Geometric Analysis and Topology Seminar
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.
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.