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Hypersurfaces with constant mean curvature and prescribed area. (English) Zbl 0945.49030
Fixing an \(n\)-dimensional rectifiable current \(T_0\) in \(\mathbb{R}^{n+1}\) having rectifiable boundary \(B=\partial T_0\), the author considers the family of rectifiable currents \(T\) with \(\partial T= B\). It is assumed that the support of \(B\) has \(n\)-dimensional Hausdorff measure \(0\). The volume enclosed by \(T-T_0\) is defined to be \(\langle Q,\Omega\rangle\), where \(Q\) is the unique \((n+1)\)-dimensional current with finite mass and boundary \(T-T_0\), and where \(\Omega\) is the volume form on \(\mathbb{R}^{n+1}\). By maximizing the volume enclosed by \(T-T_0\), subject to a constraint on the mass \(\mathbb{M}(T)\), the author proves the existence of surfaces with boundary \(B\) and having constant mean curvature off their possible singular sets of Hausdorff dimension \(n-7\). The bulk of the paper is then devoted to considering how the mean curvature of these maximizing surfaces depends on the mass constraint on \(T\). The main result is that, as the mass constraint approaches infinity, the mean curvature behaves as would be expected for a sphere.

MSC:
49Q20 Variational problems in a geometric measure-theoretic setting
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
58E12 Variational problems concerning minimal surfaces (problems in two independent variables)
53C65 Integral geometry
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