Stratified spaces and synthetic Ricci curvature bounds
[Espaces stratifiés et bornes de courbure de Ricci synthétiques]
Annales de l'Institut Fourier, Online first, 51 p.

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 dans le lieu régulier, l’angle des cônes le long de la strate de codimension deux est inférieur ou égal à 2π 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 on the regular set, the cone angle along the stratum of codimension two is smaller than or equal to 2π 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.

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DOI : https://doi.org/10.5802/aif.3393
Classification : 53C21,  54E50
Mots clés : Condition de courbure-dimension, espaces stratifiés, bornes inférieures de la courbure de Ricci
@unpublished{AIF_0__0_0_A43_0,
     author = {Bertrand, J\'er\^ome and Ketterer, Christian and Mondello, Ilaria and Richard, Thomas},
     title = {Stratified spaces and synthetic Ricci curvature bounds},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2021},
     doi = {10.5802/aif.3393},
     language = {en},
     note = {Online first},
}
Bertrand, Jérôme; Ketterer, Christian; Mondello, Ilaria; Richard, Thomas. Stratified spaces and synthetic Ricci curvature bounds. Annales de l'Institut Fourier, Online first, 51 p.

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