Inserted: 26 jul 2021
Last Updated: 14 oct 2021
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 develop an intrinsic theory of Laplacian bounds in viscosity sense and in a (seemingly new) heat flow sense, showing their equivalence also with Laplacian bounds in distributional sense;
- relying on these new tools, 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 prove sharp Laplacian bounds on the distance function from a set (locally) minimizing the perimeter; this corresponds to vanishing mean curvature in the smooth setting and encodes also information about the second variation of the area;
- we initiate a regularity theory for boundaries of sets (locally) minimizing the perimeter, obtaining sharp dimension estimates for their singular sets, quantitative estimates of independent interest and topological regularity away from the singular set.
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.