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Constant scalar curvature hypersurfaces with spherical boundary in Euclidean space. (English) Zbl 1038.53060
The authors prove the following Theorem 3. Let \(\Sigma\) be a strictly convex compact \((n-1)\)-dimensional submanifold contained in a hyperplane \(\Pi\subset{\mathbb R}^{n+1}\), and let \(\psi:M^n\to{\mathbb R}^{n+1}\) be an embedded compact hypersurface with boundary \(\Sigma\). Let us assume that, for a given \(2\leq r\leq n\), the \(r\)-mean curvature \(H_r\) of \(M\) is a non-zero constant. Then \(M\) is contained in one of the half-spaces of \({\mathbb R}^{n+1}\) determined by \(\Pi\) and \(M\) has all the symmetries of \(\Sigma\).
As a consequence, when a hypersurface has spherical boundary \(\Sigma\) and constant \(r\)th mean curvature \(H_r\) (\(r\geq2\)), then it is a hyperplanar round ball or a spherical cap (Cor. 4) and, in particular, this is the case when the scalar curvature, \(r=n\), is constant (Theorem 1).

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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