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Diameter bounds in 3d Type I Ricci flows

Panagiotis Gianniotis

TL;DR

The paper establishes a uniform diameter bound for 3D compact Ricci flows with Type I singularities by developing a quantitative neck-analysis framework. It introduces neck regions and a neck-structure theorem that reduces high-curvature regions to cylinders and their refinements, yielding an L^1 curvature bound and, via Topping-type diameter control, a uniform diameter bound. The approach relies on a refined toolkit: conjugate heat flow, entropy pinching, almost selfsimilarity, almost splitting maps, and an R-scale distance that behaves like a triangle-approximate metric; these tools extend to higher dimensions through the Ahlfors-regularity of neck-region packing. The results thereby realize a dimensional-agnostic mechanism (under natural a priori bounds) to control the global geometry near singularities and offer an affirmative answer to Perelman’s conjecture for Type I singularities in dimension three. The methods have potential implications for higher-dimensional singular Ricci flows and the understanding of singular sets via entropy-based and splitting-map techniques.

Abstract

We prove that a three dimensional compact Ricci flow that encounters a Type I singularity has uniformly bounded diameter up to the singular time, thus giving an affirmative answer - for Type I singularities - to a conjecture of Perelman. To achieve this, we introduce a concept of a neck-region for a Ricci flow, analogous to the neck-regions introduced by Jiang-Naber and Cheeger-Jiang-Naber, in the study of Ricci limit spaces. We then prove that the associated packing measure is, in a certain sense, Ahlfors regular, a result that holds in any dimension.

Diameter bounds in 3d Type I Ricci flows

TL;DR

The paper establishes a uniform diameter bound for 3D compact Ricci flows with Type I singularities by developing a quantitative neck-analysis framework. It introduces neck regions and a neck-structure theorem that reduces high-curvature regions to cylinders and their refinements, yielding an L^1 curvature bound and, via Topping-type diameter control, a uniform diameter bound. The approach relies on a refined toolkit: conjugate heat flow, entropy pinching, almost selfsimilarity, almost splitting maps, and an R-scale distance that behaves like a triangle-approximate metric; these tools extend to higher dimensions through the Ahlfors-regularity of neck-region packing. The results thereby realize a dimensional-agnostic mechanism (under natural a priori bounds) to control the global geometry near singularities and offer an affirmative answer to Perelman’s conjecture for Type I singularities in dimension three. The methods have potential implications for higher-dimensional singular Ricci flows and the understanding of singular sets via entropy-based and splitting-map techniques.

Abstract

We prove that a three dimensional compact Ricci flow that encounters a Type I singularity has uniformly bounded diameter up to the singular time, thus giving an affirmative answer - for Type I singularities - to a conjecture of Perelman. To achieve this, we introduce a concept of a neck-region for a Ricci flow, analogous to the neck-regions introduced by Jiang-Naber and Cheeger-Jiang-Naber, in the study of Ricci limit spaces. We then prove that the associated packing measure is, in a certain sense, Ahlfors regular, a result that holds in any dimension.
Paper Structure (22 sections, 34 theorems, 304 equations)

This paper contains 22 sections, 34 theorems, 304 equations.

Table of Contents

  1. Introduction
  2. Discussion of the problem
  3. Discussion of the proof
  4. Preliminaries
  5. The heat and conjugate heat equations
  6. Monotone quantities
  7. Gradient shrinking Ricci solitons
  8. $k$-independent subsets
  9. Ricci flows under a priori assumptions
  10. A priori assumptions
  11. Smooth convergence of sequences of Ricci flows
  12. Distance distortion
  13. Almost selfsimilarity
  14. Almost splitting maps
  15. Existence and small scale behaviour of almost splitting maps
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $M$ be a compact three dimensional manifold and $g(t)$, $t\in [0,T)$ be a smooth Ricci flow on $M$, which may become singular at $t=T$. Suppose that there is a constant $C_I<+\infty$ such that on $M\times [0,T)$. Then, there is a constant $C=C(g(0))<+\infty$ such that for every $t\in [0,T)$.

Theorems & Definitions (85)

  • Theorem 1.1
  • Conjecture 1.1
  • Proposition 2.1: Proposition 4.1 in G25
  • Definition 2.1
  • Remark 2.1: Remark 8.1 in G25
  • Lemma 2.1: Lemma 8.2 in G25
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • ...and 75 more