Weak Laplacian bounds and minimal boundaries in non-smooth spaces with Ricci curvature lower bounds

Andrea Mondino, Daniele Semola

Published 2021-07-26Version 1

The goal of the paper is four-fold. In the setting of non-smooth spaces with Ricci curvature lower bounds (more precisely $RCD(K,N)$ metric measure spaces): - We establish a new principle relating lower Ricci curvature bounds to the preservation of Laplacian lower bounds under the evolution via the $p$-Hopf-Lax semigroup, for general exponents $p\in[1,\infty)$. - We develop an instrinsic viscosity theory of Laplacian bounds. - We prove Laplacian bounds on the distance function from a set (locally) minimizing the perimeter; this corresponds to vanishing mean curvature in the smooth setting. - We initiate a regularity theory for boundaries of sets (locally) minimizing the perimeter. The class of $RCD(K,N)$ metric measure spaces includes as remarkable sub-classes: measured Gromov-Hausdorff limits of smooth manifolds with lower Ricci curvature bounds and finite dimensional Alexandrov spaces with lower sectional curvature bounds. Most of the results are new also in these frameworks.