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$L^1$ curvature bounds for Type I Ricci flows

Panagiotis Gianniotis, Konstantinos Leskas

TL;DR

This work establishes $L^1$-bounds for the Riemann curvature tensor along smooth closed Ricci flows with Type I curvature bounds by introducing necks of maximal symmetry and a $b$-ball decomposition that yields uniform curvature control and $(n-2)$-content estimates. The key mechanisms are entropy pinching and a quantitative splitting theorem, together with a neck-structure theorem that isolates cylindrical regions and enables a finite, multi-scale covering. These components yield a stronger curvature-radius bound and an $L^1$-type curvature bound, along with top-stratum volume estimates, significantly advancing control over singularity formation in higher-dimensional Ricci flow. The results build on and extend the cylinder-splitting framework of CJN and Jiang–Naber, with potential pathways to remove Type I assumptions in future work via neck structure and entropy methods.

Abstract

We show $L^1$-bounds of the Riemann curvature tensor on a smooth closed $n$-dimensional Ricci flow. To achieve this we introduce the notion of a neck of maximal symmetry, similar to the one in Cheeger-Jiang-Naber and Jiang-Naber and establish a decomposition result by balls with uniform curvature bounds that satisfy an appropriate $(n-2)$-content estimate.

$L^1$ curvature bounds for Type I Ricci flows

TL;DR

This work establishes -bounds for the Riemann curvature tensor along smooth closed Ricci flows with Type I curvature bounds by introducing necks of maximal symmetry and a -ball decomposition that yields uniform curvature control and -content estimates. The key mechanisms are entropy pinching and a quantitative splitting theorem, together with a neck-structure theorem that isolates cylindrical regions and enables a finite, multi-scale covering. These components yield a stronger curvature-radius bound and an -type curvature bound, along with top-stratum volume estimates, significantly advancing control over singularity formation in higher-dimensional Ricci flow. The results build on and extend the cylinder-splitting framework of CJN and Jiang–Naber, with potential pathways to remove Type I assumptions in future work via neck structure and entropy methods.

Abstract

We show -bounds of the Riemann curvature tensor on a smooth closed -dimensional Ricci flow. To achieve this we introduce the notion of a neck of maximal symmetry, similar to the one in Cheeger-Jiang-Naber and Jiang-Naber and establish a decomposition result by balls with uniform curvature bounds that satisfy an appropriate -content estimate.
Paper Structure (5 sections, 25 theorems, 182 equations)

This paper contains 5 sections, 25 theorems, 182 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quantitative splitting and regularity
  4. The decomposition theorem
  5. Applications of the decomposition theorem

Key Result

Theorem 1.1

Let $(M,g(t))_{t\in [0,T)}$ be a smooth closed Ricci flow satisfying There is a constant $C=C(n,C_I,\Lambda,\mathop{\mathrm{\mathrm{vol}}}\nolimits_{g(0)}(M),T)<+\infty$ such that for every $t\in [0,T)$.

Theorems & Definitions (55)

  • Theorem 1.1
  • Theorem 1.2: Theorem \ref{['b-ball dec']}
  • Lemma 2.1
  • proof
  • Remark 2.1
  • Lemma 2.2
  • Remark 2.2
  • Definition 2.1
  • Remark 2.3
  • Definition 2.2
  • ...and 45 more