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Kato meets Bakry-Émery

Gilles Carron, Ilaria Mondello, David Tewodrose

TL;DR

The paper proves that complete Riemannian manifolds with negative part of Ricci curvature controlled by a Dynkin-type bound admit a conformal change rendering them into $\,\mathrm{RCD}(-K/T,N)$ spaces, with constants depending only on dimension and the bound. Central to the approach is a transformation rule for the Bakry–Émery condition under time changes, together with a construction of a gauge $h$ from a Dynkin/Kato setup that yields BE bounds on the conformally changed space. This yields several new consequences: a quantitative Maintheo providing BE$( -K/T,N)$ for the weighted manifold, existence of good cutoff functions on complete manifolds, almost monotonicity of volume ratios, and local doubling/Poincaré properties, which then transfer to limit spaces. The results classify and provide rectifiability and density properties for Dynkin, Kato, and non-collapsed strong Kato limit spaces, expanding the understanding of geometric limits under Ricci-type bounds and extending prior closed-manifold results to the complete setting with explicit, dimension-dependent constants.

Abstract

We prove that any complete Riemannian manifold with negative part of the Ricci curvature in a suitable Dynkin class is bi-Lipschitz equivalent to a finite-dimensional $\mathrm{RCD}$ space, by building upon the transformation rule of the Bakry-Émery condition under time change. We apply this result to show that our previous results on the limits of closed Riemannian manifolds satisfying a uniform Kato bound carry over to limits of complete manifolds. We also obtain a weak version of the Bishop-Gromov monotonicity formula for manifolds satisfying a strong Kato bound.

Kato meets Bakry-Émery

TL;DR

The paper proves that complete Riemannian manifolds with negative part of Ricci curvature controlled by a Dynkin-type bound admit a conformal change rendering them into spaces, with constants depending only on dimension and the bound. Central to the approach is a transformation rule for the Bakry–Émery condition under time changes, together with a construction of a gauge from a Dynkin/Kato setup that yields BE bounds on the conformally changed space. This yields several new consequences: a quantitative Maintheo providing BE for the weighted manifold, existence of good cutoff functions on complete manifolds, almost monotonicity of volume ratios, and local doubling/Poincaré properties, which then transfer to limit spaces. The results classify and provide rectifiability and density properties for Dynkin, Kato, and non-collapsed strong Kato limit spaces, expanding the understanding of geometric limits under Ricci-type bounds and extending prior closed-manifold results to the complete setting with explicit, dimension-dependent constants.

Abstract

We prove that any complete Riemannian manifold with negative part of the Ricci curvature in a suitable Dynkin class is bi-Lipschitz equivalent to a finite-dimensional space, by building upon the transformation rule of the Bakry-Émery condition under time change. We apply this result to show that our previous results on the limits of closed Riemannian manifolds satisfying a uniform Kato bound carry over to limits of complete manifolds. We also obtain a weak version of the Bishop-Gromov monotonicity formula for manifolds satisfying a strong Kato bound.
Paper Structure (17 sections, 17 theorems, 138 equations)

This paper contains 17 sections, 17 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. The Bakry-Émery condition under time change
  3. The Bakry-Émery condition
  4. Time changes
  5. Transformation rule
  6. Proof of Theorem \ref{['Maintheo']}
  7. Kato condition and the bottom of the spectrum
  8. Proof of Theorem \ref{['Maintheo']}
  9. Consequences on complete manifolds
  10. Almost mononicity of the volume ratio
  11. Existence of good cut-off functions
  12. Monotonicity of heat ratios
  13. Local doubling and Poincaré
  14. Consequences for limit spaces
  15. Dynkin limit spaces
  16. ...and 2 more sections

Key Result

Theorem 1

Let $(M^n,g)$ be a complete Riemannian manifold of dimension $n\geq 2$. Assume that there exist $T>0$ and $\gamma \in (0,1/(n-2))$ such that Then there exist constants $K\ge 0$ and $N>n$, both depending on $n$ and $\gamma$ only, and $h\in \mathcal{C}^2(M)$ with $0 \le h \le C=C(n,\gamma)$, such that the weighted Riemannian manifold $(M, e^{2h}g , e^{2h} \nu_g )$ satisfies the $\mathrm{RCD}(-K/T,

Theorems & Definitions (31)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Corollary 4
  • Definition 2.1
  • Lemma 2.2
  • proof : Proof of Lemma \ref{['eq:BEHS']}.
  • Corollary 2.3
  • Proposition 3.1
  • Corollary 3.2
  • ...and 21 more