Rigidity of the 1-Bakry–Émery Inequality and Sets of Finite Perimeter in RCD Spaces

Abstract

This note is dedicated to the study of the asymptotic behaviour of sets of finite perimeter over $${{\,\mathrm{RCD}\,}}(K,N)$$ metric measure spaces. Our main result asserts existence of a Euclidean tangent half-space almost everywhere with respect to the perimeter measure and it can be improved to an existence and uniqueness statement when the ambient is non collapsed. As an intermediate tool, we provide a complete characterization of the class of $${{\,\mathrm{RCD}\,}}(0,N)$$ spaces for which there exists a nontrivial function satisfying the equality in the 1-Bakry–Emery inequality. This result is of independent interest and it is new, up to our knowledge, even in the smooth framework.