Calculus of Variations and Geometric Measure Theory

L. Ambrosio - E. Bruè - D. Semola

Rigidity of the 1-Bakry-Émery inequality and sets of finite perimeter in RCD spaces

created by bruè on 19 Dec 2018
modified by semola on 30 May 2020


Published Paper

Inserted: 19 dec 2018
Last Updated: 30 may 2020

Journal: Geom. Funct. Anal.
Volume: 29
Number: 4
Pages: 949--1001
Year: 2019
Doi: 10.1007/s00039-019-00504-5


This note is dedicated to the study of the asymptotic behaviour of sets of finite perimeter over 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 RCD(0,N) spaces for which there exists a nontrivial function satisfying the equality in the 1-Bakry-Émery inequality. This result is of independent interest and it is new, up to our knowledge, even in the smooth framework.