Subgraphs of $\rm BV$ functions on $\rm RCD$ spaces

TL;DR

This paper generalizes classical results about subgraphs of BV functions from Euclidean spaces to RCD spaces, providing explicit formulas for perimeter measures and boundary normals in this broader setting.

Contribution

It extends the theory of BV subgraphs to RCD spaces, including formulas for perimeter measures and boundary normals in this non-smooth setting.

Findings

Explicit expression of the perimeter measure of BV subgraphs on RCD spaces

Formulas for the normal vector to the boundary of BV subgraphs

Change-of-variable formulas in the RCD setting

Abstract

In this work we extend classical results for subgraphs of functions of bounded variation in $\mathbb{R}^n\times\mathbb{R}$ to the setting of $\mathsf{X}\times\mathbb{R}$, where $\mathsf{X}$ is an ${\rm RCD}(K,N)$ metric measure space. In particular, we give the precise expression of the push-forward onto $\mathsf{X}$ of the perimeter measure of the subgraph in $\mathsf{X}\times\mathbb{R}$ of a $\rm BV$ function on $\mathsf{X}$. Moreover, in properly chosen good coordinates, we write the precise expression of the normal to the boundary of the subgraph of a $\rm BV$ function $f$ with respect to the polar vector of $f$, and we prove change-of-variable formulas.