Normal family theory and Gauss curvature estimate of minimal surfaces in $\mathbb{R}^m$

Abstract

In this paper, we first extend Zalcman’s principle of normality to the families of holomorphic mappings from Riemann surfaces to a compact Hermitian manifold. We then use this principle to derive an estimate for Gauss curvatures of the minimal surfaces in $\mathbb{R}^m$ whose Gauss maps satisfy some property $\mathcal{P}$, in the spirit of Bloch’s heuristic principle in complex analysis. Consequently, we recover and simplify the known results about value distribution properties of the Gauss map of minimal surfaces in $\mathbb{R}^m$.