Geroch monotonicity and the construction of weak solutions of the inverse mean curvature flow

For surfaces evolving under the inverse mean curvature flow, Geroch observed that the Hawking mass is a Lyapunov function. For weak solutions of the flow, the corresponding monotonicity formula was proved by Huisken and Ilmanen. An analogous formula exists for approximate equations as well, and it provides uniform control of the solutions in certain Sobolev spaces. This helps to construct weak solutions under very weak assumptions on the initial data.