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Initial stability estimates for Ricci flow and three dimensional Ricci-pinched manifolds

Alix Deruelle, Felix Schulze, Miles Simon

TL;DR

The paper develops a stability theory for Ricci flow starting from Reifenberg-type initial data that are locally bi-Lipschitz to Euclidean space. By constructing an adjustment map and comparing two flows via the δ-Ricci-DeTurck formulation and Ricci-harmonic map flow, it proves that the flows converge exponentially fast once gauge is fixed, after first establishing curvature decay and pseudolocality in this non-smooth setting. It then proves polynomial and ultimately exponential convergence rates for the adjusted flows, and uses these results to analyze almost Ricci-pinched expanding solitons. As an application, the authors resolve Hamilton–Lott-type questions in dimension three, showing that complete 3-manifolds with uniformly Ricci-pinched, bounded-curvature metrics are either compact (spherical space forms) or flat, thereby establishing a sharp dichotomy in this setting.

Abstract

This paper investigates the question of stability for a class of Ricci flows which start at possibly non-smooth metric spaces. We show that if the initial metric space is Reifenberg and locally bi-Lipschitz to Euclidean space, then two solutions to the Ricci flow whose Ricci curvature is uniformly bounded from below and whose curvature is bounded by $c\cdot t^{-1}$ converge to one another at an exponential rate once they have been appropriately gauged. As an application, we show that smooth three dimensional, complete, uniformly Ricci-pinched Riemannian manifolds with bounded curvature are either compact or flat, thus confirming a conjecture of Hamilton and Lott.

Initial stability estimates for Ricci flow and three dimensional Ricci-pinched manifolds

TL;DR

The paper develops a stability theory for Ricci flow starting from Reifenberg-type initial data that are locally bi-Lipschitz to Euclidean space. By constructing an adjustment map and comparing two flows via the δ-Ricci-DeTurck formulation and Ricci-harmonic map flow, it proves that the flows converge exponentially fast once gauge is fixed, after first establishing curvature decay and pseudolocality in this non-smooth setting. It then proves polynomial and ultimately exponential convergence rates for the adjusted flows, and uses these results to analyze almost Ricci-pinched expanding solitons. As an application, the authors resolve Hamilton–Lott-type questions in dimension three, showing that complete 3-manifolds with uniformly Ricci-pinched, bounded-curvature metrics are either compact (spherical space forms) or flat, thereby establishing a sharp dichotomy in this setting.

Abstract

This paper investigates the question of stability for a class of Ricci flows which start at possibly non-smooth metric spaces. We show that if the initial metric space is Reifenberg and locally bi-Lipschitz to Euclidean space, then two solutions to the Ricci flow whose Ricci curvature is uniformly bounded from below and whose curvature is bounded by converge to one another at an exponential rate once they have been appropriately gauged. As an application, we show that smooth three dimensional, complete, uniformly Ricci-pinched Riemannian manifolds with bounded curvature are either compact or flat, thus confirming a conjecture of Hamilton and Lott.
Paper Structure (14 sections, 27 theorems, 158 equations)

This paper contains 14 sections, 27 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Outline of paper
  4. Notation
  5. Acknowldgements
  6. Curvature decay
  7. Existence of an adjustment map
  8. Rough convergence rate
  9. Polynomial convergence rate
  10. Exponential convergence rate
  11. Almost Ricci-pinched expanding gradient Ricci solitons
  12. Compactness of Ricci-pinched manifolds in dimension three
  13. Volume and distance convergence
  14. A splitting theorem of R. Hochard for the Ricci flow

Key Result

Theorem 1.2

For all $n\in {\mathbb N}$ and $\beta_0\in (0,1)$ there exists $\varepsilon_0=\varepsilon_0(n,\beta_0)>0$, depending only on $\beta_0$ and $n$, such that the following holds. Let $(M_i^n,g_i(t))_{t\in (0,T)}$, $i=1,2$, be smooth solutions to Ricci flow (not necessarily with complete time slices) bot

Theorems & Definitions (63)

  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4: Lott-Ricci-pinched
  • Remark 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 53 more