Normal family theory and Gauss curvature estimate of minimal surfaces in \(\mathbb{R}^m\)
type: Item
instance of: scholarly article
title: Normal family theory and Gauss curvature estimate of minimal surfaces in \(\mathbb{R}^m\)
zbMATH Open document ID: 1351.53072
author: Xuecheng Pang
author: Xiaojun Liu
published in: Journal of Differential Geometry
publication date: 2016-07-14T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.jdg/1463404120
review text: A fascinating result in complex function theory is Bloch's heuristic principle. According to this principle, a family of meromorphic functions which have a property \(\mathcal{P}\) in common on a domain \(U\subset\mathbb{C}\) should be normal in \(U\) if the property \(\mathcal{P}\) forces a meromorphic function defined on the whole complex plane \(\mathbb{C}\) to be constant. In the theory of minimal surfaces, a similar phenomenon holds. Every time while there was a Bernstein-type result concerning complete minimal surfaces satisfying a certain condition, there was a corresponding Gauss curvature estimate for the minimal surfaces with boundary that satisfy the same condition, i.e., \[ |K(p)|d(p)^2\leq C, \] where \(C\) is a positive constant, \(K\) the Gauss curvature and \(d(p)\) the geodesic distance from the point \(p\) to the boundary. Inspired by Bloch's principle, the authors derive various interesting results for minimal surfaces in \(\mathbb{R}^m\) whose Gauss maps satisfy some property \(\mathcal{P}\). Before stating the main result let us introduce some notation and terminology. Let \(\Sigma\) be a connected Riemann surface, \(\mathbb{P}^{m-1}\) be the complex projective space of dimension \(m-1\), \(D\) the open unit disc in \(\mathbb{C}\), and \(\mathcal{H}(\Sigma)\) the set of holomorphic maps from \(\Sigma\) to \(\mathbb{P}^{m-1}\). If \(\mathcal{P}\) is a property for holomorphic maps, we denote by \(\mathcal{P}(\Sigma)\) the set given by \[ \mathcal{P}(\Sigma) = \{g \in \mathcal{H}(\Sigma): g\,\, \text{satisfies the property}\,\, \mathcal{P}\}. \] We say that \(\mathcal{P}\) is a compact property if the following conditions are fulfilled: i) For any pair of Riemann surfaces \(\Sigma_1\), \(\Sigma_2\) and for any holomorphic map \(\phi : \Sigma_1 \to \Sigma_2\) without branch points, one has \(g\in\mathcal{P}(\Sigma_2)\Rightarrow g\circ\phi\in\mathcal{P}(\Sigma_1)\). ii) Let \(\Sigma\) be any Riemann surface and \(g \in \mathcal{H}(\Sigma)\). For any…
reviewed by: Andreas Savas-Halilaj
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53A10
zbMATH DE Number: 6603545
zbMATH Keywords: minimal surfaces
zbMATH Keywords: Bloch's heuristic principle
zbMATH Keywords: Zalcman's principle
zbMATH Keywords: normal family
zbMATH Keywords: Gauss curvature estimate
MaRDI profile type: Publication
OpenAlex ID: W2406230211
Wikidata QID: Q115166789
DOI: 10.4310/JDG/1463404120
description: scientific article; zbMATH DE number 6603545