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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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