Geometric Analysis and Topology Seminar

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

Speaker: Costante Bellettini, University College London

Location: Warren Weaver Hall 517

Date: Wednesday, March 28, 2018, 11 a.m.

Synopsis:

This talk describes a recent joint work of the speaker with N. 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. An "effective

version" of the compactness theorem has been (a posteriori) established in

collaboration with O. Chodosh and N. Wickramasekera.