The Spinor Representation of Surfaces in Space

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\times 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.