University of Georgia
Department of Mathematics

Sepecial Geometry Seminar
Monday, May 7, 2007
3:30pm, Room 303 Boyd

Adrian Butscher
University of Toronto, Scarborough

New constructions of submanifolds of the sphere which are critical
points of the volume functional.


Abstract: If one searches for k-dimensional submanifolds with critical k- dimensional volume in a Riemannian manifold, then one is led towards elliptic partial differential equations involving the mean curvature vector of the submanifold.  I will present new constructions of volume-critical submanifolds of the sphere in two contexts: hypersurfaces with constant mean curvature in spheres of any dimension; and Legendrian submanifolds in spheres of odd dimension that are stationary under variations preserving the contact structure.  These are constructed by solving the associated elliptic PDE using singular perturbation theory.   I will then highlight some of the analytic and geometric similarities between these two contexts.