Stratified spaces and synthetic Ricci curvature bounds

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

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 University of Toronto
3 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

We prove that a compact stratied space satises the Riemannian curvature-dimension condition RCD(K, N) if and only if its Ricci tensor is bounded below by K ∈ R 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 stratied 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.

Domaines: Géométrie différentielle [math.DG]

Jérôme Bertrand, Christian Ketterer, Ilaria Mondello, Thomas Richard. Stratified spaces and synthetic Ricci curvature bounds. Annales de l'Institut Fourier, 2021, 71 (1), pp.123-173. ⟨10.5802/aif.3393⟩. ⟨hal-01773881v3⟩