The space of Gauss maps of complete minimal surfaces

TL;DR

This paper investigates the topological structure of the space of Gauss maps of complete minimal surfaces, showing it is homotopy equivalent to the space of all continuous maps from the surface to the sphere, with results extended to higher dimensions.

Contribution

It proves that the Gauss map assignment is a Serre fibration and determines the homotopy type of the space of such maps, extending to higher-dimensional minimal immersions.

Findings

Gauss map assignment is a Serre fibration.

Homotopy type of the space of Gauss maps matches that of all continuous sphere maps.

Results extend to minimal immersions into higher-dimensional Euclidean spaces.

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 $n\geq 3$.