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Pré-Publication, Document De Travail Année : 2021

Limits of manifolds with a Kato bound on the Ricci curvature

Résumé

We study the structure of Gromov-Hausdorff limits of sequences of Riemannian manifolds $\{(M_\alpha^n,g_\alpha)\}_{\alpha \in A}$ whose Ricci curvature satisfies a uniform Kato bound. We first obtain Mosco convergence of the Dirichlet energies to the Cheeger energy and show that tangent cones of such limits satisfy the $\mathrm{RCD}(0,n)$ condition. When assuming a non-collapsing assumption, we introduce a new family of monotone quantities, which allows us to prove that tangent cones are also metric cones. We then show the existence of a well-defined stratification in terms of splittings of tangent cones. We finally prove volume convergence to the Hausdorff $n$-measure.

Dates et versions

hal-03143374 , version 1 (16-02-2021)

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Citer

Gilles Carron, Ilaria Mondello, David Tewodrose. Limits of manifolds with a Kato bound on the Ricci curvature. 2021. ⟨hal-03143374⟩
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