Constant mean curvature surfaces with planar boundary
In this paper, constant mean curvature surfaces having a convex closed planar curve \(\Gamma\) as boundary in the Euclidean 3-space are studied. It is proved the following: Given \(\Gamma\) as above, there is a constant \(V_\Gamma\) depending only on \(\Gamma\), such that any constant mean curvature compact surface with boundary \(\Gamma\) and volume less than or equal to \(V_\Gamma\) must be a graph. It is computed that \(V_\Gamma =2 \pi/3\) in the case when the surface is a disc and \(\Gamma\) is a circle. Besides, it is given an existence theorem for graphs, namely: If \(L\) is the length of \(\Gamma\) and \(H\) is a nonnegative real number such that \(LH< \sqrt {3\pi}\), then there exists a graph with constant mean curvature \(H\) and boundary \(\Gamma\). Moreover, it is given an optimal bound to ensure that a constant mean curvature compact embedded surface with convex planar boundary is above the boundary plane. Let \(\Sigma\) be a surface as above, \(A\) and \(H\) its area and mean curvature, respectively. If \(AH^2\leq 2\pi\), then \(\Sigma\) lies above the boundary plane. Moreover, if \(AH^2 \leq\pi\), then \(\Sigma\) is a graph.
scientific article; zbMATH DE number 987000
scientific article