In this talk we consider a family of monotonicity formulas which originate from seemingly different contexts in geometric analysis. On the one side, we have monotonicity formulas for the Laplace-Beltrami operator, tracing back to the work of
Colding (“New Monotonicity Formulas for Ricci Curvature and Applications. I”, 2012) and Colding-Minicozzi (“On Uniqueness of Tangent Cones for Einstein Manifolds”, 2014). On the other side, we have the monotonicity of the Hawking mass
along the inverse mean curvature flow, proved by Huisken and Ilmanen (“The Inverse Mean Curvature Flow and the Riemannian Penrose Inequality”, 2001). The connection between these two formulas lies in monotonicity formulas of geometric quantities for the level sets of solutions to the p-Laplacian (formally the IMCF corresponds to the case p = 1). These general formulas require some regularity: we show that almost every level set is a curvature varifold for which a Gauss-Bonnet-type theorem is established. To prove the convergence to the inverse mean curvature flow we show that, as p → 1, almost every level sets converge in the sense of curvature varifolds and gradients strongly converge in L^q for every finite q.
Our monotonicy formulas imply several geometric inequalities, from the Willmore and the p-Minkowski inequalities to the Riemannian Penrose inequality.