Calculus of Variations and Geometric Measure Theory
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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 on 11 Sep 2019


Accepted Paper

Inserted: 19 dec 2018
Last Updated: 11 sep 2019

Journal: Geom. Funct. Anal.
Pages: 45
Year: 2018


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.


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