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Strong uniqueness of tangent flows at cylindrical singularities in Ricci flow

Hanbing Fang, Yu Li

TL;DR

This work addresses the problem of tangent-flow uniqueness at cylindrical singularities in Ricci flow by proving a Lojasiewicz-type inequality for the pointed $\mathcal{W}$-entropy and employing a gauge-fixed, modified Ricci flow near cylindrical models. The authors develop a radius-function framework to quantify almost-cylindrical regions, derive remainder estimates for entropy functionals, and establish a discrete Lojasiewicz inequality with exponent $\beta\in(0,3/4)$, which yields strong uniqueness of cylindrical tangent flows and extends to quotient cylinders. Their approach provides quantitative convergence results toward cylindrical shrinkers and advances the understanding of singularity structure in Ricci flow, with implications for noncompact shrinkers and the geometry of Ricci-flow limit spaces. The results bridge rigidity of Ricci shrinkers with dynamical stability via entropy methods, offering tools for refined blow-up analysis and singularity modeling.

Abstract

In this paper, we establish a Lojasiewicz inequality for the pointed $\mathcal{W}$-entropy in the Ricci flow, under the assumption that the geometry near the base point is close to a standard cylinder $\mathbb{R}^k \times S^{n-k}$ or the quotient thereof. As an application, we prove the strong uniqueness of the cylindrical tangent flow at the first singular time of the Ricci flow. Specifically, we show that the modified Ricci flow near the singularity converges to the cylindrical model under a fixed gauge.

Strong uniqueness of tangent flows at cylindrical singularities in Ricci flow

TL;DR

This work addresses the problem of tangent-flow uniqueness at cylindrical singularities in Ricci flow by proving a Lojasiewicz-type inequality for the pointed -entropy and employing a gauge-fixed, modified Ricci flow near cylindrical models. The authors develop a radius-function framework to quantify almost-cylindrical regions, derive remainder estimates for entropy functionals, and establish a discrete Lojasiewicz inequality with exponent , which yields strong uniqueness of cylindrical tangent flows and extends to quotient cylinders. Their approach provides quantitative convergence results toward cylindrical shrinkers and advances the understanding of singularity structure in Ricci flow, with implications for noncompact shrinkers and the geometry of Ricci-flow limit spaces. The results bridge rigidity of Ricci shrinkers with dynamical stability via entropy methods, offering tools for refined blow-up analysis and singularity modeling.

Abstract

In this paper, we establish a Lojasiewicz inequality for the pointed -entropy in the Ricci flow, under the assumption that the geometry near the base point is close to a standard cylinder or the quotient thereof. As an application, we prove the strong uniqueness of the cylindrical tangent flow at the first singular time of the Ricci flow. Specifically, we show that the modified Ricci flow near the singularity converges to the cylindrical model under a fixed gauge.
Paper Structure (8 sections, 72 theorems, 550 equations)

This paper contains 8 sections, 72 theorems, 550 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Remainder estimates for integral quantities
  4. Variations of the $\mathcal{W}$-functional near cylinders
  5. Radius functions on weighted Riemannian manifolds
  6. Lojasiewicz inequalities for cylindrical geometries
  7. Strong uniqueness for cylindrical tangent flows
  8. Generalization to quotient cylindrical geometries

Key Result

Theorem 1.1

Let $(Z, d_Z, \mathfrak{t})$ be the completion of a closed Ricci flow $\mathcal{X}=\{M^n, (g(t))_{t \in [-T,0)}\}$ with entropy bounded below by $-Y$. For any $z \in Z_0$, if a tangent flow at $z$ is isometric to $\bar{\mathcal{C}}^k$, then every tangent flow at $z$ is isometric to $\bar{\mathcal{C}

Theorems & Definitions (152)

  • Theorem 1.1: Uniqueness of the cylindrical tangent flow
  • Theorem 1.2: Strong uniqueness of the cylindrical tangent flow
  • Theorem 1.3: Lojasiewicz inequality
  • Corollary 1.4
  • Definition 1.5
  • Example 2.1: Model space: weighted cylinders
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • ...and 142 more