The geometric inequalities (Penrose Inequalities) in general relativity which relate to the ADM mass, angular momentum and charge have been proven for a large class of the axially symmetric, asymptotically flat, maximal initial data of the Einstein-Maxwell equations. In this talk, we will introduce how to reduce the general formulation for the non-maximal initial data, to the known maximal case, whenever a system of elliptic equations admits a solution. Each equation in the system will be analyzed individually, and the solvability of the system in the near maximal case will be discussed. The talk is based on joint work with Marcus Khuri.

The evolution of a family of metrics on a closed surface can always be described as a combination of a conformal change, a change obtained by pulling back with suitable diffeomorphisms and an evolution in "horizontal direction". As we shall discuss in this talk, curves that move only in horizontal direction turn out to be very well controlled even when the underlying conformal structure degenerates. As an application we will show that solutions of Teichmüller harmonic map flow can be continued canonically past any finite time singularity and consequently that this flow admits global weak solution for arbitrary initial data and target manifolds. Joint work with Peter Topping.