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On the structure of noncollapsed Ricci flow limit spaces

Hanbing Fang, Yu Li

TL;DR

The work establishes a weak compactness theory for the moduli space of closed Ricci flows with uniformly bounded Nash entropy, using a spacetime distance $d^*$ to obtain pointed Gromov–Hausdorff limits. The limit space $(Z,d_Z, rak t)$ splits into a regular part $\mathcal{R}$ carrying a Ricci flow spacetime structure and a singular set of codimension at least $4$, with smooth convergence on the regular part. The authors connect the limit to Bamler's $\\mathbb{F}$-limits, showing that each limit point $z\in Z$ has an associated metric flow $\mathcal X^z$ that embeds isometrically into $Z$, and they demonstrate reproduction of heat kernels and strong regularity properties. Tangent flows are shown to be noncollapsed Ricci flow limit spaces, i.e., Ricci shrinker spaces, and the singular set obeys quantitative stratification and dimension bounds, providing a robust geometric picture of Ricci flow limits under entropy bounds.

Abstract

We establish a weak compactness theorem for the moduli space of closed Ricci flows with uniformly bounded entropy, each equipped with a natural spacetime distance, under pointed Gromov-Hausdorff convergence. Furthermore, we develop a structure theory for the corresponding Ricci flow limit spaces, showing that the regular part, where convergence is smooth, admits the structure of a Ricci flow spacetime, while the singular set has codimension at least four.

On the structure of noncollapsed Ricci flow limit spaces

TL;DR

The work establishes a weak compactness theory for the moduli space of closed Ricci flows with uniformly bounded Nash entropy, using a spacetime distance to obtain pointed Gromov–Hausdorff limits. The limit space splits into a regular part carrying a Ricci flow spacetime structure and a singular set of codimension at least , with smooth convergence on the regular part. The authors connect the limit to Bamler's -limits, showing that each limit point has an associated metric flow that embeds isometrically into , and they demonstrate reproduction of heat kernels and strong regularity properties. Tangent flows are shown to be noncollapsed Ricci flow limit spaces, i.e., Ricci shrinker spaces, and the singular set obeys quantitative stratification and dimension bounds, providing a robust geometric picture of Ricci flow limits under entropy bounds.

Abstract

We establish a weak compactness theorem for the moduli space of closed Ricci flows with uniformly bounded entropy, each equipped with a natural spacetime distance, under pointed Gromov-Hausdorff convergence. Furthermore, we develop a structure theory for the corresponding Ricci flow limit spaces, showing that the regular part, where convergence is smooth, admits the structure of a Ricci flow spacetime, while the singular set has codimension at least four.
Paper Structure (15 sections, 165 theorems, 697 equations, 1 figure)

This paper contains 15 sections, 165 theorems, 697 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Spacetime distance and Ricci flow limit spaces
  4. $\mathbb{F}$-limits of Ricci flows
  5. Smooth convergence on the regular part
  6. Extended metric flows
  7. Ricci shrinker spaces and tangent flows
  8. Stratification and dimension of the singular set
  9. Application: the first singular time of the Ricci flow
  10. Almost splitting maps
  11. Further discussions
  12. Change of basis for conjugate heat kernel measures
  13. Comparison of spacetime distances
  14. Eigenvalues and almost splitting
  15. Spines of Ricci shrinker spaces

Key Result

Theorem 1.3

Given any sequence $\mathcal{X}^i=\{M_i^n,(g_i(t))_{t \in \mathbb{I}^{++}}\} \in \mathcal{M}(n, Y, T)$ with base points $p_i^* \in M_i \times \mathbb{I}$(when $T=+\infty$, we additionally assume $\limsup_{i \to \infty} \mathfrak{t}_i(p_i^*)>-\infty$), by taking a subsequence if necessary, we obtain where $d^*_i$ denotes the restriction of the $d^*$-distance on $M_i \times \mathbb{I}$, and $\mathf

Figures (1)

  • Figure 1: Singular set is a segment at $t_3$; $\iota_x(\mathcal{R}^x_t) = \iota_y(\mathcal{R}^y_t)$ for $t < t_3$.

Theorems & Definitions (346)

  • Definition 1.1: Moduli space
  • Definition 1.2
  • Theorem 1.3: Weak compactness
  • Theorem 1.4
  • Theorem 1.5: Smooth convergence
  • Definition 1.6
  • Theorem 1.7
  • Definition 1.8: Tangent flow
  • Theorem 1.9: Characterization of Ricci shrinker spaces
  • Theorem 1.10
  • ...and 336 more