A Bernstein type theorem for entire Willmore graphs. (English) Zbl 1269.53010
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.”
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.”
Reviewer: Esther Sanabria (Valencia)
MSC:
| 53A05 | Surfaces in Euclidean and related spaces |
| 53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
| 35J30 | Higher-order elliptic equations |
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