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An $ε$-Regularity Theorem for Non-collapsed Ricci Flow

Harry Fluck, Max Hallgren

TL;DR

This work advances the regularity theory of non-collapsed Ricci flows by proving an $ε$-regularity theorem that controls curvature scales under a Nash-entropy non-collapsing condition and a finite-energy, near-splitting hypothesis. It develops a robust framework—centered on $ m F$-convergence, metric flows, and parabolic regularizations of soliton-type potentials—to obtain uniform convergence of heat-type observables along convergent flows and to analyze singular models arising in Fano and Kähler-Ricci settings. The authors deduce sharp Minkowski-type estimates for the singular set, showing parabolic codimension $4$ control, and apply these results to singular Kähler-Ricci solitons, broadening the applicability beyond previously known Kähler-specific arguments. Collectively, the results provide finer regularity and dimensional information about singularities in Ricci flows, enabling more precise stratifications and potential generalizations to broader geometric flows.

Abstract

In this article we prove an $ε$-regularity theorem for non-collapsed Ricci flows, and use this to prove new estimates for singularity models of Fano Kähler-Ricci flows. In the course of our proof, we find a criterion for uniform convergence of solutions to the heat equation along a sequence of $\mathbb{F}$-converging Ricci flows, and apply this to new parabolic regularizations of some natural geometric quantities.

An $ε$-Regularity Theorem for Non-collapsed Ricci Flow

TL;DR

This work advances the regularity theory of non-collapsed Ricci flows by proving an -regularity theorem that controls curvature scales under a Nash-entropy non-collapsing condition and a finite-energy, near-splitting hypothesis. It develops a robust framework—centered on -convergence, metric flows, and parabolic regularizations of soliton-type potentials—to obtain uniform convergence of heat-type observables along convergent flows and to analyze singular models arising in Fano and Kähler-Ricci settings. The authors deduce sharp Minkowski-type estimates for the singular set, showing parabolic codimension control, and apply these results to singular Kähler-Ricci solitons, broadening the applicability beyond previously known Kähler-specific arguments. Collectively, the results provide finer regularity and dimensional information about singularities in Ricci flows, enabling more precise stratifications and potential generalizations to broader geometric flows.

Abstract

In this article we prove an -regularity theorem for non-collapsed Ricci flows, and use this to prove new estimates for singularity models of Fano Kähler-Ricci flows. In the course of our proof, we find a criterion for uniform convergence of solutions to the heat equation along a sequence of -converging Ricci flows, and apply this to new parabolic regularizations of some natural geometric quantities.

Paper Structure

This paper contains 11 sections, 21 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. Notation
  4. Setup and Basic Concepts
  5. Integral Almost Properties
  6. Metric Flows and the $\mathbb{F}$-Topology
  7. Chern-Simons Invariants and Chern-Gauss-Bonnet
  8. Strong $(k,\delta,r)$-Soliton Potentials
  9. Uniform Convergence of Almost-Soliton Potentials
  10. Proof of Theorem \ref{['thm:epsreg']}
  11. Minkowski Estimates for the Singular Set

Key Result

Theorem 1.1

For any $Y<\infty$, there exists $\epsilon_0=\epsilon_0(Y)>0$ such that the following holds. Let $(M^n,(g_t)_{t\in I})$ be a closed Kähler Ricci flow of (real) dimension $n$ and $(x_0,t_0)\in M\times I$, $r>0$ satisfy $\mathcal{N}_{x_0,t_0}(r^2)\geq -Y$. If $(x_0,t_0)$ is $(n-4,\epsilon_0,r)$-split then $r_{\operatorname{Rm}}(x_0,t_0)\geq \epsilon_0 r$. If $n=4$, this also holds without the Kähle

Theorems & Definitions (80)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 1.4
  • Remark 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Theorem 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 70 more