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

2025/04/03 by Kyeongsu Choi, Choi, Kyeongsu, Yi Chun Lai +1 · 1 citation
Mathematics · Physics and Astronomy · #53E20 #Advanced Differential Geometry Research #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2504.02804

openalex publication_date 2025/04/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Sⁿ singularity converges at least at the rate (-t)^(n+1)/(n-1).

Cited by

• On the rate of convergence of cylindrical singularity in mean curvature flow

Related

• Heat kernel on Ricci shrinkers
• The Entropy of Ricci Flows with Type-I Scalar Curvature Bounds
• Singularities of Connection Ricci Flow and Ricci Harmonic Flow
• Analysis of Type I Singularities in the Harmonic Ricci Flow
• Closed Ricci Flows with Singularities Modeled on Asymptotically Conical Shrinkers