UC San Diego

In this dissertation, we study singular isoperimetric hypersurfaces and singular constant mean curvature hypersurfaces in closed Riemannian manifolds. It is well known that isoperimetric regions in a smooth, compact (n + 1)-manifold have smooth boundaries, except possibly on a closed set of codimension at most 8. For n ≥ 7, we construct an (n + 1)-dimensional compact, smooth manifold whose unique isoperimetric region, containing half the volume of the manifold, exhibits isolated singularities. These are the first known examples of singular isoperimetric regions.We then explore the twisted Jacobi field of singular constant mean curvature hypersurfaces under certain regularity assumptions. This exploration provides a direction for studying the generic regularity of isoperimetric and constant mean curvature hypersurfaces in eight dimensions.