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.

I will report on new and recent work with Leandro Arosio (Universita di Roma, Tor Vergata) on chaos and other aspects of dynamics in certain highly symmetric complex manifolds. For example, we prove that for many linear algebraic groups G, chaotic automorphisms are generic among volume-preserving holomorphic automorphisms of G. I will start with some background on holomorphic dynamics in general, on chaos, and on the highly symmetric complex manifolds to which our results apply.