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Limits of manifolds with a Kato bound on the Ricci curvature

Gilles Carron, Ilaria Mondello, David Tewodrose

Abstract

We study the structure of Gromov-Hausdorff limits of sequences of Riemannian manifolds $\{(M_α^n,g_α)\}_{α\in A}$ 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.

Limits of manifolds with a Kato bound on the Ricci curvature

Abstract

We study the structure of Gromov-Hausdorff limits of sequences of Riemannian manifolds 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 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 -measure.

Paper Structure

This paper contains 44 sections, 72 theorems, 639 equations.

Table of Contents

  1. Preliminaries
  2. The doubling condition
  3. Dirichlet spaces
  4. The Poincaré inequality and PI Dirichlet spaces
  5. Notions of convergence
  6. Convergence of spaces
  7. Convergence of points and functions
  8. Mosco convergence
  9. A compactness result for Dirichlet spaces
  10. Dirichlet spaces satisfying an $\mathop{\mathrm{RCD}}\nolimits$ condition
  11. Kato limits
  12. Dynkin limits
  13. Kato limits
  14. Ahlfors regularity
  15. Non-collapsed strong Kato limits
  16. ...and 29 more sections

Key Result

Theorem A

Let $(X,\mathsf{d}, o)$ be the pointed Gromov-Hausdorff limit of a sequence of closed Riemannian manifolds $\left\{(M_\alpha^n,\mathsf{d}_{g_\alpha}, o_\alpha)\right\}_\alpha$ satisfying the uniform Kato bound where $f\colon [0,1]\rightarrow \mathbb R_+$ is a non-decreasing function such that and the non-collapsing condition Then the following holds.

Theorems & Definitions (162)

  • Theorem A
  • Theorem B
  • Definition 1.1
  • Proposition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Proposition 1.6
  • Proposition 1.7
  • Definition 1.8
  • ...and 152 more