Constancy of the dimension in codimension one and locality of the unit normal on $\mathrm{RCD}(K,N)$ spaces

TL;DR

This paper investigates the geometric and measure-theoretic properties of sets with finite perimeter on $ ext{RCD}(K,N)$ spaces, establishing boundary measure concentration, normal vector localization, and Gauss-Green formulas, which are foundational for regularity theory.

Contribution

It proves boundary measure concentration on the regular set, provides representation formulas for perimeters of set unions and intersections, and refines Gauss-Green formulas for divergence measure fields on $ ext{RCD}(K,N)$ spaces.

Findings

Boundary measure concentrates on the $n$-regular set.

Representation formulas for perimeter of unions and intersections.

Refined Gauss-Green formulas for divergence measure fields.

Abstract

The aim of this paper is threefold. We first prove that, on $\mathrm{RCD}(K,N)$ spaces, the boundary measure of any set with finite perimeter is concentrated on the $n$-regular set $\mathcal{R}_n$, where $n\le N$ is the essential dimension of the space. After, we discuss localization properties of the unit normal providing representation formulae for the perimeter measure of intersections and unions of sets with finite perimeter. Finally, we study Gauss-Green formulae for essentially bounded divergence measure vector fields, sharpening the analysis in \cite{BuffaComiMiranda19}.\\ These tools are fundamental for the development of a regularity theory for local perimeter minimizers on $\mathrm{RCD}(K,N)$ spaces in \cite{MondinoSemola21}.