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Long-time Ricci flow existence and topological rigidity from manifolds with pinched scale-invariant integral curvature

Albert Chau, Adam Martens

Abstract

We prove long-time existence of the Ricci flow starting from complete manifolds with bounded curvature and scale-invariant integral curvature sufficiently pinched with respect to the inverse of its Sobolev constant. Moreover, if the curvature is sub-critical $L^p$ integrable, this flow converges locally smoothly to a limiting metric $g(\infty)$ on $M$ with $(M,g(\infty))$ isometric to the standard flat $\mathbb{R}^n$, which implies topological rigidity of $M$. This generalizes work of Chen, who proved analogous results for asymptotically flat manifolds. We also prove a long-time Ricci flow existence (and likewise topological rigidity) result for unbounded curvature initial data, assuming the initial data is a locally smooth limit of bounded curvature manifolds as described above.

Long-time Ricci flow existence and topological rigidity from manifolds with pinched scale-invariant integral curvature

Abstract

We prove long-time existence of the Ricci flow starting from complete manifolds with bounded curvature and scale-invariant integral curvature sufficiently pinched with respect to the inverse of its Sobolev constant. Moreover, if the curvature is sub-critical integrable, this flow converges locally smoothly to a limiting metric on with isometric to the standard flat , which implies topological rigidity of . This generalizes work of Chen, who proved analogous results for asymptotically flat manifolds. We also prove a long-time Ricci flow existence (and likewise topological rigidity) result for unbounded curvature initial data, assuming the initial data is a locally smooth limit of bounded curvature manifolds as described above.
Paper Structure (6 sections, 20 theorems, 195 equations)

This paper contains 6 sections, 20 theorems, 195 equations.

Table of Contents

  1. Introduction
  2. Boundedness of $\int |\operatorname{Rm}|^p\,dV_t$ along the Ricci flow
  3. Sobolev Inequality along the Ricci flow
  4. Proof of Theorem \ref{['main']}
  5. Proof of Theorem \ref{['main2']}
  6. Obtaining a bound for $L(n)$

Key Result

Theorem 1.1

ChenEric For $n\geq 3$, there exists a dimensional constant $\delta(n)>0$ such that the following holds. Let $(M^n, g)$ be an asymptotically flat manifold of order $\tau>0$. Suppose that $(M^n,g)$ satisfies the Sobolev inequality for all $u\in W^{1,2}$, and the scale-invariant curvature pinching Then the complete Ricci flow $g(t)$ with initial condition $g(0)=g$ exists for all times $t\in [0,\in

Theorems & Definitions (41)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • ...and 31 more