Approximation by smooth embedded hypersurfaces with positive mean curvature

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Abstract

Here we initiate the study of the following problem. Let Ω be a compact domain in a Riemannian manifold such that ∂Ω is of minimum area for the contained volume. Can ∂Ω be approximated by smooth hypersurfaces of positive mean curvature? It reduces to the question of whether or not a stable (or minimizing) hypercone in a Euclidian space can be approximated by smooth hypersurfaces of positive mean curvature. The positive solution to the problem may be useful for studying the curvature and topology of Ω. We show in this paper that such approximation is possible provided that the given minimal cone satisfies some additional hypothesis.

Original languageEnglish (US)
Pages (from-to)197-208
Number of pages12
JournalBulletin of the Australian Mathematical Society
Volume36
Issue number2
DOIs
StatePublished - Oct 1987

ASJC Scopus subject areas

  • Mathematics(all)

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