UC Irvine

This dissertation concerns the mean curvature flow, a geometric evolution equation for submanifolds, with an emphasis on singularity models and weak solutions. One highlight is we extend the mean curvature flow with surgery from the two-convexity assumption to a low entropy setting, entropy in the sense of Colding-Minicozzi, only assuming mean convexity. This allows us for instance to show that low entropy self shrinkers must be isotopic to round spheres with a weaker upper bound on the entropy than was previously known. Other applications of the surgery flow are given as well such as a extrinsic finiteness theorem, for 2-convex hypersurfaces and mean convex hypersurfaces of low entropy and an application to the regularity theory of the level set flow in some special cases.

Additionally constructions to give pathological examples of flows are also given. These include examples of ancient solutions to the mean curvature flow from minimal surfaces which are not solitons as well as constructions which show that the set of hypersurfaces which shrink to rounds points is quite complicated. There are also some structural results for singularity models, such as a theorem concerning the topology of self shrinkers and rotationaly symmetric ancient mean convex mean curvature flows, as well as an extension of the generic mean curvature flow of Colding and Minicozzi to flows in curved ambient spaces.