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Comparison estimates on nonsmooth spaces with integrable Ricci lower bounds via localization

Emanuele Caputo, Francesco Nobili, Tommaso Rossi

TL;DR

This work extends classical comparison principles to nonsmooth metric measure spaces carrying a variable Ricci curvature lower bound in the ${\sf CD}(k,N)$ sense, under an integrable deficit ${\rho^k_p}$. The authors develop a localization framework along transport rays, disintegrating the reference measure and reducing key problems to one-dimensional model intervals with deficit control via the mean curvature deficit $\psi$. They establish a quantitative Bishop-Gromov inequality, a corresponding Myers diameter bound, and a Cheng type eigenvalue comparison under small averaged deficits, all within a unified nonsmooth setting. The results rely on regularity of conditional measures, the 1D ${\sf CD}$ analysis, and reintegration techniques to transfer 1D estimates back to the ambient space, thereby extending integral curvature bounds and classical comparison results to Ricci limit and singular spaces.

Abstract

We study comparison estimates on metric measure spaces admitting a synthetic variable Ricci curvature lower bound. We obtain geometric and functional inequalities assuming that the deficit of the lower bound from a given constant is sufficiently integrable. More precisely, we extend to the nonsmooth setting the Bishop-Gromov comparison, the Myers' diameter estimate and the Cheng's comparison principle for Dirichlet eigenvalues. Our analysis relies on the localization method and on one-dimensional comparison estimates for nonsmooth weighted intervals.

Comparison estimates on nonsmooth spaces with integrable Ricci lower bounds via localization

TL;DR

This work extends classical comparison principles to nonsmooth metric measure spaces carrying a variable Ricci curvature lower bound in the sense, under an integrable deficit . The authors develop a localization framework along transport rays, disintegrating the reference measure and reducing key problems to one-dimensional model intervals with deficit control via the mean curvature deficit . They establish a quantitative Bishop-Gromov inequality, a corresponding Myers diameter bound, and a Cheng type eigenvalue comparison under small averaged deficits, all within a unified nonsmooth setting. The results rely on regularity of conditional measures, the 1D analysis, and reintegration techniques to transfer 1D estimates back to the ambient space, thereby extending integral curvature bounds and classical comparison results to Ricci limit and singular spaces.

Abstract

We study comparison estimates on metric measure spaces admitting a synthetic variable Ricci curvature lower bound. We obtain geometric and functional inequalities assuming that the deficit of the lower bound from a given constant is sufficiently integrable. More precisely, we extend to the nonsmooth setting the Bishop-Gromov comparison, the Myers' diameter estimate and the Cheng's comparison principle for Dirichlet eigenvalues. Our analysis relies on the localization method and on one-dimensional comparison estimates for nonsmooth weighted intervals.

Paper Structure

This paper contains 22 sections, 30 theorems, 224 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Metric measure spaces with variable Ricci bounds
  4. Non-branching and existence of transport maps
  5. Localization of variable Ricci curvature bounds
  6. Disintegration of the reference measure
  7. Regularity of the disintegration
  8. Localization of variable curvature bounds
  9. One-dimensional comparison estimates
  10. Bishop-Gromov comparison
  11. Technical lemmas
  12. Proof of Bishop-Gromov comparison
  13. Consequences of Bishop-Gromov comparison
  14. Myers' diameter estimate
  15. Star-shaped partition
  16. ...and 7 more sections

Key Result

Theorem 1.1

For every $K\in \mathbb{R},N \in (1,\infty), p>N/2$, there exists an explicit non-decreasing positive function $\left(0,\pi\sqrt{\tfrac{N-1}{K^+}}\right)\ni R\mapsto C_{K,N,p}(R)$ (see eq:CpKNR) such that the following holds. Let $({\rm X},{\sf d},\mathfrak m)$ be an essentially non-branching ${\sf Moreover, if $T = B_R(x)$, it holds Finally, the function $C_{K,N,p}(R)$ is globally bounded if $K

Theorems & Definitions (68)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1: Distortion coefficients
  • Definition 2.2: Admissible variable lower bound
  • Definition 2.3: Generalized distortion coefficients
  • Definition 2.4: ${\sf CD}(k,N)$ space
  • Proposition 2.5
  • proof
  • Definition 2.6
  • ...and 58 more