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

Abstract

The goal of this paper is to establish a monotonicity formula for perimeter minimizing sets in $\mathrm{RCD}(0,N)$ 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 $\mathrm{RCD}$ spaces and the existence of blow-down cones for global perimeter minimizers in Riemannian manifolds with non-negative Ricci curvature and Euclidean volume growth.