Mean curvature flow self-shrinkers with genus and asymptotically conical ends

Abstract

This doctoral dissertation is on the theory of Minimal Surfaces and of singularities in Mean Curvature Flow, for smooth submanifolds Y" in an ambient Riemannian (n+ 1)-manifold Nn+1, including: (1) New asymptotically conical self-shrinkers with a symmetry, in R"+1. (1') Classification of complete embedded self-shrinkers with a symmetry, in IR"+1, and of asymptotically conical ends with a symmetry. (2) Construction of complete, embedded self-shrinkers E2 C R3 of genus g, with asymptotically conical infinite ends, via minimal surface gluing. (3) Construction of closed embedded self-shrinkers y2 C R3 with genus g, via minimal surface gluing. In the work there are two central geometric and analytic themes that cut across (1)-(3): The notion of asymptotically conical infinite ends in (1)-(1') and (2), and in (2) and (3) the gluing methods for minimal surfaces which were developed by Nikolaos Kapouleas. For the completion of (2) it was necessary to initiate the development of a stability theory in a setting with unbounded geometry, the manifolds in question having essentially singular (worse than cusp-like) infinities. This was via a Schauder theory in weighted Hölder spaces for the stability operator, which is a Schrodinger operator of Ornstein-Uhlenbeck type, on the self-shrinkers viewed as minimal surfaces. This material is, for the special case of graphs over the plane, included as part of the thesis. The results in (1)-(1') are published as the joint work [KMø 1] with Stephen Kleene, and the result in (2) was proven in collaboration with Kleene-Kapouleas, and appeared in [KKMø 0]. The results in (3) are contained in the preprint [Mø1].