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Regularity of isoperimetric hypersurfaces in Riemannian manifolds. (English) Zbl 1063.49031
A hypersurface \(S\) in a Riemannian manifold \(M\) is called an isoperimetric hypersurface if \(S\) minimizes the area among all hypersurfaces in \(M\) enclosing the same volume. Technically these hypersurfaces are produced by minimizing the mass (area) within certain classes of integer multiplicity rectifiable currents. This immediately raises the question of regularity of isoperimetric hypersurfaces to which the paper gives an essential new contribution: if the dimension \(n\) of \(S\) is at most \(6\) and if \(M\) is a smooth manifold, then also \(S\) is a smooth hypersurface. If the metric of \(M\) is merely Lipschitz, then \(S\) is still Hölder differentiable. For general dimensions \(n\) it is shown that \(S\) is smooth except for a possible set of singular points whose Hausdorff-dimension is at most \(n- 7\).

MSC:
49Q20 Variational problems in a geometric measure-theoretic setting
49N60 Regularity of solutions in optimal control
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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