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 $M\to {\mathbf{Q}}^{n-2}\subset {\mathbb{CP}}^{n-1}$. Denote by ${\mathrm{CMI}}_{\mathrm{full}}(M,{\mathbb{R}}^n)$ and ${\mathcal{O}}_{\mathrm{full}}(M,{\mathbf{Q}}^{n-2})$ the spaces of full conformal minimal immersions $M\to {\mathbb{R}}^n$ and full holomorphic maps $M\to {\mathbf{Q}}^{n-2}$, respectively, endowed with the compact-open topology. In this paper we show that the Gauss map assignment $\mathcal{G}:{\mathrm{CMI}}_{\mathrm{full}}(M,{\mathbb{R}}^n)\to {\mathcal{O}}_{\mathrm{full}}(M,{\mathbf{Q}}^{n-2})$, 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 $\mathcal{G}$ is a quotient map. The same results hold for the map $(\mathcal{G},Flux):{\mathrm{CMI}}_{\mathrm{full}}(M,{\mathbb{R}}^n)\to {\mathcal{O}}_{\mathrm{full}}(M,{\mathbf{Q}}^{n-2})\times H^1(M,{\mathbb{R}}^n)$, where $Flux:{\mathrm{CMI}}_{\mathrm{full}}(M,{\mathbb{R}}^n)\to H^1(M,{\mathbb{R}}^n)$ is the flux assignment. As application, we establish that the set of maps $G\in {\mathcal{O}}_{\mathrm{full}}(M,{\mathbf{Q}}^{n-2})$ such that the family ${\mathcal{G}}^{-1}(G)$ 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 $M\to {\mathbf{Q}}^{n-2}$.

arXiv commentTo appear in Ann. Mat. Pura Appl