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Splitting maps in Type I Ricci flows

Panagiotis Gianniotis

TL;DR

The paper addresses the structure of singularities in Type I Ricci flows by developing a parabolic analogue of harmonic almost splitting maps. It introduces entropy-pinched, sharp $(k, ) $-splitting maps adapted to selfsimilar Ricci flows, and proves that such maps persist down to smaller scales under almost selfsimilarity, provided a summability condition on entropy-pinching holds. A spectral-gap mechanism for the drift Laplacian on shrinking solitons yields Hessian decay and almost-linear growth results, which together with the geometric transformation theorem enable controlled, scale-consistent transformations of splitting maps. The work advances the understanding of the local geometry near singularities in Ricci flows and lays groundwork for a parabolic analogue of Cheeger–Colding–Naber theory in the Type I setting, with potential implications for the study of singular sets and limit spaces.

Abstract

We study the existence and small scale behaviour of almost splitting maps along a Ricci flow satisfying Type I curvature bounds. These are special solutions of the heat equation that serve as parabolic analogues of harmonic almost splitting maps, which have proven to be an indespensable tool in the study of the structure of the singular set of non-collapsed Ricci limit spaces. In this paper, motivated by the recent work of Cheeger-Jiang-Naber in the Ricci limit setting, we construct sharp splitting maps on Ricci flows that are almost selfsimilar, and then investigate their small scale behaviour. We show that, modulo linear transformations, an almost splitting map at a large scale remains a splitting map even at smaller scales, provided that the Ricci flow remains sufficiently self-similar. Allowing these linear transformations means that a priori an almost splitting map might degenerate at small scales. However, we show that under an additional summability hypothesis such degeneration doesn't occur.

Splitting maps in Type I Ricci flows

TL;DR

The paper addresses the structure of singularities in Type I Ricci flows by developing a parabolic analogue of harmonic almost splitting maps. It introduces entropy-pinched, sharp -splitting maps adapted to selfsimilar Ricci flows, and proves that such maps persist down to smaller scales under almost selfsimilarity, provided a summability condition on entropy-pinching holds. A spectral-gap mechanism for the drift Laplacian on shrinking solitons yields Hessian decay and almost-linear growth results, which together with the geometric transformation theorem enable controlled, scale-consistent transformations of splitting maps. The work advances the understanding of the local geometry near singularities in Ricci flows and lays groundwork for a parabolic analogue of Cheeger–Colding–Naber theory in the Type I setting, with potential implications for the study of singular sets and limit spaces.

Abstract

We study the existence and small scale behaviour of almost splitting maps along a Ricci flow satisfying Type I curvature bounds. These are special solutions of the heat equation that serve as parabolic analogues of harmonic almost splitting maps, which have proven to be an indespensable tool in the study of the structure of the singular set of non-collapsed Ricci limit spaces. In this paper, motivated by the recent work of Cheeger-Jiang-Naber in the Ricci limit setting, we construct sharp splitting maps on Ricci flows that are almost selfsimilar, and then investigate their small scale behaviour. We show that, modulo linear transformations, an almost splitting map at a large scale remains a splitting map even at smaller scales, provided that the Ricci flow remains sufficiently self-similar. Allowing these linear transformations means that a priori an almost splitting map might degenerate at small scales. However, we show that under an additional summability hypothesis such degeneration doesn't occur.
Paper Structure (31 sections, 53 theorems, 423 equations)

This paper contains 31 sections, 53 theorems, 423 equations.

Table of Contents

  1. Introduction
  2. Results and main ideas
  3. Structure of the paper
  4. Preliminaries
  5. The heat and conjugate heat equation
  6. Entropy and no local collapsing
  7. Gradient shrinking Ricci solitons
  8. A priori assumptions on Ricci flows
  9. Self-similar Ricci flows
  10. Convergence and selfsimilarity
  11. Almost selfsimilar Ricci flows
  12. $\mathcal{W}$-entropy drop and almost selfsimilar Ricci flows
  13. The heat equation under Ricci flow
  14. Polynomial growth estimates
  15. Concentration estimates
  16. ...and 16 more sections

Key Result

Theorem 1.1

Fix $\varepsilon>0$, $\mu\in (0,1/6)$. Let $(M^n,g(t),p)_{t\in [-10\delta^{-2},0]}$ be a smooth compact Ricci flow satisfying for every $t\in [-10\delta^{-2},0)$, and let $R\geq \frac{D'}{\mu}$, where $D'=D'(n,C_I,\Lambda)<+\infty$. Suppose that $(M,g(t),q)_{t\in(-2\delta^{-2},0)}$ is $(k,\delta^2)$-selfsimilar at scale $1$ at $q\in B(p,-1,R)$, for some $1\leq k\leq n$, and if $q\not = p$ suppose

Theorems & Definitions (118)

  • Definition 1.1: $(k,\delta)$-splitting map
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1: Theorem 1.10 in HeinNaber
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.2
  • Definition 2.3
  • Corollary 2.1
  • ...and 108 more