Geometry and analysis of Ricci curvature and mean curvature flows

Abstract

In this thesis, we study the geometry and analysis of spaces with Ricci curvature bounded below from the following three perspectives and the asymptotically conical singularities of mean curvature flows in the following two perspectives. For the spaces with Ricci curvature bounded below, firstly we study the unique continuation problem on RCD spaces, which is a long-standing open problem, with little known even in the setting of Alexandrov spaces. Together with Qin Deng, we proved that on RCD(K,2) spaces both harmonic functions and caloric functions satisfy weak unique continuation properties. Furthermore we constructed counter-examples showing that strong unique continuation in general fails for harmonic and caloric functions on RCD(K,N) spaces where N is greater or equal to 4. Secondly, we consider constructing a canonical diffeomorphism between the n-sphere and a n-dimensional space with Ricci curvature bounded from below by n-1 which is close to the n-sphere in the Gromov-Hausdorff sense. Together with Bing Wang we proved that the first (n+1)-eigenfunctions of Laplacian provides a bi-Holder diffeomorphism and we further give a counter-example showing that the bi-Holder estimate is sharp and cannot be improved to a bi-Lipschitz estimate. Thirdly, we study the Margulis Lemma on RCD spaces. Together with Qin Deng, Jaime Santos-Rodríguez and Sergio Zamora, we extend the Margulis Lemma for manifolds with lower Ricci curvature bounds to the RCD setting. As one of our main tools, we obtain improved regularity estimates for Regular Lagrangian flows on these spaces. For the asymptotically conical singularities of mean curvature flows, firstly together with Tang-Kai Lee, we proved asymptotically conical self-shrinkers as tangent flows of MCFs are unique, generalizing the result in the case of hypersurface proven by Chodosh-Schulze. Secondly, together with Tang-Kai Lee we prove that given any asymptotically conical shrinker, there exists an embedded closed hypersurface such that the mean curvature flow starting from it develops a type I singularity at time 1 at the origin modeled on the given shrinker.