# Stable constant-mean-curvature hypersurfaces: regularity and compactness.

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Costante Bellettini

created by malchiodi on 14 Nov 2017

24 nov 2017
-- 14:00
[open in google calendar]

Scuola Normale Superiore, Aula Tonelli

**Abstract.**

This talk describes a recent joint work of the speaker with Neshan Wickramasekera (Cambridge). The work develops a regularity theory, with an associated compactness theorem, for weakly defined hypersurfaces (codimension 1 integral varifolds) of a smooth Riemannian manifold that are stationary and stable on their regular parts for volume preserving ambient deformations. The main regularity theorem gives two structural conditions on such a hypersurface that imply that, away from a set of codimension 7 or higher, the hypersurface is locally either a single smoothly embedded disk or precisely two smoothly embedded disks intersecting tangentially. Easy examples show that neither structural hypothesis can be relaxed. An important special case is when the varifold corresponds to the boundary of a Caccioppoli set, in which case the structural conditions can be considerably weakened.