On the Gauss map assignment for minimal surfaces and the Osserman curvature estimate

Abstract

The Gauss map of a conformal minimal immersion of an open Riemann surface $M$ into $\mathbb{R}^n$, $n\ge 3$, is a holomorphic map $M\to{\bf Q}^{n-2}\subset \mathbb{CP}^{n-1}$. Denote by ${\rm CMI}_{\rm full}(M,\mathbb{R}^n)$ and $\mathscr{O}_{\rm full}(M,{\bf Q}^{n-2})$ the spaces of full conformal minimal immersions $M\to\mathbb{R}^n$ and full holomorphic maps $M\to{\bf Q}^{n-2}$, respectively, endowed with the compact-open topology. In this paper we show that the Gauss map assignment $\mathscr{G}:{\rm CMI}_{\rm full}(M,\mathbb{R}^n)\to \mathscr{O}_{\rm full}(M,{\bf Q}^{n-2})$, taking a full conformal minimal immersion to its Gauss map, is an open map. This implies, in view of a result of Forstneric and the authors, that $\mathscr{G}$ is a quotient map. The same results hold for the map $(\mathscr{G},Flux):{\rm CMI}_{\rm full}(M,\mathbb{R}^n)\to \mathscr{O}_{\rm full}(M,{\bf Q}^{n-2})\times H^1(M,\mathbb{R}^n)$, where $Flux:{\rm CMI}_{\rm full}(M,\mathbb{R}^n)\to H^1(M,\mathbb{R}^n)$ is the flux assignment. As application, we establish that the set of maps $G\in \mathscr{O}_{\rm full}(M,{\bf Q}^{n-2})$ such that the family $\mathscr{G}^{-1}(G)$ of all minimal surfaces in $\mathbb{R}^n$ with the Gauss map $G$ satisfies the classical Osserman curvature estimate, is meagre in the space of holomorphic maps $M\to {\bf Q}^{n-2}$.