The space of Gauss maps of complete minimal surfaces (Q6609492)

The Gauss map of a minimal surface in \(\mathbb{R}^{3}\), parametrised as a conformal minimal immersion from an open Riemann surface \(M\) into \(\mathbb{R}^{3}\), may be viewed as a meromorphic function on \(M\). It is a long-standing unsolved problem in the global theory of minimal surfaces to usefully characterise those meromorphic functions that are the Gauss map of a complete minimal surface. In this paper, the authors 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.