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Removing scalar curvature assumption for Ricci flow smoothing

Adam Martens

TL;DR

This work removes the scalar curvature lower bound from CHL’s quantitative short-time Ricci flow framework, proving that the local CHL bounds alone (volume growth, local entropy, and curvature concentration) suffice for short-time existence with scale-invariant estimates. The core method replaces the scalar-curvature control with curvature-concentration control via a Volume Perturbation Lemma and a modified Expanding Balls Lemma, enabling a convergence-based construction of a smooth Ricci flow on noncompact manifolds. It also develops a smoothing framework: under local hypotheses, pmG limits yield a limit manifold with a Ricci flow, and a limiting distance $d_0$ and measure $\mu$ describing the initial geometry. In the global (all scales) setting, blow-down arguments produce long-time flows and rigidity results, including topological rigidity $M \cong \mathbb{R}^n$ and lower-volume-growth bounds, while posing questions about the redundancy of volume upper bounds and the rate of distance expansion. Altogether, the paper broadens Ricci flow smoothing under weaker initial bounds and strengthens compactness and rigidity results for noncompact manifolds.

Abstract

In recent work of Chan-Huang-Lee, it is shown that if a manifold enjoys uniform bounds on (a) the negative part of the scalar curvature, (b) the local entropy, and (c) volume ratios up to a fixed scale, then there exists a Ricci flow for some definite time with estimates on the solution assuming that the local curvature concentration is small enough initially (depending only on these a priori bounds). In this work, we show that the bound on scalar curvature assumption (a) is redundant. We also give some applications of this quantitative short-time existence, including a Ricci flow smoothing result for measure space limits, a Gromov-Hausdorff compactness result, and a topological and geometric rigidity result in the case that the a priori local bounds are strengthened to be global.

Removing scalar curvature assumption for Ricci flow smoothing

TL;DR

This work removes the scalar curvature lower bound from CHL’s quantitative short-time Ricci flow framework, proving that the local CHL bounds alone (volume growth, local entropy, and curvature concentration) suffice for short-time existence with scale-invariant estimates. The core method replaces the scalar-curvature control with curvature-concentration control via a Volume Perturbation Lemma and a modified Expanding Balls Lemma, enabling a convergence-based construction of a smooth Ricci flow on noncompact manifolds. It also develops a smoothing framework: under local hypotheses, pmG limits yield a limit manifold with a Ricci flow, and a limiting distance and measure describing the initial geometry. In the global (all scales) setting, blow-down arguments produce long-time flows and rigidity results, including topological rigidity and lower-volume-growth bounds, while posing questions about the redundancy of volume upper bounds and the rate of distance expansion. Altogether, the paper broadens Ricci flow smoothing under weaker initial bounds and strengthens compactness and rigidity results for noncompact manifolds.

Abstract

In recent work of Chan-Huang-Lee, it is shown that if a manifold enjoys uniform bounds on (a) the negative part of the scalar curvature, (b) the local entropy, and (c) volume ratios up to a fixed scale, then there exists a Ricci flow for some definite time with estimates on the solution assuming that the local curvature concentration is small enough initially (depending only on these a priori bounds). In this work, we show that the bound on scalar curvature assumption (a) is redundant. We also give some applications of this quantitative short-time existence, including a Ricci flow smoothing result for measure space limits, a Gromov-Hausdorff compactness result, and a topological and geometric rigidity result in the case that the a priori local bounds are strengthened to be global.
Paper Structure (8 sections, 10 theorems, 96 equations)

This paper contains 8 sections, 10 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Modified distance distortion
  3. Proof of results
  4. Proof of Theorem \ref{['shortimeexistencemain']}
  5. Proof of Theorem \ref{['smoothingcorollary']}
  6. Existence and properties of $d_0$
  7. Existence and properties of $\mu$
  8. Proof of Theorem \ref{['longtimecorollary']}

Key Result

Theorem 1.1

For all $n\geq 4,A, \tau, \mathfrak{v}_0>0$, there are $C_0, \sigma, T>0$ depending only on $n,A,\tau,\mathfrak{v}_0$ such that the following holds. Let $(M^n,g)$ be a complete noncompact manifold (not necessarily bounded curvature) such that for some $\varepsilon\leq \sigma$ and all $x\in M$, the m Then there exists a complete Ricci flow $g(t)$ on $M\times [0,T]$ with $g(0)=g$ such that for all $

Theorems & Definitions (32)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Lemma 2.1: Volume Perturbation
  • Remark 2.2
  • proof : Proof of Lemma \ref{['volumeperturbation']}
  • Lemma 2.3: Expanding Balls
  • proof
  • Claim 3.1
  • ...and 22 more