The Weak Inverse Mean Curvature Flow: Existence Theories and Applications

Abstract

This thesis is devoted to the study of the weak inverse mean curvature flow. We first extend its existence theory by showing that:1. if the ambient manifold satisfies a certain isoperimetric inequality, then for any bounded initial data, there exists a unique proper solution; 2. if the manifold satisfies an Euclidean isoperimetric inequality, then the proper solution has the optimal growth of $(n-1)\log r-C$ at infinity; 3. any initial value problem has a unique innermost solution; 4. in a smooth bounded domain, the initial value problem has a unique solution that satisfies an outer obstacle boundary condition (developing such a boundary condition is a part of this work). 5. the solution in item 4 can be approximated using $p$-harmonic functions with Dirichlet boundary conditions. We further provide two applications of our theory, by showing: 6. a topological gap property for the $\pi_2$\,--\,systole of closed 3-manifolds with positive scalar curvature; 7. that a contractible 3-manifold with positive scalar curvature and bounded geometry must be diffeomorphic to $\mathbb R^3$.