Appears in collection : Not Only Scalar Curvature Seminar

The (weak) inverse mean curvature flow is known as a useful tool for studying the geometry of scalar curvature. In recent work with Otis Chodosh and Yi Lai, we used this tool to prove that complete contractible PSC 3-manifolds with bounded geometry are diffeomorphic to $\mathbb{R}^3$. In this talk, we will investigate the analytic side of this theory. We will discuss the existence and properties of the desirable class of solutions -- which are usually called the innermost inverse mean curvature flow. When being run in the interior of a smooth domain, this class of solutions has the geometric meaning that all hypersurfaces stay tangential to the boundary. We will also discuss its relation to p-harmonic functions with Dirichlet boundary condition.