Geometric Analysis Seminar
An interesting problem in differential geometry is to try to understand how many constant mean curvature surfaces (CMCs) there are in a given manifold. Zhou and Zhu developed a min-max theory for constructing CMCs, and used it to show that any manifold M of dimension between 3 and 7 contains a smooth, almost-embedded CMC hypersurface of mean curvature h for every h > 0. In this talk, I will explain how this min-max theory can be used to construct CMC doublings of certain minimal surfaces in 3-manifolds. Such CMC doublings were previously constructed for minimal hypersurfaces in M^n with n > 3 by Pacard and Sun using gluing methods.