Journal of Differential Geometry

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 $\mathbb{R}^m$.

Article information

Source
J. Differential Geom., Volume 103, Number 2 (2016), 297-318.

Dates
First available in Project Euclid: 16 May 2016

https://projecteuclid.org/euclid.jdg/1463404120

Digital Object Identifier
doi:10.4310/jdg/1463404120

Mathematical Reviews number (MathSciNet)
MR3504951

Zentralblatt MATH identifier
1351.53072

Citation

Liu, Xiaojun; Pang, Xuecheng. Normal family theory and Gauss curvature estimate of minimal surfaces in $\mathbb{R}^m$. J. Differential Geom. 103 (2016), no. 2, 297--318. doi:10.4310/jdg/1463404120. https://projecteuclid.org/euclid.jdg/1463404120