[Paper Review] The Spinor Representation of Surfaces in Space

Rob Kusner, Nick Schmitt|arXiv (Cornell University)|Oct 8, 1996

Algebraic Geometry and Number TheoryMathematics18 references19 citations

TL;DR

This paper introduces a spinor representation for conformal immersions of Riemann surfaces into R³, generalizing the classical Weierstrass representation by using meromorphic spinor sections (s₁, s₂) to parametrize minimal surfaces. The key contribution is a direct link between the spin structure on a surface and the regular homotopy class of its immersion, with the Arf invariant of a Z₂-quadratic form determining whether the immersion is regularly homotopic to an embedding—providing a complete classification of W-critical spheres and projective planes via this framework.

ABSTRACT

The spinor representation is developed for conformal immersions of Riemann surfaces into space. We adapt the approach of Dennis Sullivan, which treats a spin structure on a Riemann surface M as a complex line bundle S whose square is the canonical line bundle K=T(M). Given a conformal immersion of M into \bbR^3, the unique spin strucure on S^2 pulls back via the Gauss map to a spin structure S on M, and gives rise to a pair of smooth sections (s_1,s_2) of S. Conversely, any pair of sections of S generates a (possibly periodic) conformal immersion of M under a suitable integrability condition, which for a minimal surface is simply that the spinor sections are meromorphic. A spin structure S also determines (and is determined by) the regular homotopy class of the immersion by way of a \bbZ_2-quadratic form q_S. We present an analytic expression for the Arf invariant of q_S, which decides whether or not the correponding immersion can be deformed to an embedding. The Arf invariant also turns out to be an obstruction, for example, to the existence of certain complete minimal immersions. The later parts of this paper use the spinor representation to investigate minimal surfaces with embedded planar ends. In general, we show for a spin structure S on a compact Riemann surface M with punctures at P that the space of all such (possibly periodic) minimal immersions of M\setminus P into \bbR^3 (upto homothety) is the the product of S^1 imes H^3 with the Grassmanian of 2-planes in a complex vector space \calK of meromorphic sections of S. An important tool -- a skew-symmetric form Ωdefined by residues of a certain meromorphic quadratic differential on M -- lets us compute how \calK varies as M and P are varied. Then we apply this to determine the moduli spaces of planar-ended minimal spheres and real projective planes, and also to construct a new family of minimal tori and a minimal Klein bottle with 4 ends. These surfaces compactify in S^3 to yield surfaces critical for the \Moebius invariant squared mean curvature functional W. On the other hand, Robert Bryant has shown all W-critical spheres and real projective planes arise this way. Thus we find at the same time the moduli spaces of W-critical spheres and real projective planes via the spinor representation.

Motivation & Objective

• To develop a spinor representation for conformal immersions of Riemann surfaces into R³, extending Dennis Sullivan’s approach.
• To establish a direct correspondence between spin structures on a surface and the regular homotopy class of its conformal immersion into R³.
• To characterize when a minimal immersion is regularly homotopic to an embedding using the Arf invariant of a Z₂-quadratic form associated with the spin structure.
• To classify W-critical spheres and real projective planes via the spinor representation, showing they arise from embedded planar ends and specific spinor data.
• To construct new examples of minimal surfaces, including tori and Klein bottles with embedded planar ends, using residue-based bilinear forms on meromorphic spinor sections.

Proposed method

• Represent a conformal immersion via a pair of meromorphic sections (s₁, s₂) of a spin structure S on a Riemann surface M, with the immersion given by Re∫(s₁²−s₂², i(s₁²+s₂²), 2s₁s₂).
• Define a Z₂-quadratic form q_S associated with each spin structure S, which classifies the regular homotopy class of the immersion.
• Compute the Arf invariant of q_S analytically using a residue-based formula involving a meromorphic quadratic differential on M.
• Use a skew-symmetric bilinear form Ω defined via residues of a meromorphic quadratic differential to analyze the variation of the space K of meromorphic spinor sections as M and punctures P vary.
• Apply the spinor representation to minimal surfaces with embedded planar ends by analyzing the moduli space as a product of S¹×H³ and a Grassmannian of 2-planes in K.
• Derive period conditions on the torus double cover of a Klein bottle to ensure closedness of the immersion, reducing to integrals of s₁² and s₁s₂ over a real cycle.

Experimental results

Research questions

• RQ1 How can the spinor representation be used to parametrize conformal immersions of Riemann surfaces into R³?
• RQ2 What is the precise relationship between spin structures on a surface and the regular homotopy class of its immersion into R³?
• RQ3 How does the Arf invariant of the associated Z₂-quadratic form determine whether a minimal immersion is regularly homotopic to an embedding?
• RQ4 Can the spinor representation classify all W-critical spheres and real projective planes in S³?
• RQ5 What are the moduli spaces of minimal surfaces with embedded planar ends, and how do they vary with the underlying Riemann surface and punctures?

Key findings

• The Arf invariant of the Z₂-quadratic form q_S associated with a spin structure S determines whether the corresponding minimal immersion is regularly homotopic to an embedding.
• For minimal surfaces with embedded planar ends, the moduli space of immersions (up to homothety) is isomorphic to S¹×H³×Gr(2, K), where K is the space of meromorphic spinor sections.
• The bilinear form Ω, defined via residues of a meromorphic quadratic differential, annihilates K and governs how K varies with changes in the Riemann surface and punctures.
• All W-critical spheres and real projective planes in S³ arise from the spinor representation of minimal immersions with embedded planar ends, and their moduli spaces are completely classified by this method.
• A new family of minimal tori with four embedded planar ends and a minimal Klein bottle with four ends are constructed, with the latter compactifying in S³ to yield W-critical surfaces.
• For a Klein bottle with finite total curvature, the double cover is a rectangular torus C/Λ with Λ generated by 2ω₁∈R and 2ω₃∈iR, and the deck transformation is I(u)=ū+ω₁, with admissible spin structures determined by ℘(u)−℘(ω₂) and ℘(u)−℘(ω₃).