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 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 → 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 → R n for arbitrary n ≥ 3.