Diameter bounds in 3d Type I Ricci flows

2025/10/15 by Gianniotis, Panagiotis · 2 citations
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2510.14019

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.

Cited by

• L¹ curvature bounds for Type I Ricci flows
• The Bounded Diameter Conjecture and Sharp Geometric Estimates for Mean Curvature Flow

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