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Area minimizing hypersurfaces with prescribed volume and boundary. (English) Zbl 0787.53056
The authors treat “Rellich’s conjecture” in higher dimension which originally is the non-uniqueness of hypersurfaces of prescribed mean curvature with given boundary. This conjecture states that for a given \((n-1)\)-dimensional boundary \(B\) in \(\mathbb{R}^{n+1}\) there are numbers \(c_ - < 0 < c_ +\) such that for every \(H \in [c_ -,c_ +]\setminus \{0\}\) there exist at least two different \(n\)-dimensional surfaces with boundary \(B\) and constant mean curvature \(H\). One small and one large solution in analogy to spherical caps with the same boundary and the same \(H\). This problem has been solved by Struwe and Brezis/Coron for two- dimensional parametric surfaces in \(\mathbb{R}^ 3\). To treat the higher dimensional situation the authors work in the setting of geometric measure theory and study the corresponding volume constrained problem for locally rectifiable integer multiplicity \(n\)-currents in \(\mathbb{R}^{n+1}\).
Besides many results for existence and regularity and asymptotic behaviour for large volumes for minimizers, one of the main results is the following theorem: There exist infinitely many values \(H > 0\) and infinitely many values \(H < 0\) accumulating at 0 such that one has at least two indecomposable \(n\)-currents with constant mean curvature \(H\) and boundary \(B\) with certain regularity properties.

MSC:
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
49Q20 Variational problems in a geometric measure-theoretic setting
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