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 \(M\rightarrow \textbf{Q}^{n-2}\subset \mathbb {C}\mathbb {P}^{n-1}\). Denote by \(\textrm{CMI}_{\textrm{full}}(M,\mathbb {R}^n)\) and \(\mathscr {O}_{\textrm{full}}(M,\textbf{Q}^{n-2})\) the spaces of full conformal minimal immersions \(M\rightarrow \mathbb {R}^n\) and full holomorphic maps \(M\rightarrow \textbf{Q}^{n-2}\), respectively, endowed with the compact-open topology. In this paper we show that the Gauss map assignment \(\mathscr {G}:\textrm{CMI}_{\textrm{full}}(M,\mathbb {R}^n)\rightarrow \mathscr {O}_{\textrm{full}}(M,\textbf{Q}^{n-2})\), taking a full conformal minimal immersion to its Gauss map, is an open map. This implies, in view of a result of Forstnerič and the authors, that \(\mathscr {G}\) is a quotient map. The same results hold for the map \((\mathscr {G},\textrm{Flux}):\textrm{CMI}_{\textrm{full}}(M,\mathbb {R}^n)\rightarrow \mathscr {O}_{\textrm{full}}(M,\textbf{Q}^{n-2})\times H^1(M,\mathbb {R}^n)\), where \(\textrm{Flux}:\textrm{CMI}_{\textrm{full}}(M,\mathbb {R}^n)\rightarrow H^1(M,\mathbb {R}^n)\) is the flux assignment. As application, we establish that the set of maps \(G\in \mathscr {O}_{\textrm{full}}(M,\textbf{Q}^{n-2})\) 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 \(M\rightarrow \textbf{Q}^{n-2}\).