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 in | Geometric and functional analysis Vol. 29; no. 4; pp. 949 - 1001 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Cham
Springer International Publishing
01.08.2019
Springer Nature B.V |
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| Online Access | Get full text |
| ISSN | 1016-443X 1420-8970 |
| DOI | 10.1007/s00039-019-00504-5 |
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| Abstract | 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. |
|---|---|
| AbstractList | 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. 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. |
| Author | Brué, Elia Semola, Daniele Ambrosio, Luigi |
| Author_xml | – sequence: 1 givenname: Luigi surname: Ambrosio fullname: Ambrosio, Luigi email: luigi.ambrosio@sns.it organization: Scuola Normale Superiore – sequence: 2 givenname: Elia surname: Brué fullname: Brué, Elia organization: Scuola Normale Superiore – sequence: 3 givenname: Daniele surname: Semola fullname: Semola, Daniele organization: Scuola Normale Superiore |
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Constancy of the dimension for RCD(K,N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin N Gigli (504_CR31) 2018; 251 R Jiang (504_CR37) 2016; 44 504_CR18 504_CR16 504_CR14 L Ambrosio (504_CR6) 2014; 266 E Le Donne (504_CR39) 2011; 36 504_CR11 K-T Sturm (504_CR49) 2006; 196 L Ambrosio (504_CR3) 2002; 10 X Tolsa (504_CR51) 2015; 54 L Ambrosio (504_CR15) 2014; 7 G De Philippis (504_CR22) 2018; 5 A Petrunin (504_CR46) 2011; 4 A Nakayasu (504_CR45) 2014; 24 J Lott (504_CR40) 2009; 169 H-W Fleming (504_CR27) 1966; 121 504_CR28 504_CR25 504_CR23 H Federer (504_CR26) 1960; 72 D Preiss (504_CR47) 1987; 125 E De Giorgi (504_CR19) 1954; 36 N Gigli (504_CR34) 2015; 111 N Gigli (504_CR29) 2014; 2 M Kell (504_CR38) 2018; 18 G De Philippis (504_CR21) 2016; 26 E De Giorgi (504_CR20) 1955; 4 504_CR4 504_CR1 504_CR36 504_CR35 X Menguy (504_CR42) 2001; 11 L Ambrosio (504_CR2) 2001; 159 L Ambrosio (504_CR7) 2015; 367 B White (504_CR56) 1989; 64 N Gigli (504_CR33) 2015; 705 L Ambrosio (504_CR9) 2014; 163 N Weaver (504_CR55) 2000; 178 L Ambrosio (504_CR12) 2018; 53 J Cheeger (504_CR17) 1999; 9 M-K Von Renesse (504_CR54) 2008; 259 N Gigli (504_CR32) 2016; 271 504_CR44 M Erbar (504_CR24) 2015; 201 L Ambrosio (504_CR10) 2015; 43 K-T Sturm (504_CR50) 2006; 196 C Villani (504_CR53) 2009 L Ambrosio (504_CR5) 2014; 256 L Ambrosio (504_CR13) 2009; 19 504_CR8 M Miranda Jr (504_CR43) 2003; 82 K.-T Sturm (504_CR48) 1996; 75 N Gigli (504_CR30) 2015; 236 J Azzam (504_CR52) 2015; 25 V Magnani (504_CR41) 2015; 145 |
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RCD
(
K
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N
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metric measure spaces. Our main result asserts... 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... |
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| Title | Rigidity of the 1-Bakry–Émery Inequality and Sets of Finite Perimeter in RCD Spaces |
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