University of Toronto

Department of Mathematics email:

**Abstract**: If one searches for k-dimensional submanifolds with
critical k-dimensional volume in a Riemannian manifold, then one is led towards
non-linear 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 non-linear elliptic PDE using singular perturbation theory in its
geometric formulation known as "gluing". I will then highlight some of the
analytic and geometric similarities between these two contexts.