Subgraphs of BV functions on RCD spaces

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 \({\textsf{X}}\times \mathbb R\), where \({\textsf{X}}\) is an \({\textrm{RCD}}(K,N)\) metric measure space. In particular, we give the precise expression of the push-forward onto \({\textsf{X}}\) of the perimeter measure of the subgraph in \({\textsf{X}}\times \mathbb R\) of a \({\textrm{BV}}\) function on \({\textsf{X}}\). Moreover, in properly chosen good coordinates, we write the precise expression of the normal to the boundary of the subgraph of a \({\textrm{BV}}\) function f with respect to the polar vector of f, and we prove change-of-variable formulas.