}) ∈ ℂ × ℝ | τ_{1}τ_{3} + σ_{1}σ_{3} − 20 = 0, σ_{3} = −\overline{σ}_{1}\},

where σ_{1}, σ_{2}, σ_{3} are the symmetric polynomials of the ends defined in section 22. The desired moduli space is a quotient space of Γ, since permutations of the ends give rise to the same surface.

Since the parameters σ_{1} and σ_{2} depend on the particular normalization of the ends made in section 22, new parameters should be chosen, namely the three direction cosines of the angles between the ends 0, a_{1} and a_{2}, viewed as vectors in S^{2} ⊂ ℝ^{3}. To convert the equation (22.14) to these new parameters let ϕ : ℂ ⟶ S^{2} ⊂ ℝ^{3} be inverse stereographic projection defined by

ϕ(a) = (\frac{2 Re a}{|a|^{2} + 1}, \frac{2 Im a}{|a|^{2} + 1}, \frac{|a|^{2} − 1}{|a|^{2} + 1}) .

With the usual inner product ⟨ , ⟩ in ℝ^{3}, the direction cosines are

c_{1} = ⟨ϕ(0), ϕ(a_{1})⟩ = \frac{1 − |a_{1}|^{2}}{1 + |a_{1}|^{2}} ,

c_{2} = ⟨ϕ(0), ϕ(a_{2})⟩ = \frac{1 − |a_{2}|^{2}}{1 + |a_{2}|^{2}} ,

c_{3} = ⟨ϕ(a_{1}), ϕ(a_{2})⟩ = \frac{(1 − |a_{1}|^{2})(1 − |a_{2}|^{2})}{(1 + |a_{1}|^{2})(1 + |a_{2}|^{2}) + 4Re a_{1}\overline{a}_{2}} .

The above three equations may be written

|a_{1}|^{2} = \frac{1 − c_{1}}{1 + c_{1}} ,
|a_{2}|^{2} = \frac{1 − c_{2}}{1 + c_{2}} ,
Re a_{1}\overline{a}_{2} = \frac{c_{3} − c_{1}c_{2}}{(1 + c_{1})(1 + c_{2})}.

Using the normalization of the ends above, and writing a_{1} = γr_{1}, a_{2} = \overline{β}r_{2} (γ ∈ S^{1} ⊂ ℂ; r_{1},r_{2} ∈ ℂ) yields

r_{1} = \begin{matrix}√ & \overline{1 − c_{1}} \\ √ & \overline{1 + c_{1}} \end{matrix} ,
r_{2} = \begin{matrix}√ & \overline{1 − c_{2}} \\ √ & \overline{1 + c_{2}} \end{matrix} ,
γ^{2} = \frac{c_{3} − c_{1}c_{2} + ix}{\sqrt{1 − c_{1}^{2}}\sqrt{1 − c_{2}^{2}}} ,

where

x^{2} = 1 − c_{1}^{2} − c_{2}^{2} − c_{3}^{2} + 2c_{1}c_{2}c_{3}.

To convert the determinant of equation (22.14) from the variables σ_{1}, σ_{2}, σ_{3} to c_{1}, c_{2}, c_{3}, compute

σ_{1} = \frac{2γ(−c_{1} + 2c_{1}c_{2}^{2} − c_{2}c_{3} + ic_{2}x)}{\sqrt{1 − c_{1}^{2}}(1 − c_{2}^{2})},

σ_{2} = \frac{2(c_{3} − 3c_{1}c_{2})}{(1 − c_{1}^{2})(1 − c_{2}^{2})},
σ_{3} = −\overline{σ}_{1},

and the determinant becomes, up to a non-zero multiple,

(c_{1}^{2} + 3)(c_{2}^{2} + 3)(c_{3}^{2} + 3) − 32(c_{1}c_{2}c_{3} + 1).

The surface

Γ = {(c_{1}, c_{2}, c_{3}) ∈ ℝ^{3} | (c_{1}^{2} + 3)(c_{2}^{2} + 3)(c_{3}^{2} + 3) − 32(c_{1}c_{2}c_{3} + 1) = 0}

in the cube

C = {(x, y, z) ∈ ℝ^{3}| − 1 < x, y, z < 1}

is a tetrahedron-like object but with smoothed edges and (omitted) vertices at (1, 1, 1), (1, −1, −1), (−1, 1, −1), and (−1, −1, 1).

The moduli space Π_{3} is a quotient of Γ which arises from permutations of the ends. A choice c = (c_{1}, c_{2}, c_{3}) determines a set of six ends on the double-covering sphere. The group of rotations of the cube is the order-24 permutation group S_{4} generated by two kinds of elements:

• permuting the three numbers (c_{1}, c_{2}, c_{3}),

• negating any two of the three numbers (c_{1}, c_{2}, c_{3}). Action under this group determines the same six ends. Hence Π_{3} = Γ/S_{4} is a representation of the moduli space of minimal projective planes with three embedded planar ends.

Draw the two diagonals on each face of the cube C dividing each face into four triangles. Consider the the 24 tetrahedra whose bases are these triangles, and whose common vertex is the origin. Each of these tetrahedra is a fundamental domain under the action of S_{4} on the cube. This can also be seen by noting that any (c′_{1}, c′_{2}, c′_{3}) in the cube C has in its orbit under S_{4} a point (c_{1}, c_{2}, c_{3}) satisfying c_{1} ≥ c_{2} ≥ |c_{3}| ≥ 0. Let

T = {(c_{1}, c_{2}, c_{3}) ∈ C | c_{1} ≥ c_{2} ≥ |c_{3}| ≥ 0}

be one of these tetrahedra. Then D = T ∩ Γ is a fundamental domain in Γ for the group S_{4}, with boundary

∂D = ∂T ∩ Γ = ({c_{1} = c_{2}} ∪ {c_{2} = c_{3}} ∪ {c_{2} = −c_{3}}) ∩ (T ∩ Γ).

D can be shown to be topologically a closed disk with the point corresponding to the corner (1, 1, 1) of the cube removed.

Proof of (ii): The minimal Möbius strip with total curvature −6π, found in [18], has spinor representation

G(w)\sqrt{dw} = \sqrt{i}(−(w + 1)/w^{2}, w − 1)\sqrt{dw}

Let (σ_{1}(s), σ_{2}(s)) : ℝ^{+} ⟶ Γ be a proper curve. It follows from the reality of σ_{2} that

lim_{s→∞} \frac{1}{σ_{1}(s)} = lim_{s→∞} \frac{1}{σ_{2}(s)} = lim_{s→∞} \frac{σ_{1}(s)}{σ_{2}(s)} = 0,

and by a permutation of the ends we can assume

lim_{s→∞} \frac{\overline{σ_{1}(s)}}{σ_{1}(s)} = 1.

Further,

lim_{s→∞} |\frac{τ_{1}(s)}{σ_{1}(s)}| = 1,

since

|\frac{τ_{1}}{σ_{1}} |^{2} = −\frac{τ_{1}τ_{3}}{σ_{1}σ_{3}} = 1 − \frac{20}{|σ_{1}^{2}|}.

Now choose a function α : ℝ^{+} ⟶ S^{1} ⊂ ℂ such that

lim_{s→∞} (\frac{τ_{1}(s)}{σ_{1}(s)} − α(s)) = 0,

and so

lim_{s→∞} (\frac{τ_{3}(s)}{σ_{1}(s)} − \overline{α(s)}) = 0.

Let X be defined by

X(z)\sqrt{dz} = \frac{\sqrt{i}}{σ_{1}}(t_{1}, t_{2}),

where t_{1}, t_{2} are as in section 22. A careful reparametrization and rotation of the surface generated by X(z)\sqrt{dz} converges uniformly in compact sets to the Möbius strip given above: Let z = αw, and

A_{α} = ( \begin{matrix}a^{3/2} & 0 \\ 0 & α^{−3/2} \end{matrix} ) .

Then

A_{α}X(z)\sqrt{dz} = A_{α}\sqrt{α}X(αw)\sqrt{dw}

is the appropriate reparametrization and rotation. This amounts to showing

lim_{s→∞} A_{α(s)}\sqrt{α(s)}X(α(s)w) = G(w)

uniformly in compact sets not containing the ends, which follows by a calculation using the limits above.

Proof of (iii): To find the surfaces in Π_{3} which have non-trivial symmetry groups as surfaces in space, let G = ℤ_{2} × PSU(2) ≅O(3) be the group of conformal and anticonformal diffeomorphisms of ℂ ∪ {∞} = S^{2} with the property that any ξ ∈ G commutes with I. Via stereographic projection, G can be thought of as the isometry group of S^{2} ⊂ ℝ^{3}, so ξ ∈ G satisfies ⟨a, b⟩ = ⟨ξa, ξb⟩. The group of symmetries of the minimal surface in space induces a subgroup H ⊂ G acting on the domain S^{2}. Moreover, the subgroup H ⊂ G which permutes the ends is isomorphic to the subgroup K ⊆ S_{4} which fixes the point (c_{1}, c_{2}, c_{3}) representing the ends, since ξ ∈ H preserves the inner product defining the cosines c_{1}, c_{2}, c_{3}.

The point of all this is that the symmetry group of a surface represented by (c_{1}, c_{2}, c_{3}) ∈ Π_{3} can be determined by finding the subgroup of S_{4} which fixes (c_{1}, c_{2}, c_{3}). Using this method, the surfaces other than the Möbius strip at (1, 1, 1) are

• elements of ∂D, each with a line of reflective symmetry,

• (\sqrt{5}/3, 0, 0) ∈ ∂D with symmetry group ℤ_{2} × ℤ_{2},

• (c, c, −c) ∈ ∂D with symmetry group S_{3} = D_{3} The last (and most symmetric) of these is a surface described in [14]. □

24 Genus one

The remaining sections concern minimal immersions in the regular homotopy classes of tori and Klein bottles with embedded planar ends. In sections 25 and 26, the skew-symmetric form Ω is computed for the twisted and the untwisted tori. This computation is then used to show the nonexistence and existence of various examples. In section 27 it is shown that no such tori exist with three ends, and in section 28, is found a real two-dimensional family of immersions with four ends exists on each conformal type of torus. After some general results about Klein bottles in section 29, a minimal Klein bottle with embedded planar ends is constructed in section 30.

25 Ω on the twisted torus

For the non-example in section 27, and for the example in section 28, it is necessary to compute a basis for F for the twisted torus (see section 8), and the matrix for Ω in this basis. On the torus ℂ/{2ω_{1}, 2ω_{3}} with the standard coordinate u, let S be the spin structure corresponding to the twisted torus, that is, represented by the holomorphic differential φ_{0}^{2} = du. Let P = [a_{1}] + … + [a_{n}] and set ω_{2} = ω_{1} + ω_{3} throughout the remainder of Part III.

To show that H = {cφ_{0} | c ∈ ℂ}, let t ∈ H. Then 0 ≤ (t) = (t/φ_{0}) + (φ_{0}) = (t/φ_{0}). Hence t/φ_{0} is a holomorphic function on the torus, so it is constant.

A basis for F is {t_{0}, t_{1}, …, t_{n−1}}, where

t_{0} = φ_{0},
t_{i} = (ζ(u − a_{i}) − ζ(u) + ζ(a_{i})) φ_{0},
= \frac{1}{2} (\frac{℘′(u) + ℘′(a_{i})}{℘(u) − ℘(a_{i})}) φ_{0}

(see equation (B.17)). These are in F because

(t_{0}) = 0 ≥ −P,
(t_{i}) = [x_{i}] + [y_{i}] − [a_{i}] − [0] ≥ −P

where x_{i} and y_{i} are the zeros of ℘′(u) + ℘′(a_{i}) other than −a_{i}. These sections are independent because they have distinct poles, and hence span F since dim F = n. To compute Ω in this basis, first compute the expansions of t_{i} at a_{0}, …, a_{n−1} (assume i, j ≠ 0):

\begin{aligned}t_{i} & = & (−u^{−1} + o(u))φ_{0}, \\ t_{i} & = & ((t_{i}/φ_{0})(a_{j}) + o(u))φ_{0} (i ≠ j), \\ t_{i} & = & (u − a_{i})^{−1}φ_{0}. \end{aligned}

Using equation (16.12), we have

Ω(t_{i}, t_{j}) = \{ \begin{matrix}\frac{t_{i}}{φ_{0}} |_{a_{j}} & (i ≠ 0; j ≠ 0; i ≠ j), \\ 0 & (otherwise). \end{matrix}

26 Ω on the untwisted tori

As above, it is also necessary to exhibit a basis for F on the untwisted tori (see section 8), as well as the matrix for Ω in this basis. On the torus ℂ/{2ω_{1}, 2ω_{3}} with the standard conformal coordinate u, fix r ∈ {1, 2, 3} and choose the spin structure on the untwisted torus, represented by

φ_{r}^{2} = \frac{du}{℘_{r}(u)},

where ℘_{r}(u) = ℘(u) − ℘(ω_{r}). Let P = ∑[a_{i}] with the a_{i} ∈ T \ {0, ω_{r}} distinct.

For this choice of spin structure, H = 0. To show this, first note first that (φ_{r}) = [0] − [ω_{r}]. If t ∈ H, then

0 ≤ (t) = (t/φ_{r}) + (φ_{r}) = (t/φ_{r}) + [0] − [ω_{r}].

It follows that (t/φ_{r}) ≥ [ω_{r}]−[0]. But since t/φ_{r} is a function, the degree of its divisor is 0. Hence (t/φ_{r}) = [ω_{r}] − [0]. But this is impossible by Abel’s theorem on the torus: for an elliptic function f, if (f) = ∑ n_{i}[p_{i}] (as a formal sum) then ∑ n_{i}p_{i} = 0 (as a sum in ℂ).

A basis for F is {t_{1}, …, t_{n}}, where

t_{i}(u) = (ζ(u − a_{i}) − ζ(u) − ζ(ω_{r} − a_{i}) + ζ(ω_{r})) φ_{r}
= \frac{1}{2} (\frac{℘_{r}(u)℘′_{r}(a_{i}) + ℘′_{r}(u)℘_{r}(a_{i})}{℘_{r}(a_{i})(℘_{r}(u) − ℘_{r}(a_{i}))}) φ_{r}

(see equation (B.17)). These are in F because (φ_{r}) = [0] − [ω_{r}], so (t_{i}) = [a_{i} − ω_{r}] − [a_{i}] ≥ −P, and are independent because their poles are distinct, so they span F since dim F = n. The expansions of t_{i} at a_{1}, …, a_{n} are

\begin{aligned}t_{i} & = & ((t_{i}/φ_{r})(a_{j}) + o(u − a_{j}))φ_{r} (i ≠ j), \\ t_{i} & = & ((u − a_{i})^{−1} + o(u − a_{i}))φ_{r}. \end{aligned}

Using the local expression (16.12) for Ω, we have

Ω(t_{i}, t_{j}) = \{ \begin{matrix}\frac{t_{i}}{φ_{r}} |_{a_{j}} & (i ≠ j), \\ 0 & (i = i). \end{matrix}

A particularly simple situation arises when the ends come in pairs a and −a. Assume n = 2m and a_{m+i} = −a_{i} (i = 1, …, m). In this case, a simpler basis is {t̂_{1}, …, t̂_{m}, t̂_{m+1}, …, t̂_{2m}}, where for 1 ≤ i ≤ m,

\begin{aligned}t̂_{i} & = & \frac{℘_{r}(a_{i})}{℘′_{r}(a_{i})} (t_{i} − t_{m+i}) φ_{r} & = & (\frac{℘_{r}(u)}{℘_{r}(u) − ℘_{r}(a_{i})}) φ_{r}, \\ t̂_{m+i} & = & (t_{i} + t_{m+i})φ_{r} & = & (\frac{℘′_{r}(u)}{℘_{r}(u) − ℘_{r}(a_{i})}) φ_{r}. \end{aligned}

In this basis, the matrix for Ω becomes

( \begin{matrix}0 & W \\ − W^{t} & 0 \end{matrix} ) ,

where W is given by

W_{ij} = \{ \begin{matrix}\frac{4}{℘_{r}(a_{i}) − ℘_{r}(a_{j})} & (i < j), \\ \frac{4}{℘_{r}(a_{j}) − ℘_{r}(a_{i})} & (i > j), \\ \frac{℘_{r}(a_{i})^{2} − c_{p}c_{q}}{℘_{r}(a_{i})(℘_{r}(a_{i}) − c_{p})(℘_{r}(a_{i}) − c_{q})} & (i = j) \end{matrix}

and c_{p} = e_{p} − e_{r}, c_{q} = e_{q} − e_{r}, {p, q, r} = {1, 2, 3}. Note that the entries of W are entirely free of ℘′_{r}.

A useful property of the basis above is as follows: let L : M ⟶ M be defined as L(u) = −u; then for i ≤ m and j ≥ m + 1,

L^{∗}(t̂_{i}t̂_{j}) = t̂_{i}t̂_{j},

so

∫_{γ_{k}} t̂_{i}t̂_{j} = ∫_{γ_{k}} L^{∗}(t̂_{i}t̂_{j}) = ∫_{L(γ_{k})} t̂_{i}t̂_{j} = − ∫_{γ_{k}} t̂_{i}t̂_{j}.

and so

∫_{γ_{k}} t̂_{i}t̂_{j} = 0 (i ≤ m; j ≥ m + 1; k = 1, 3).

27 Non-existence of tori with three planar ends

An outline of the proof of the non-existence of three-ended tori, twisted or untwisted, is given.

\underline{Theorem 26.} There does not exist a complete minimal branched immersion of a torus into space with finite total curvature and three embedded planar ends.

Sketch of proof: The proof is divided into two cases: for the twisted torus there exist immersions with periods, but the periods cannot be made purely imaginary; for the untwisted torus, there are not even periodic examples.

First consider the more difficult case of of the twisted torus. With everything as in section 25, let {0, a_{1}, a_{2}} be the set of ends, and let p_{i} = ℘(a_{i}), p′_{i} = ℘′(a_{i}). The condition dim K ≥ 2 puts the following condition on the placement of the ends:

g_{2} = 4(p_{1}^{2} + p_{1}p_{2} + p_{2}^{2}),

where g_{2} is the constant in the differential equation (℘′)^{2} = 4℘^{3} − g_{2}℘ − g_{3}. To see this, first note that ker Ω = K ⊕ H and dim H = 1. Hence if dim K = 2 then Ω ≡ 0.

Assume first that a_{1} + a_{2} ≠ 0. Then p_{1} − p_{2} ≠ 0, and the entries of Ω indicate that p′_{1} + p′_{2} = 0 Hence

(p′_{1})^{2} = 4p_{1}^{3} − g_{2}p_{1} − g_{3}

and

(p′_{2})^{2} = 4p_{2}^{3} − g_{2}p_{2} − g_{3}

are equal, and the desired condition follows. The condition also obtains in the case that a_{1} + a_{2} = 0; this can be shown as a limiting case of the above.

Changing basis now to simplify the period equations, let

t̂_{1} = t_{1} + εt_{2},
t̂_{2} = t_{1} + ε^{2}t_{2},

where ε = (−1 + \sqrt{3})/2. With γ_{1}, γ_{3} the closed curves parallel to ω_{1}, ω_{3} respectively (as in Theorem 28), the integrals relevant to the periods are (for k = 1, 3)

∫_{γ_{k}} t̂_{1}^{2} = −6q_{1}ω_{k},
∫_{γ_{k}} t̂_{1}t̂_{2} = −6η_{k},
∫_{γ_{k}} t̂_{2}^{2} = −6q_{2}ω_{k},

where

q_{1} = −((ε − ε^{2})p_{1} + (ε − 1)p_{2})/3,
q_{2} = −((ε^{2} − ε)p_{1} + (ε^{2} − 1)p_{2})/3,
q_{1}q_{2} = (p_{1}^{2} + p_{1}p_{2} + p_{2}^{2})/3 = g_{2}/12.

A choice of a pair of independent sections from K can be normalized by the action of ℝ^{∗} × SU(2) to be

s_{1} = z_{1}t̂_{1} + t̂_{2},
s_{2} = z_{2}t̂_{1},

with z_{1}, z_{2} ∈ ℂ. Then the period equations (10.9) can be written

( \begin{matrix}2z_{1} \\ z_{1}^{2}q_{1} + q_{2} \end{matrix} ) − B ( \begin{matrix}& 0 \\ \overline{q} & 1 & \overline{z} & 2 2 \end{matrix} ) = 0,
( \begin{matrix}z_{2} \\ q_{1}z_{1}z_{2} \end{matrix} ) + B ( \begin{matrix}& & \overline{z} & 2 \\ \overline{q} & 1 & \overline{z} & 1 & \overline{z} & 2 2 \end{matrix} ) = 0,

where

B = A^{−1}\overline{A} = ( \begin{matrix}a & b \\ c & d \end{matrix} ); A = ( \begin{matrix}η_{1} & ω_{1} \\ η_{3} & ω_{3} \end{matrix} ) .

Changing from the variables (z_{1}, z_{2}) to (w, z_{2}), this system is equivalent to the system

w^{2} + b^{2}q_{1}q_{2} − d^{2} = 0,
2w + 2d − b^{2}q_{1}\overline{q}_{1}\overline{z}_{2}^{2} = 0,
wz_{2} + \overline{z}_{2} = 0.

From these it follows that

w\overline{w} − 1 = 0,
aw^{2} − \overline{a} = 0,
−\overline{a} − ab^{2}q_{1}q_{2} + ad^{2} = 0.

This last condition, depending only on the conformal type of the torus and not on w, z_{1}, and z_{2}, is a degeneracy condition for the period equations. It also follows, by an examination of the sign of a(w − \overline{a}) ∈ ℝ, that

|a| > 1.

A delicate argument, which is omitted here, using the expansions [16]

g_{2} = \frac{π^{4}}{12ω_{1}^{4}} (1 + 240 ∑_{n=1}^{∞} σ_{3}(n)q^{n}) ,
η_{1} = \frac{π^{2}}{12ω_{1}} (1 − 24 ∑_{n=1}^{∞} σ_{1}(n)q^{n}) ,

where

σ_{k}(n) = ∑_{d|n} d^{k}; q = e^{2iπτ}; τ = ω_{3}/ω_{1}

shows that the degeneracy condition is not satisfied under the constraint |a| > 1 over the whole moduli space of Riemann tori. Hence no examples with three ends can be found in the case of the twisted tori.

The case of the untwisted tori is much easier. Fix r ∈ {1, 2, 3} and let φ_{r} be as in section 26. Let {a_{1}, a_{2}, a_{3}} be the ends, translated so that they avoid {0, ω_{r}}, and let {t_{1}, t_{2}, t_{3}} be the basis for F given in the same section. The condition that dim K = dim ker Ω ≤ 2 forces Ω to be zero. This means, for example, that t_{1}/φ_{r} have zeros at a_{2} and a_{3}. But the zeros of t_{1}/φ_{r} are ω_{r} and a_{1} − ω_{r}, so one of a_{2}, a_{3} has to be ω_{r}, contrary to the assumption. □

28 Minimal tori with four embedded planar ends

Here the existence of families of four-ended tori is established.

\underline{Theorem 27.} For each conformal type of torus there exists a real two-dimensional family of complete minimal immersions of the torus punctured at four points into space with finite total curvature and embedded planar ends. Each of the tori is twisted.

Proof. To exhibit the family, it is first necessary to determine the placement of the four ends. The ends in fact must be, up to a translation, at the four half-lattice points. To show this, on the torus ℂ/{2ω_{1}, 2ω_{3}}, assume the four ends are {0, a_{1}, a_{2}, a_{3}}, where a_{1}, a_{2}, a_{3} are distinct points in the torus to be determined. With φ_{0}^{2} = du, the matrix for Ω in the basis {1, t_{1}, t_{2}, t_{3}} = {φ, f_{1}φ_{0}, f_{2}φ_{0}, f_{3}φ_{0}} of section 25 is

Ω = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & f_{1}(a_{2}) & f_{1}(a_{3}) \\ 0 & f_{2}(a_{1}) & 0 & f_{2}(a_{3}) \\ 0 & f_{3}(a_{1}) & f_{3}(a_{2}) & 0 \end{matrix} ) .

If ker Ω = H ⊕ K is two-dimensional, then dim K = 1, since dim H = 1, so K is not big enough to generate a minimal surface. Hence to produce surfaces, rank Ω, being even, must be zero. In this case, all the entries of the above matrix are zero; a look at t_{i} shows that ℘′(a_{i}) + ℘′(a_{j}) = 0 for all i ≠ j. It follows that ℘′(a_{1}) = ℘′(a_{2}) = ℘′(a_{3}) = 0, so {a_{1}, a_{2}, a_{3}} = {ω_{1}, ω_{2}, ω_{3}}.

With the ends fixed at {0, ω_{1}, ω_{2}, ω_{3}}, F = ker Ω = H ⊕ K, so {t_{1}, t_{2}, t_{3}} is a basis for K. The simple zeros and poles of t_{1}, t_{2}, and t_{3} are illustrated below.

[figure: ∞; ∞; ∞; ∞; 0; 0; 0; 0; ∞; 0; ∞; 0; t_{1}; t_{2}; t_{3}]
Figure 6: Zeros and poles of t_{1}, t_{2}, and t_{3}

To solve the period problem outlined in section 10 it is convenient to choose a The Spinor Representation of Minimal Surfaces 49 new basis {t̂_{1}, t̂_{2}, t̂_{3}} for K which “diagonalizes” the period equations. Let

( t̂_{1} ; t̂_{2} ; t̂_{3} ) = ( \begin{matrix}1 & −1 & −1 \\ −1 & 1 & −1 \\ −1 & −1 & 1 \end{matrix} ) ( \begin{matrix}t_{1} \\ t_{2} \\ t_{3} \end{matrix} ) ,

or

t̂_{1}(u) = (ζ(u) + ζ(u − ω_{1}) − ζ(u − ω_{2}) − ζ(u − ω_{3}) + 2ζ(ω_{1}))φ_{0},
t̂_{2}(u) = (ζ(u) − ζ(u − ω_{1}) + ζ(u − ω_{2}) − ζ(u − ω_{3}) + 2ζ(ω_{2}))φ_{0},
t̂_{3}(u) = (ζ(u) − ζ(u − ω_{1}) − ζ(u − ω_{2}) + ζ(u − ω_{3}) + 2ζ(ω_{3}))φ_{0}.

The simple zeros and poles of t̂_{1}, t̂_{2}, and t̂_{3} are illustrated below. To compute the

[figure: ∞; ∞; ∞; ∞; ∞; ∞; ∞; ∞; ∞; ∞; ∞; ∞; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; t̂_{1}; t̂_{2}; t̂_{3}]
Figure 7: Zeros and poles of t̂_{1}, t̂_{2}, and t̂_{3}

periods, use equation (B.18) to write

t̂_{i}^{2}(u) = (℘(u) + ℘(u − ω_{1}) + ℘(u − ω_{2}) + ℘(u − ω_{3}) − 4℘(ω_{i})) du,
(t̂_{1}t̂_{2})(u) = (℘(u) − ℘(u − ω_{1}) − ℘(u − ω_{2}) + ℘(u − ω_{3})) du,
(t̂_{1}t̂_{3})(u) = (℘(u) − ℘(u − ω_{1}) + ℘(u − ω_{2}) − ℘(u − ω_{3})) du,
(t̂_{2}t̂_{3})(u) = (℘(u) + ℘(u − ω_{1}) − ℘(u − ω_{2}) − ℘(u − ω_{3})) du.

With γ_{1}, γ_{3} the closed curves on the torus respectively parallel to ω_{1}, ω_{3}, the periods are

P_{k}^{ij} = ∫_{γ_{k}} t̂_{i}t̂_{j}du = \{ \begin{matrix}−8(η_{k} + ω_{k}e_{i}) & if i = j \\ 0 & if i ≠ j \end{matrix} (k = 1, 3),

where e_{i} = ℘(ω_{i}) and η_{k} = ζ(ω_{k}) (see appendix B). In general, with

t_{1} = x_{1}t̂_{1} + x_{2}t̂_{2} + x_{3}t̂_{3}
t_{2} = y_{1}t̂_{1} + y_{2}t̂_{2} + y_{3}t̂_{3}

the period equations (10.9) are

∑_{1≤i,j≤3} P_{k}^{ij}x_{i}x_{j} = \overline{∑_{1≤i,j≤3} P_{k}^{ij}y_{i}y_{j}} (k = 1, 3)
∑_{1≤i,j≤3} P_{k}^{ij}x_{i}y_{j} ∈ iℝ (k = 1, 3).

Now let (i, j, k) be a permutation of (1, 2, 3) and make the particular choice

s_{1} = x_{i}t̂_{i} + x_{j}t̂_{j},
s_{2} = t̂_{k}.

The second period equation above is satisfied for all x_{i}, x_{j}, and the first period equation can be written in the form

( \begin{matrix}x_{i}^{2} \\ x_{j}^{2} \end{matrix} ) = ( \begin{matrix}1 & 1 \\ e_{i} & e_{j} \end{matrix} )^{−1} B ( \begin{matrix}1 \\ \overline{e} & k \end{matrix} )

where η_{i} = ζ(ω_{i}) and e_{i} = ℘(ω_{i}) and B is defined in section 27. The condition that the surface be immersed is that s_{1} and s_{2} have no common zeros. The zeros of s_{2} are at {ω_{k}/2, ω_{k}/2 + ω_{1}, ω_{k}/2 + ω_{2}, ω_{k}/2 + ω_{3}}, and

t̂_{m}^{2}(ω_{k}/2) = t̂_{m}^{2}(ω_{k}/2 + ω_{l}) = 4(e_{k} − e_{i}) (m = i, j; l = 1, 2, 3).

A necessary condition that the surface branch is that

(e_{k} − e_{i})x_{i}^{2} − (e_{k} − e_{j})x_{j}^{2} = 0,

or

( g_{2}/2 − 3e_{k}^{2} −3e_{k} ) B ( \begin{matrix}1 \\ \overline{e} & k \end{matrix} ) = 0.

With the choice {i, j, k} = {1, 2, 3} it can be shown that this condition is not satisfied in the standard fundamental region of the moduli space of tori. The proof uses the q-expansion for g_{2} and η given in section 27, as well as the expansion

e_{1} = \frac{π^{2}}{6ω_{1}^{2}} (1 + 24 ∑_{n=1}^{∞} τ(n)q^{n}) ,

where

τ(n) = ∑_{{d|n \atop d odd}} d.

Thus we have found a single immersion of every conformal type of torus punctured at the half-lattice points. Since the period conditions amount to at most six real conditions on 12 variables, there is a real 6-parameter family of surfaces, which modulo the action of the group in equation (9.8) leaves a 2-parameter family. The existence of the real two-dimensional family follows from the fact that the condition of being immersed is an open analytic condition. □

29 Klein bottles: conformal type, spin structure and periods

Theorem 28 shows that the torus underlying a Klein bottle must have the conformal type of the complex plane modulo a rectangular lattice, and it computes the order-two deck transformation for the covering of the Klein bottle by the torus. The theorem further shows that the torus which doubly covers the immersed Klein bottle must be untwisted. (This can also be seen from purely topological considerations.)

\underline{Theorem 28.} Let X : K′ ⟶ ℝ^{3} be a complete minimal immersion of a punctured Klein bottle with finite total curvature, π : T ⟶ K = \overline{K′} the oriented two-sheeted covering by a torus T, and I : T ⟶ T the order-two orientation-reversing deck transformation for this cover. Then we have the following.

(i) T is conformally equivalent to ℂ/Λ, where Λ is a rectangular lattice with generators 2ω_{1} ∈ ℝ and 2ω_{3} ∈ iℝ.

(ii) On this torus, the deck transformation I may be chosen to be I(u) = ū + ω_{1}. (iii) With this choice, the admissible spin structures are those represented by (℘(u)−

℘(ω_{2}))du and (℘(u) − ℘(ω_{3}))du. (iv) If (s_{1}, s_{2}) is the spin representative of X ∘ π on T, the period conditions reduce

to the conditions ∫_{γ_{1}} s_{1}^{2} = 0 and ∫_{γ_{1}} s_{1}s_{2} = 0 along a closed curve γ_{1} parallel to ω_{1}.

Proof of (i) and (ii): Let Λ_{0} be a lattice such that T = ℂ/Λ_{0}. Since every conformal map from T to T must be linear in the standard coordinate u on ℂ and since I is anticonformal, I(u) = αū+β for some α, β ∈ ℂ. The periodicity of I and I^{−1} implies that α\overline{Λ_{0}} ⊆ Λ_{0} and \overline{α}^{−1}\overline{Λ_{0}} ⊆ Λ_{0}. These together imply that α\overline{Λ_{0}} = Λ_{0}. Choose γ ∈ ℂ satisfying |γ| = 1 and \overline{γ}/γ = α; the rotated lattice Λ = γΛ_{0} satisfies \overline{Λ} = Λ (a socalled real lattice). Hence Λ is either rectangular with generators 2ω_{1} ∈ ℝ, 2ω_{3} ∈ iℝ, or Λ is rhombic with generators 2ω_{1} and 2ω_{3} = 2\overline{ω}_{1}. On ℂ/Λ we have I(u) = αū + β for some new α, β ∈ ℂ. As before, α\overline{Λ} = Λ, but \overline{Λ} = Λ, so α = ±1. If α = −1, replacing Λ by iΛ preserves its reality, and changes α to 1.

With α = 1, the condition that I is involutive is that β + \overline{β} ∈ Λ. By the change of coordinate u ↦ u − i Im β, it can be assumed that β ∈ ℝ. Then the involutive condition is that 2β ∈ Λ. If β ∈ Λ then 0 is a fixed point of I. Hence β ≡ ω_{1} (rectangle) or β = ω_{1} + ω_{3} (rhombus). In the latter case, ω_{1} is a fixed point of I, so the only admissible case is the rectangle, with I(u) = \overline{u} + ω_{1}.

Proof of (iii): The compatibility condition in Theorem 11 demands that I^{∗}I^{∗}(s) = −s for any section s of the spin structure. A computation shows that this condition is met only for the two spin structures named.

Proof of (iv): Let γ_{1} and γ_{3} be respectively the closed curves t ↦ ω_{1}t/|ω_{1}| + c_{1} and t ↦ ω_{3}t/|ω_{3}| + c_{2}, (0 ≤ t ≤ 2), where c_{1}, c_{2} ∈ ℂ are chosen so that the curves do not pass through any ends. Then I(γ_{1}) = γ_{1}, I(γ_{3}) = −γ_{3}. The periods conditions are

∫_{γ_{k}} s_{1}^{2} = \overline{∫_{γ_{k}} s_{2}^{2}} (k = 1, 3),
∫_{γ_{k}} s_{1}s_{2} ∈ iℝ (k = 1, 3).

With I as above, under the double-cover assumption

(s_{1}, s_{2}) = ±(i\overline{I^{∗}s_{2}}, −i\overline{I^{∗}s_{1}}),

we have

∫_{γ_{3}} s_{1}^{2} = ∫_{γ_{3}} −\overline{I^{∗}s_{2}^{2}} = −\overline{∫_{I(γ_{3})} s_{2}^{2}} = \overline{∫_{γ_{1}} s_{2}^{2}}
∫_{γ_{3}} s_{1}s_{2} = ∫_{γ_{3}} \overline{I^{∗}s_{1}s_{2}} = \overline{∫_{I(γ_{3})} s_{1}s_{2}} = −\overline{∫_{γ_{3}} s_{1}s_{2}},

so the period conditions are automatically satisfied for k = 3. Moreover, we also have

∫_{γ_{1}} s_{1}^{2} = ∫_{γ_{1}} −\overline{I^{∗}s_{2}^{2}} = −\overline{∫_{I(γ_{1})} s_{2}^{2}} = −\overline{∫_{γ_{1}} s_{2}^{2}}

∫_{γ_{1}} s_{1}s_{2} = ∫_{γ_{1}} \overline{I^{∗}s_{1}s_{2}} = \overline{∫_{I(γ_{1})} s_{1}s_{2}} = \overline{∫_{γ_{1}} s_{1}s_{2}}

and the first two period conditions (10.9) become

∫_{γ_{1}} s_{1}^{2} = 0
∫_{γ_{1}} s_{1}s_{2} = 0

(this amounts to three real conditions because under the above assumption, the second integral is automatically real). □

30 Minimal Klein bottles with embedded planar ends

A minimal Klein bottle is constructed in this section. Its compactification is a W - critical surface with energy W = 16π, which lies in the amphichiral regular homotopy class K_{0} = B#\overline{B} of Klein bottles (cf. [13], [24]). Clearly there are no minimal Klein bottles with two embedded ends and we conjecture there are none with three embedded planar ends.

\underline{Theorem 29.} There exists a minimal immersion of the Klein bottle with four embedded planar ends.

To construct this example, let T = ℂ/{2ω_{1}, 2ω_{3}} be a square lattice with

ω_{3} = iω_{1}
ω_{2} = −ω_{1} − ω_{3}
℘(ω_{1}) = 1
℘(ω_{2}) = 0
℘(ω_{3}) = −1.

Let I: T ⟶ T be the deck transformation

I(u) = ū + ω_{1}

as in Theorem 28 i). Let a ∈ T be a point (yet to be determined) such that

I(a) = −a,

and let E = {a_{1}, …, a_{8}} ⊂ T be the points in Table 3. We want to construct a minimal immersion X: (T \ E)/I ⟶ ℝ^{3},

X(z) = Re ∫ ^{z}(s_{1}^{2} − s_{2}^{2}, i(s_{1}^{2} + s_{2}^{2}), 2s_{1}s_{2}),

where s_{1}, s_{2} are sections of the spin structure S determined by φ, where

φ^{2} = \frac{du}{℘(u) − ℘(ω_{2})} = \frac{du}{℘(u)}.

54 Kusner and Schmitt Table 3: Values of ℘ and ℘′ at ends of Klein bottle

\begin{matrix}u & ℘(u) & ℘′(u) & I(u) \\ a_{1} = a & r & r′ & a_{5} \\ a_{2} = a + ω_{2} & −1/r & r′/r^{2} & a_{6} \\ a_{3} = −ia & −r & −ir′ & a_{4} \\ a_{4} = −ia + ω_{2} & 1/r & −ir′/r^{2} & a_{3} \\ a_{5} = −a_{1} & r & −r′ & a_{1} \\ a_{6} = −a_{2} & −1/r & −r′/r^{2} & a_{2} \\ a_{7} = −a_{3} & −r & ir′ & a_{8} \\ a_{8} = −a_{4} & 1/r & ir′/r^{2} & a_{7} \end{matrix}

Step 1: Determination of the ends Let {t_{1}, …, t_{8}},

t_{α} = \frac{℘(u)}{℘(u) − ℘(a_{α})}φ

(1 ≤ α ≤ 4)

t_{α+4} = \frac{℘′(u)}{℘(u) − ℘(a_{α})}φ

be a basis for F, as in section 26. The skew-symmetric matrix for Ω in this basis is

( \begin{matrix}0 & W \\ − W^{t} & 0 \end{matrix} ) ,

where W is given by

W = ( \begin{matrix}\frac{r^{2} + 1}{r(r^{2} − 1)} & \frac{4r}{r^{2} + 1} & \frac{2}{r} & \frac{4r}{r^{2} − 1} \\ \frac{−4r}{r^{2} + 1} & \frac{r(r^{2} + 1)}{r^{2} − 1} & \frac{4r}{r^{2} − 1} & −2r \\ \frac{−2}{r} & \frac{−4r}{r^{2} − 1} & \frac{−(r^{2} + 1)}{r(r^{2} − 1)} & \frac{−4r}{r^{2} + 1} \\ \frac{−4r}{r^{2} − 1} & 2r & \frac{4r}{r^{2} + 1} & \frac{−r(r^{2} + 1)}{r^{2} − 1} \end{matrix} ) .

The desired sections s_{1}, s_{2} lie in ker Ω, so a necessary condition for existence is that

0 = det W = \frac{(3r^{8} − 4r^{6} + 50r^{4} − 4r^{2} + 3)^{2}}{(r^{4} − 1)^{2}} = \frac{9(r^{4} + mr^{2} + 1)^{2}(r^{4} + \overline{m}r^{2} + 1)^{2}}{(r^{4} − 1)^{2}} ,

where m = −2(1 − 4\sqrt{2}i)/3. Let r be the root of r^{4} + mr^{2} + 1 in the fourth quadrant; with this choice, the domain T \E is shown below.

[figure: a + ω_{2}; −a; −ia + ω_{2}; ia; −ia; ia + ω_{2}; a; −a + ω_{2}; 0; ω_{1}; ω_{2}; ω_{3}]

Figure 8: The eight ends in the double cover of the Klein bottle

Step 2: Choosing sections s_{1}, s_{2} of S; the period equations

With r fixed as above, rank Ω is 4, and a basis for ker Ω is {ŝ_{1}, ŝ_{2}, ŝ_{3}, ŝ_{4}} where

ŝ_{1} = ∑_{a=1}^{4} c_{1}^{α}t_{α}
ŝ_{2} = ∑_{a=1}^{4} c_{2}^{α}t_{α}
ŝ_{3} = i\overline{I^{∗}ŝ_{1}}
ŝ_{4} = i\overline{I^{∗}ŝ_{2}},
c_{1} = (2(r^{2} − 1)^{2}, (r^{2} + 1)(r^{2} − 3), (r^{2} + 1)(3r^{2} − 1), −2(r^{2} − 1)^{2})
c_{2} = ((r^{2} + 1)(3r^{2} − 1), −2(r^{2} − 1)^{2}, 2(r^{2} − 1)^{2}, (r^{2} + 1)(r^{2} − 3))

and I^{∗} is a choice of a lift of the deck transformation I to the spin structure S.

Let

s_{1} = x_{1}ŝ_{1} + x_{2}ŝ_{2}
x_{1}, x_{2} ∈ ℂ
s_{2} = i\overline{I^{∗}s_{1}} = \overline{x}_{1}ŝ_{3} + \overline{x}_{2}ŝ_{4}.

We want to find x_{1}, x_{2} such that the real part of all periods are zero. By Theorem 28(iv) and section 29, the period equations reduce to the single equation

0 = ∫_{γ_{1}} s_{1}^{2} = x_{1}^{2}P_{1}^{11} + 2x_{1}x_{2}P_{1}^{12} + x_{2}^{2}P_{1}^{22},

56 Kusner and Schmitt where

P_{k}^{αβ} = ∫_{γ_{k}} ŝ_{α}ŝ_{β}

along the curve γ_{k}: t ⟼ tω_{k} (−1 ≤ t ≤ 1). Step 3: Explicit solution of the period equation

The period equation above can be solved once P_{k}^{αβ} are known. To compute these, let

ŝ_{1}^{2} = \frac{1}{2} (− ∑_{α=1}^{4} A_{α}℘(u − a_{α}) + B) du, A = ∑ A_{α}

ŝ_{1}ŝ_{2} = \frac{1}{2} (− ∑_{α=1}^{4} C_{α}℘(u − a_{α}) + D) du, C = ∑ C_{α}

as in equation (B.18). Then

P_{1}^{11} = ∫_{γ_{1}} ŝ_{1}^{2} = Aη_{1} + Bω_{1},
P_{1}^{12} = ∫_{γ_{1}} ŝ_{1}ŝ_{2} = Cη_{1} + Dω_{1},
P_{3}^{11} = Aη_{3} + Bω_{3} = i(−Aη_{1} + Bω_{1})
P_{3}^{12} = Cη_{3} + Dω_{3} = i(−Cη_{1} + Dω_{1})
η_{k} = −\frac{1}{2} ∫_{γ_{k}} ℘(u)du.

Let J: T → T be defined by J(u) = iu, and let J^{∗} be a lift of J to S. Then ŝ_{1} = \sqrt{i}J^{∗}ŝ_{2}, ŝ_{2} = \sqrt{i}J^{∗}ŝ_{1}

for some choice of \sqrt{i}. Then

P_{1}^{12} = ∫_{γ_{1}} ŝ_{1}ŝ_{2} = ∫_{γ_{1}} iJ^{∗}ŝ_{1}ŝ_{2} = i ∫_{J_{(γ_{1})}} J^{∗}ŝ_{1}ŝ_{2} = i ∫_{γ_{3}} ŝ_{1}ŝ_{2} = iP_{3}^{12},

so D = 0. Again,

P_{1}^{22} = ∫_{γ_{1}} ŝ_{2}^{2} = ∫_{γ_{1}} iJ^{∗}ŝ_{1}^{2} = i ∫_{J_{(γ_{1})}} ŝ_{1}^{2} = i ∫_{γ_{3}} ŝ_{1}^{2} = iP_{3}^{11},

so P_{1}^{22} = Aη_{1} − Bω_{1}.

Having computed P_{1}^{11}, P_{1}^{12}, P_{1}^{22} in terms of A, B, C, we compute A, B, C by expanding ŝ_{α}ŝ_{β}/du in two ways and equating coefficients. On the one hand, by the definition of ŝ_{α}, we have

ŝ_{a}ŝ_{β}/du = ∑_{γ,δ} \frac{c_{α}^{γ}c_{β}^{δ}℘(u)}{(℘(u) − ℘(a_{γ}))(℘(u) − ℘(a_{δ}))} (1 ≤ α, β ≤ 2; 1 ≤ γ, δ ≤ 4).

Using the formula (for ℘′(u_{0}) finite and non-zero)

\frac{1}{℘(u) − ℘(u_{0})} = \frac{1/℘′(u_{0})}{u − u_{0}} + ⋯,

we get the expansion at a_{γ}

ŝ_{a}ŝ_{β}/du = \frac{c_{a}^{γ}c_{β}^{γ}℘(a_{γ})/(℘′(a_{γ}))^{2}}{(u − a_{γ})^{2}} .

On the other hand we have the expansions at a_{γ}

ŝ_{1}^{2}/du = \frac{−A_{γ}/2}{(u − a_{γ})^{2}}
ŝ_{1}ŝ_{2}/du = \frac{−C_{γ}/2}{(u − a_{γ})^{2}} .

Equating coefficients,

A_{γ} = −2(c_{1}^{γ})^{2}℘(a_{γ})/(℘′(a_{γ}))^{2}
C_{γ} = −2c_{1}^{γ}c_{2}^{γ}℘(a_{γ})/(℘′(a_{γ}))^{2},

so

A = ∑ A_{i} = −32r^{2}(r^{4} + 4r^{2} + 1)/3
C = ∑ C_{i} = −2(r^{4} − 1)^{2}.

To compute B, note that s_{1} has a zero at 0 to get

B = ∑ A_{γ}℘(a_{γ}) = 4r(r^{2} + 1)^{3}.

This solves the period equation.

Finally, that the immersion is unbranched is the condition that s_{1}, s_{2} have no common zeros. This amounts to the condition that if u_{0} is a zero of s_{1}, then I(u_{0}) is not. By using the identity

\overline{I^{∗}℘} = \frac{℘ + 1}{℘ − 1},

this can be checked by setting s_{1} to zero, and solving numerically the cubic in ℘ which results.

Appendix

A The Pfaffian

Here we recall some basic facts about skew-symmetric forms.

\underline{Definition.} A bilinear form A on a vector space V of dimension n is skewsymmetric if

A(v_{1}, v_{2}) + A(v_{2}, v_{1}) = 0 for all v_{1}, v_{2} ∈ V ,

or alternatively, if the matrix A for A satisfies

A + A^{t} = 0.

The space of skew-symmetric bilinear forms is ⋀^{2}(V^{∗}). The pfaffian is a function on skew-symmetric forms whose square is the determinant.

\underline{Definition.} For A ∈ ⋀^{2}(V^{∗}), the pfaffian of A is

pf (A) = \{ \begin{matrix}\frac{1}{m!} \overbrace{(A ∧ … ∧ A)}^{m times} & if dim(V) = 2m is even, \\ 0 & if dim(V) is odd. \end{matrix}

For a matrix (a_{ij}) of A ∈ ⋀^{2}(V^{∗}) in the basis {e_{1}, …, e_{m}} the pfaffians for m = 2, m = 4, and m = 6 are respectively

a_{12},
a_{12}a_{34} − a_{13}a_{24} + a_{14}a_{23},
a_{12}a_{34}a_{56} − a_{12}a_{35}a_{46} + a_{12}a_{36}a_{45} − a_{13}a_{24}a_{56} + a_{13}a_{25}a_{46}−
a_{13}a_{26}a_{45} + a_{14}a_{23}a_{56} − a_{14}a_{25}a_{36} + a_{14}a_{26}a_{35} − a_{15}a_{23}a_{46}+
a_{15}a_{24}a_{36} − a_{15}a_{26}a_{34} + a_{16}a_{23}a_{45} − a_{16}a_{24}a_{35} + a_{16}a_{25}a_{34}.

The general pfaffian of a 2m × 2m matrix has (2m)!/(2m!) = 1 · 3 · 5 · · · · · (2m − 1) terms.

\underline{Lemma.} The rank of a skew-symmetric matrix is even.

Proof. Let A be an m × m skew-symmetric matrix with rank r. The proof is by induction on m. In the case m = 1, then A = (0) with even rank 0. Assume for some n that the lemma is true for all skew-symmetric matrices smaller than A. If n is odd, then

det A = det A^{t} = det(−A) = (−1)^{n} det A = − det A,

so det A = 0 and A has a non-zero kernel. If n is even, then A also has a non-zero kernel unless it has full — hence even — rank r = n. So in either case we may assume A has a non-zero kernel.

Let v_{1}, …, v_{n−r} be a basis for ker A, and extend to a basis v_{1}, …, v_{n−r}, w_{1}, …, w_{r} for ℂ^{n}. Let P be the n × n matrix with these vectors as columns. Then P^{t}AP is of the form

P^{t}AP = ( \begin{matrix}0 & 0 \\ 0 & A_{0} \end{matrix} ) ,

where A_{0} is an r × r matrix of rank r < n. Moreover,

(P^{t}AP)^{t} = P^{t}A^{t}P = −(P^{t}AP),

so P^{t}AP, and hence A_{0} is skew-symmetric. By the induction hypothesis, r = rank A is even, since it is the rank of the smaller skew-symmetric matrix A_{0}. □

B Elliptic functions

For reference, here are some standard notations and facts about elliptic functions used in this paper (see for example [6], [7]).

Lattices. A non-degenerate lattice Λ is real if Λ = \overline{Λ}. There are two kinds of real lattices:

(i) rectangular: generators ω_{1} ∈ ℝ and ω_{3} ∈ iℝ can be chosen for Λ. (ii) rhombic: generators ω_{1} and ω_{3} = \overline{ω}_{1} can be chosen for Λ. For any lattice with generators ω_{1}, ω_{3}, let ω_{2} = −ω_{1} − ω_{3}.

The Weierstrass ℘ function: Given a lattice Λ generated by ω_{1} and ω_{3}, the elliptic function ℘ on ℂ/Λ satisfies the differential equation

(℘′)^{2} = 4℘^{3} − g_{2}℘ − g_{3} = 4(℘ − e_{1})(℘ − e_{2})(℘ − e_{3}),

where

e_{i} = ℘(ω_{i}) (i = 1, 2, 3),
e_{1} + e_{2} + e_{3} = 0,
g_{2} = −4(e_{1}e_{2} + e_{1}e_{3} + e_{2}e_{3}),
g_{3} = 4e_{1}e_{2}e_{3}.

The function ℘ has a double pole at 0 and two simple zeros which come together only on the square lattice; ℘′ has a triple pole at 0 and three simple poles at ω_{1}, ω_{2}, ω_{3}.

The function ℘ is even; ℘′ is odd. On a horizontal rectangular lattice, ℘(\overline{u}) = \overline{℘(u)}; on a horizontal square lattice, ℘(iu) = −℘(u).

The expansion for ℘ at 0 is

℘(u) = \frac{1}{u^{2}} + \frac{g_{2}}{20}u^{2} + ….

A useful property of ℘ is the following special case of the addition formula ({i, j, k} is any permutation of {1, 2, 3}):

℘(u ± ω_{i}) = e_{i} + \frac{(e_{i} − e_{j})(e_{i} − e_{k})}{℘(u) − e_{i}}. (B.15)

The Weierstrass ζ function: The ζ function is defined by

ζ(u) = − ∫ ℘(u)du,

with the constant of integration chosen so that lim_{u→0} ζ(u)−u^{−1} = 0. With η_{i} = ζ(ω_{i}) (i = 1, 2, 3), properties of ζ include:

η_{1} + η_{2} + η_{3} = 0,

ζ(u + 2ω_{i}) = ζ(u) + 2η_{i} (i = 1, 2, 3), ζ is an odd function.

Legendre’s relation is that

η_{1}ω_{3} − η_{3}ω_{1} = iπ/2. (B.16)

A form of the quasi-addition formula for ζ is

ζ(u − v) − ζ(u) + ζ(v) = \frac{1}{2} (\frac{℘′(u) + ℘′(v)}{℘(u) − ℘(v)}) . (B.17)

A useful property of elliptic functions which can also be stated in more generality is the following: Let f be an elliptic function with poles of order at most 2, with no residues, and with principal parts

\frac{a_{1}}{(u − α_{1})^{2}} , …, \frac{a_{n}}{(u − α_{n})^{2}} .

Then

f(u) = b + ∑ a_{i}℘(u − a_{i}) (B.18)

for some b, because the difference f(u) − ∑ α_{i}℘(u − α_{i}) has no poles and hence is constant.

References

[1] Arbarello, E., Cornalba, M., Griffiths, P. A., and Harris, J. Geometry of algebraic curves. New York: Springer-Verlag, 1985.

[2] Bryant, R. A duality theorem for Willmore surfaces. J. Diff. Geom., 20:23–53, 1984.

[3] Bryant, R. Surfaces in conformal geometry. Proceedings of Symposia in Pure Mathematics 48:227–240, 1988.

[4] Callahan, M., Hoffman, D., Meeks III, W. H. Embedded minimal surfaces with an infinite number of ends. Invent. Math. 96:459–505, 1989.

[5] Costa, C. Complete minimal surfaces in ℝ^{3} of genus one and four planar embedded ends. Preprint, 1990.

[6] DuVal, P. Elliptic functions and elliptic curves. Cambridge: Cambridge University Press, 1973.

[7] Erdelyi, A., ed. Higher transcendental functions. New York: McGraw-Hill, 1953.

[8] Gilbert, J. E., and Murray, A. M. A. Clifford algebras and Dirac operators in harmonic analysis. Cambridge: Cambridge University Press, 1991.

[9] Griffiths, P., and Harris, J. Principles of algebraic geometry. New York: Wiley-Interscience, 1978.

[10] Gunning, R. Lectures on Riemann surfaces. Princeton: Princeton University Press, 1966.

[11] Gunning, R. Riemann surfaces and generalized theta functions. New York: Springer Verlag, 1976.

[12] Kauffman, L. H. On knots. Princeton: Princeton University Press, 1987.

[13] Kusner, R. Comparison surfaces for the Willmore problem. Pacific Journal of Mathematics 138:317–345, 1989.

[14] Kusner, R. Conformal geometry and complete minimal surfaces. Bull. Amer. Math. Soc. 17:291–295, 1987.

[15] Kusner, R. Global geometry of extremal surfaces in three-space. Dissertation, Univ. of California, Berkley, 1988.

[16] Lang, S. Elliptic functions. New York: Springer-Verlag, 1987.

[17] Lawson, H. B., and Michaelsohn, M. L. Spin Geometry Princeton: Princeton U. Press, 1989.

[18] Meeks III, W. H. The classification of complete minimal surfaces in ℝ^{3} with total curvature greater than −8π. Duke Mathematical Journal 48:523–535, 1981.

[19] Meeks III, W. H. and Patrusky, J. Representing homology classes by embedded circles on a compact surface. Ill. J. Math. 22:262-269, 1978.

[20] Milnor, J. Spin structures on manifolds. Enseign. Math. 9:198–203, 1963.

[21] Mumford, D. Tata lectures on theta, v. 1 and 2. Boston: Birkhäuser, 1983.

[22] Osserman, R. A survey of minimal surfaces. New York: Van Nostrand Reinhold, 1969.

[23] Peng, C. K. Some new examples of minimal surfaces in ℝ^{3} and its applications. Preprint, MSRI, 1986.

[24] Pinkall, U. Regular homotopy classes of immersed surfaces. Topology, 24:421– 434, 1985.

[25] Rees, E. Notes on geometry. New York: Springer-Verlag, 1983.

[26] Schmitt, N. Minimal surfaces with embedded planar ends. Dissertation, Univ. of Massachusetts, Amherst, 1993.

[27] Sullivan, D. The spinor representation of minimal surfaces in space. Notes, 1989.

Graphics

The following computer graphics were produced at the Center for Geometry, Analysis, Numerics and Graphics using the MESH program authored by Jim Hoffman. For more information about MESH, contact web@gang.umass.edu.