Higher genus minimal surfaces in $S^3$ and stable bundles

Abstract

We consider compact minimal surfaces $f\colon M\to S^3$ of genus 2 which are homotopic to an embedding. We assume that the associated holomorphic bundle is stable. We prove that these surfaces can be constructed from a globally defined family of meromorphic connections by the DPW method. The poles of the meromorphic connections are at the Weierstrass points of the Riemann surface of order at most 2. For the existence proof of the DPW potential we give a characterization of stable extensions $0\to S^{-1}\to V\to S\to 0$ of spin bundles $S$ by its dual $S^{-1}$ in terms of an associated element of $P H^0(M;K^2).$ We also consider the family of holomorphic structures associated to a minimal surface in $S^3.$ For surfaces of genus $g\geq2$ the holonomy of the connections is generically non-abelian and therefore the holomorphic structures are generically stable.