A Bernstein type theorem for entire Willmore graphs
type: Item
instance of: scholarly article
title: A Bernstein type theorem for entire Willmore graphs
zbMATH Open document ID: 1269.53010
author: Tobias Lamm
author: Jingyi Chen
published in: The Journal of Geometric Analysis
publication date: 2013-02-18T00:00:00Z
review text: Let \(\Sigma\) be a closed two-dimensional surface immersed by \(f:\Sigma \rightarrow \mathbb{R}^3\) in the space. The Willmore functional is \[ W(f)=\frac{1}{4}\int_{\Sigma}{H^2\,d{\mu}_g}, \] where \(g\) in the induced metric, \(H\) the mean curvature of \(\Sigma\) and \(d{\mu}_g\) is the area element. Critical points of \(W\) are called Willmore surfaces and they are solutions of the Willmore equation: \[ {\Delta}_g H+\frac{1}{2} H^3-2HK=0, \] where \({\Delta}_g \) is the Laplace-Beltrami operator and \(K\) the Gauss curvature of \(\Sigma.\) In this paper, the authors are interested in entire Willmore graphs, i.e. surfaces defined by \( \Sigma=\{(x,u(x)) / x \in \mathbb{R}^2 \}, \) with \(u:\mathbb{R}^2 \rightarrow \mathbb{R}\), and they show the following Bernstein-type theorem: ``Every two-dimensional entire graphical solution of the Willmore equation with square integrable second fundamental form is a plane.''
Mathematics Subject Classification ID: 53A05
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 35J30
zbMATH DE Number: 6136851
zbMATH Keywords: Willmore surfaces
zbMATH Keywords: Bernstein-type result
MaRDI profile type: Publication
full work available at URL: s12220-011-9264-2
OpenAlex ID: W1965720776
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DOI: 10.1007/S12220-011-9264-2
description: scientific article; zbMATH DE number 6136851