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

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.