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Synthetic notions of Ricci flow for metric measure spaces

Matthias Erbar, Zhenhao Li, Timo Schultz

TL;DR

This work develops synthetic notions of Ricci flow for time-dependent metric measure spaces by marrying optimal transport with heat-flow analysis. It introduces two principal approaches: (i) dynamic entropy convexity/concavity along Wasserstein geodesics and (ii) short-time transport estimates for the heat flow, defining weak/rough, sub-/super-, and N-refined flows. On smooth manifolds, these notions recover the classical weighted Ricci flow, while in non-smooth settings they provide a robust framework for studying Ricci-flow-like evolution through singularities and changing geometry. The paper also establishes infinitesimal characterisations via RFex, $\eta$, and $\vartheta$, compares the various notions, and furnishes concrete examples (cones, spherical suspensions, Gaussian weights) to illustrate both consistencies and limitations of the synthetic theory.

Abstract

We develop different synthetic notions of Ricci flow in the setting of time-dependent metric measure spaces based on ideas from optimal transport. They are formulated in terms of dynamic convexity and local concavity of the entropy along Wasserstein geodesics on the one hand and in terms of global and short-time asymptotic transport cost estimates for the heat flow on the other hand. We show that these properties characterise smooth (weighted) Ricci flows. Further, we investigate the relation between the different notions in the non-smooth setting of time-dependent metric measure spaces.

Synthetic notions of Ricci flow for metric measure spaces

TL;DR

This work develops synthetic notions of Ricci flow for time-dependent metric measure spaces by marrying optimal transport with heat-flow analysis. It introduces two principal approaches: (i) dynamic entropy convexity/concavity along Wasserstein geodesics and (ii) short-time transport estimates for the heat flow, defining weak/rough, sub-/super-, and N-refined flows. On smooth manifolds, these notions recover the classical weighted Ricci flow, while in non-smooth settings they provide a robust framework for studying Ricci-flow-like evolution through singularities and changing geometry. The paper also establishes infinitesimal characterisations via RFex, , and , compares the various notions, and furnishes concrete examples (cones, spherical suspensions, Gaussian weights) to illustrate both consistencies and limitations of the synthetic theory.

Abstract

We develop different synthetic notions of Ricci flow in the setting of time-dependent metric measure spaces based on ideas from optimal transport. They are formulated in terms of dynamic convexity and local concavity of the entropy along Wasserstein geodesics on the one hand and in terms of global and short-time asymptotic transport cost estimates for the heat flow on the other hand. We show that these properties characterise smooth (weighted) Ricci flows. Further, we investigate the relation between the different notions in the non-smooth setting of time-dependent metric measure spaces.
Paper Structure (26 sections, 40 theorems, 238 equations)

This paper contains 26 sections, 40 theorems, 238 equations.

Table of Contents

  1. Introduction
  2. Setting and main results
  3. Examples
  4. Heuristics
  5. Preliminaries
  6. Optimal transport and displacement convexity
  7. Metric measure spaces and curvature-dimension conditions
  8. Dynamic convexity
  9. Time-dependent metric measure spaces and heat flows
  10. Flows of smooth Riemannian manifolds
  11. Synthetic Ricci flow
  12. Weak Ricci-flow
  13. Rough Ricci-flow
  14. Characterizing synthetic notions by infinitesimal quantities
  15. Smooth flows
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $(M,d_t, e^{-f_t}\mathrm{vol}_{g_t})_{t\in I}$ be a smooth flow. Then the following are equivalent:

Theorems & Definitions (84)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Definition 2.2: dynamic convexity
  • Definition 2.3: Upper regular
  • Definition 2.4: log-Lipschitz control on metrics
  • Theorem 2.6
  • Lemma 2.7: Regularity of dual heat flows
  • ...and 74 more