The Gauss map and the h-principle

SEMINARIO DE GEOMETRÍA

Conferenciante: Finnur Lárusson (University of Adelaida)

Abstract: The Gauss map of a minimal surface in \(\mathbb R^3\), parametrised as a conformal minimal immersion from an open Riemann surface \(M\) into \(\mathbb R^3\), is a meromorphic function on \(M\). Although the Gauss map has been a central object of interest in the theory of minimal surfaces since the mid-19th century, it was only recently proved by Alarcón, Forstnerič and López, using new complex-analytic methods, that every meromorphic function on \(M\) is a Gauss map. It remains an open problem to usefully characterise those meromorphic functions that are the Gauss map of a complete minimal surface. I will describe recent joint work with Antonio Alarcón, in which we take a new approach to this problem. We investigate the space of meromorphic functions on \(M\) that are the Gauss map of a complete minimal surface from a homotopy-theoretic viewpoint, using a new h-principle as a key tool. My talk will include a brief general introduction to h-principles and their applications.