A Bernstein type theorem for entire Willmore graphs
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.''
scientific article; zbMATH DE number 6136851
scientific article