Rectifiability of the reduced boundary for sets of finite perimeter over RCD$(K,N)$ spaces

Abstract

This note is devoted to the study of sets of finite perimeter over RCD$(K,N)$ metric measure spaces. Its aim is to complete the picture about the generalization of De Giorgi's theorem within this framework. Starting from the results of [2] we obtain uniqueness of tangents and rectifiability for the reduced boundary of sets of finite perimeter. As an intermediate tool, of independent interest, we develop a Gauss-Green integration by parts formula tailored to this setting. These results are new and non-trivial even in the setting of Ricci limits.

References in corpus (10)

• Metric measure spaces with Riemannian Ricci curvature bounded from below
• Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds
• Riemannian Ricci curvature lower bounds in metric measure spaces with $\sigma$-finite measure
• Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in RCD(K,\infty) metric measure spaces
• Alexandrov meets Lott--Villani--Sturm
• On the topology and the boundary of N-dimensional RCD(K,N) spaces
• Independence on $p$ of weak upper gradients on RCD spaces
• Rectifiability of Singular Sets in Noncollapsed Spaces with Ricci Curvature bounded below
• Metric-measure boundary and geodesic flow on Alexandrov spaces
• Sobolev and BV spaces on metric measure spaces via derivations and integration by parts

Cited by in corpus (2)

• Isoperimetric sets in spaces with lower bounds on the Ricci curvature
• Monotonicity formulas for harmonic functions in $\mathrm{RCD}(0,N)$ spaces