The space of Gauss maps of complete minimal surfaces

Abstract

The Gauss map of a conformal minimal immersion of an open Riemann surface $M$ into ${\mathbb{R}}^3$ is a meromorphic function on $M$. In this paper, we prove that the Gauss map assignment, taking a full conformal minimal immersion $M\to {\mathbb{R}}^3$ to its Gauss map, is a Serre fibration. We then determine the homotopy type of the space of meromorphic functions on $M$ that are the Gauss map of a complete full conformal minimal immersion, and show that it is the same as the homotopy type of the space of all continuous maps from $M$ to the 2-sphere. We obtain analogous results for the generalised Gauss map of conformal minimal immersions $M\to {\mathbb{R}}^n$ for arbitrary $n\ge 3$.