Monotonicity formula and stratification of the singular set of perimeter minimizers in RCD spaces

TL;DR

This paper develops a monotonicity formula for perimeter minimizers in RCD spaces, leading to dimension estimates of singular sets and the existence of blow-down cones in nonnegatively curved Riemannian manifolds.

Contribution

It introduces a monotonicity formula for perimeter minimizers in RCD spaces and applies it to estimate singular set dimensions and establish blow-down cone existence.

Findings

Sharp Hausdorff dimension estimates for singular strata

Existence of blow-down cones in nonnegatively curved manifolds

Rigidity results for perimeter minimizing sets

Abstract

The goal of this paper is to establish a monotonicity formula for perimeter minimizing sets in RCD(0,N) metric measure cones, together with the associated rigidity statement. The applications include sharp Hausdorff dimension estimates for the singular strata of perimeter minimizing sets in non collapsed RCD spaces and the existence of blow-down cones for global perimeter minimizers in Riemannian manifolds with nonnegative Ricci curvature and Euclidean volume growth.