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On the equivalence of distributional and synthetic Ricci curvature lower bounds

Andrea Mondino, Vanessa Ryborz

TL;DR

The paper proves an equivalence between distributional and synthetic lower bounds for Bakry–Émery Ricci curvature on weighted manifolds with a continuous metric and low-regularity connections. It develops a comprehensive non-smooth calculus, including a weak Bochner formula, and establishes a robust first- and second-order framework that aligns distributional Ricci bounds with the RCD/CD notions via BE inequalities. The main contributions are (i) showing that $\mathsf{RCD}^*(K,N)$ (or $\mathsf{RCD}(K,\infty)$) implies a distributional $\mathrm{Ric}_{\mu,N}\ge K g$ and (ii) proving the reverse implication under a volume-growth condition, thereby unifying distributional and synthetic curvature notions on spaces with low regularity. This bridges classical geometric analysis with modern metric-measure theory, enabling curvature-based methods on spaces with minimal smoothness and weighted measures.

Abstract

The goal of the paper is to prove the equivalence of distributional and synthetic Ricci curvature lower bounds for a weighted Riemannian manifold with continuous metric tensor having Christoffel symbols in $L^2_{\rm loc}$, and with weight in $C^0\cap W^{1,2}_{\rm loc}$.

On the equivalence of distributional and synthetic Ricci curvature lower bounds

TL;DR

The paper proves an equivalence between distributional and synthetic lower bounds for Bakry–Émery Ricci curvature on weighted manifolds with a continuous metric and low-regularity connections. It develops a comprehensive non-smooth calculus, including a weak Bochner formula, and establishes a robust first- and second-order framework that aligns distributional Ricci bounds with the RCD/CD notions via BE inequalities. The main contributions are (i) showing that (or ) implies a distributional and (ii) proving the reverse implication under a volume-growth condition, thereby unifying distributional and synthetic curvature notions on spaces with low regularity. This bridges classical geometric analysis with modern metric-measure theory, enabling curvature-based methods on spaces with minimal smoothness and weighted measures.

Abstract

The goal of the paper is to prove the equivalence of distributional and synthetic Ricci curvature lower bounds for a weighted Riemannian manifold with continuous metric tensor having Christoffel symbols in , and with weight in .
Paper Structure (21 sections, 60 theorems, 332 equations)

This paper contains 21 sections, 60 theorems, 332 equations.

Table of Contents

  1. Introduction
  2. Distributional calculus with a Riemannian metric of low regularity
  3. Notation and basic definitions
  4. Some regularisation results and their application to Sobolev Riemannian metrics
  5. A weak formulation of Bochner's formula for metrics of low regularity
  6. Smooth calculus on manifolds
  7. Bochner's formula for vector fields on smooth weighted manifolds
  8. A Bochner formula for weighted manifolds of lower regularity
  9. First order calculus
  10. The Cheeger energy
  11. Optimal transport and curvature dimension conditions
  12. TEXT and Sobolev spaces on weighted Riemannian manifolds of low regularity
  13. Cheeger energy on a manifold with continuous Riemannian metric and weight
  14. Second order calculus
  15. Some elements of the theory for general TEXT-spaces
  16. ...and 6 more sections

Key Result

Theorem 1

Let $M$ be a smooth manifold, $g$ a continuous Riemannian metric with Christoffel symbols in $L^2_{\rm{loc}}$, and $V \in C^{0}\cap W^{1,2}_{\rm{loc}}(M)$ a positive function on $M$. Define the weighted measure $\mu$ as ${\rm d}\mu:= e^{-V}{\rm d}\mathrm{vol}_g$. Let $N \in [n, \infty]$ and $K \in

Theorems & Definitions (125)

  • Theorem 1: see Theorem \ref{['equivalence_result']}
  • Remark 1.1: On the smoothness assumption on $M$
  • Remark 1.2: On the volume growth assumption
  • Definition 2.1: Action of a vector field on a distribution, lefloch2007definition
  • Definition 2.2: lefloch2007definition, Def. 4.1
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5: A Friedrichs type lemma
  • proof
  • ...and 115 more