BV functions and sets of finite perimeter in sub-Riemannian manifolds

TL;DR

This paper develops a notion of BV functions and finite perimeter sets in sub-Riemannian manifolds, generalizing existing concepts and proving a blowup theorem under certain conditions.

Contribution

It introduces a BV function framework on manifolds with a volume form and quadratic forms, extending to sub-Riemannian spaces and proving a new blowup theorem.

Findings

Definition of BV functions on sub-Riemannian manifolds

Equivalence with metric measure space BV in certain cases

Proven blowup theorem for finite perimeter sets

Abstract

We give a notion of BV function on an oriented manifold where a volume form and a family of lower semicontinuous quadratic forms $G_p: T_pM \to [0,\infty]$ are given. When we consider sub-Riemannian manifolds, our definition coincide with the one given in the more general context of metric measure spaces which are doubling and support a Poincar\'e inequality. We then focus on finite perimeter sets, i.e., sets whose characteristic function is BV, in sub-Riemannian manifolds. Under an assumption on the nilpotent approximation, we prove a blowup theorem, generalizing the one obtained for step-2 Carnot groups in [24].