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Convergence of Ricci flow and long-time existence of Harmonic map heat flow

Kyeongsu Choi, Yi Lai

TL;DR

This work develops a global parabolic gauge for Ricci flow by coupling the flow to a harmonic map heat flow (HMHF) into a compact integrable shrinker, enabling long-time existence of the HMHF for ancient and immortal flows and yielding a unique tangent flow without diffeomorphism ambiguity. The authors establish a generalized slice theorem, relate HMHF perturbations to the linearized Ricci–DeTurck problem via commutator identities, and show exponentially fast convergence of rescaled Ricci flows to shrinkers under carefully chosen gauges. They classify ancient flows asymptotic to compact integrable shrinkers, showing an ess-index$(L)$-parameter family of ancient flows converging exponentially to the shrinker, and prove optimal convergence rates at shrinkage singularities dictated by the first negative essential eigenvalue of the stability operator $L= abla_f+2 ext{Rm}*$. In particular, spherical singularities converge at least at rate $( -t)^{1+ rac{2}{n-1}}$ in unrescaled time, corresponding to the eigenvalue $- rac{2}{n-1}$, while the rescaled flow decays exponentially with the same spectral data. Together, these results provide a robust framework to classify ancient Ricci flows and quantify singularity formation, with implications for tangent-flow uniqueness and rate estimates in geometric analysis.

Abstract

For an ancient Ricci flow asymptotic to a compact integrable shrinker, or a Ricci flow developing a finite-time singularity modelled on the shrinker, we establish the long-time existence of a harmonic map heat flow between the Ricci flow and the shrinker for all times. This provides a global parabolic gauge for the Ricci flow and implies the uniqueness of the tangent flow without modulo any diffeomorphisms. We present two main applications: First, we construct and classify all ancient Ricci flows asymptotic to any compact integrable shrinker, showing that they converge exponentially. Second, we obtain the optimal convergence rate at singularities modelled on the shrinker, characterized by the first negative eigenvalue of the stability operator for the entropy. In particular, we show that any Ricci flow developing a round $\mathbb S^n$ singularity converges at least at the rate $(-t)^{\frac{n+1}{n-1}}$.

Convergence of Ricci flow and long-time existence of Harmonic map heat flow

TL;DR

This work develops a global parabolic gauge for Ricci flow by coupling the flow to a harmonic map heat flow (HMHF) into a compact integrable shrinker, enabling long-time existence of the HMHF for ancient and immortal flows and yielding a unique tangent flow without diffeomorphism ambiguity. The authors establish a generalized slice theorem, relate HMHF perturbations to the linearized Ricci–DeTurck problem via commutator identities, and show exponentially fast convergence of rescaled Ricci flows to shrinkers under carefully chosen gauges. They classify ancient flows asymptotic to compact integrable shrinkers, showing an ess-index-parameter family of ancient flows converging exponentially to the shrinker, and prove optimal convergence rates at shrinkage singularities dictated by the first negative essential eigenvalue of the stability operator . In particular, spherical singularities converge at least at rate in unrescaled time, corresponding to the eigenvalue , while the rescaled flow decays exponentially with the same spectral data. Together, these results provide a robust framework to classify ancient Ricci flows and quantify singularity formation, with implications for tangent-flow uniqueness and rate estimates in geometric analysis.

Abstract

For an ancient Ricci flow asymptotic to a compact integrable shrinker, or a Ricci flow developing a finite-time singularity modelled on the shrinker, we establish the long-time existence of a harmonic map heat flow between the Ricci flow and the shrinker for all times. This provides a global parabolic gauge for the Ricci flow and implies the uniqueness of the tangent flow without modulo any diffeomorphisms. We present two main applications: First, we construct and classify all ancient Ricci flows asymptotic to any compact integrable shrinker, showing that they converge exponentially. Second, we obtain the optimal convergence rate at singularities modelled on the shrinker, characterized by the first negative eigenvalue of the stability operator for the entropy. In particular, we show that any Ricci flow developing a round singularity converges at least at the rate .

Paper Structure

This paper contains 23 sections, 55 theorems, 311 equations.

Table of Contents

  1. Introduction
  2. Long-time existence of Harmonic map heat flow
  3. Classification of ancient Ricci flows
  4. Optimal convergence rate for Ricci flow singularities
  5. Acknowledgements
  6. Commutators and eigentensors
  7. Existence of ancient and immortal Ricci flows
  8. A generalized slice theorem
  9. Fast convergence of Ricci flows
  10. Spectral analysis using the Entropy
  11. Fast convergence of ancient Ricci flows
  12. Fast convergence of Ricci flows to singularities
  13. Classification and optimal convergence rate
  14. Classification of ancient flows
  15. Rate of convergence to singularity
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $(M,\bar{g})$ be a compact integrable shrinker. Let $g_\tau$ be an ancient (resp. immortal) rescaled Ricci flow asymptotic to $\bar{g}$. Then there exists an ancient (resp. immortal) HMHF between $g_{\tau}$ and $\bar{g}$ that converges exponentially to $\mathrm{id}$ as $\tau$ goes to negative (r

Theorems & Definitions (118)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: Existence of ancient Ricci flows
  • Theorem 1.4: Classification
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7: Kotschwar kotschwar2022maximal
  • Theorem 1.8: Rate of convergence at singularity
  • Corollary 1.9
  • Lemma 2.1
  • ...and 108 more