Stratified spaces and synthetic Ricci curvature bounds

Bertrand, Jérôme; Ketterer, Christian; Mondello, Ilaria; Richard, Thomas

doi:10.5802/aif.3393

Stratified spaces and synthetic Ricci curvature bounds
[Espaces stratifiés et bornes de courbure de Ricci synthétiques]

Bertrand, Jérôme ¹ ; Ketterer, Christian ² ; Mondello, Ilaria ³ ; Richard, Thomas ³

¹ Université Paul Sabatier Institut de mathémathiques 118 Route de Narbonne 31062 Toulouse Cedex 9 (France)
² University of Toronto Dept. of mathematics 40 St George St Toronto, Ontario M5S 2E4 (Canada)
³ Université de Paris Est Créteil Laboratoire d’analyse et mathématiques appliquées 61 Avenue du Général de Gaulle 94010 Créteil Cedex (France)

Annales de l'Institut Fourier, Tome 71 (2021) no. 1, pp. 123-173.

Résumé
Abstract

Nous prouvons qu’un espace stratifié compact satisfait la condition de courbure-dimension riemannienne $RCD (K, N)$ si et seulement si son tenseur de Ricci est borné inférieurement par $K \in ℝ$ dans le lieu régulier, l’angle des cônes le long de la strate de codimension deux est inférieur ou égal à $2 π$ et sa dimension est au plus égale à $N$ . Ceci donne lieu à une large classe de nouveaux exemples d’espaces métriques mesurés satisfaisant la condition de courbure-dimension $RCD (K, N)$ , qui inclut notamment les suspensions sphériques, les orbifolds, les variétés de Kähler–Einstein avec un diviseur, les variétés d’Einstein avec des singularités le long d’une courbe. Nous obtenons aussi de nouveaux résultats analytiques et géométriques sur les espaces stratifiés, comme l’inégalité volumique de Bishop–Gromov, le théorème de comparaison pour le Laplacien de la distance, l’inégalité isopérimétrique de Lévy–Gromov. Notre résultat implique en outre une caractérisation similaire des espaces stratifiés compacts de courbure minorée au sens d’Alexandrov.

We prove that a compact stratified space satisfies the Riemannian curvature-dimension condition $RCD (K, N)$ if and only if its Ricci tensor is bounded below by $K \in ℝ$ on the regular set, the cone angle along the stratum of codimension two is smaller than or equal to $2 π$ and its dimension is at most equal to $N$ . This gives a new wide class of geometric examples of metric measure spaces satisfying the $RCD (K, N)$ curvature-dimension condition, including for instance spherical suspensions, orbifolds, Kähler–Einstein manifolds with a divisor, Einstein manifolds with conical singularities along a curve. We also obtain new analytic and geometric results on stratified spaces, such as Bishop–Gromov volume inequality, Laplacian comparison, Lévy–Gromov isoperimetric inequality. Our result also implies a similar characterization of compact stratified spaces carrying a lower curvature bound in the sense of Alexandrov.

Reçu le : 2019-02-28
Révisé le : 2019-08-26
Accepté le : 2019-12-20
Publié le : 2021-06-07

DOI : 10.5802/aif.3393

Classification : 53C21, 54E50
Keywords: Curvature-dimension condition, stratified spaces, Ricci curvature lower bounds
Mot clés : Condition de courbure-dimension, espaces stratifiés, bornes inférieures de la courbure de Ricci

Affiliations des auteurs :

Bertrand, Jérôme ¹ ; Ketterer, Christian ² ; Mondello, Ilaria ³ ; Richard, Thomas ³

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Stratified spaces and synthetic {Ricci} curvature bounds},
     journal = {Annales de l'Institut Fourier},
     pages = {123--173},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Bertrand, Jérôme; Ketterer, Christian; Mondello, Ilaria; Richard, Thomas. Stratified spaces and synthetic Ricci curvature bounds. Annales de l'Institut Fourier, Tome 71 (2021) no. 1, pp. 123-173. doi : 10.5802/aif.3393. https://aif.centre-mersenne.org/articles/10.5802/aif.3393/

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