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

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...

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Published inGeometric and functional analysis Vol. 29; no. 4; pp. 949 - 1001
Main Authors Ambrosio, Luigi, Brué, Elia, Semola, Daniele
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.08.2019
Springer Nature B.V
Subjects
Analysis
Asymptotic properties
Euclidean geometry
Half spaces
Mathematics
Mathematics and Statistics
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Summary: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.
Bibliography:ObjectType-Article-1
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ISSN:1016-443X
1420-8970
DOI:10.1007/s00039-019-00504-5