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Ricci-DeTurck Flow from Initial Metric with Morrey-type Integrability Condition

Man-Chun Lee, Stephen Shang Yi Liu

TL;DR

The paper develops a rough existence theory for the Ricci-DeTurck flow from metrics $g_0$ with Morrey-type integrability, establishing short-time existence with quantitative estimates and convergence to $g_0$. It uses parabolic bootstrapping and heat-kernel techniques to obtain interior gradient and higher-derivative bounds, even in the presence of a controlled singular set $\Sigma$. A key contribution is showing that distributional lower bounds on scalar curvature are preserved under the flow when $g_0$ satisfies Morrey-type regularity, leading to a removable singularity-type conclusion in the compact case. The work generalizes prior results (notably Jiang-Sheng-Zhang) and provides tools for smoothing rough metrics while maintaining scalar curvature rigidity, with implications for geometric analysis under low regularity.

Abstract

In this work, we study the short-time existence theory of Ricci-DeTurck flow starting from rough metrics which satisfy a Morrey-type integrability condition. Using the rough existence theory, we show the preservation and improvement of distributional scalar curvature lower bounds provided the singular set for such metrics is not too large. As an application, we use the Ricci flow smoothing to study the removable singularity for scalar curvature rigidity in the compact case under Morrey regularity conditions. Our result supplements those of Jiang-Sheng-Zhang.

Ricci-DeTurck Flow from Initial Metric with Morrey-type Integrability Condition

TL;DR

The paper develops a rough existence theory for the Ricci-DeTurck flow from metrics with Morrey-type integrability, establishing short-time existence with quantitative estimates and convergence to . It uses parabolic bootstrapping and heat-kernel techniques to obtain interior gradient and higher-derivative bounds, even in the presence of a controlled singular set . A key contribution is showing that distributional lower bounds on scalar curvature are preserved under the flow when satisfies Morrey-type regularity, leading to a removable singularity-type conclusion in the compact case. The work generalizes prior results (notably Jiang-Sheng-Zhang) and provides tools for smoothing rough metrics while maintaining scalar curvature rigidity, with implications for geometric analysis under low regularity.

Abstract

In this work, we study the short-time existence theory of Ricci-DeTurck flow starting from rough metrics which satisfy a Morrey-type integrability condition. Using the rough existence theory, we show the preservation and improvement of distributional scalar curvature lower bounds provided the singular set for such metrics is not too large. As an application, we use the Ricci flow smoothing to study the removable singularity for scalar curvature rigidity in the compact case under Morrey regularity conditions. Our result supplements those of Jiang-Sheng-Zhang.
Paper Structure (7 sections, 9 theorems, 84 equations)

This paper contains 7 sections, 9 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. a-priori estimates
  3. Parabolic boot-strapping
  4. Gradient Estimate
  5. Short-time existence
  6. Applications: preserving distributional scalar curvature lower bound
  7. Preservation of distributional lower bound

Key Result

Theorem 1.1

Suppose $(M^n, h)$ is a complete Riemannian manifold satisfying eqn:h-remark-curvature-estimates. Suppose $g_0$ is a $C^0_{loc}\cap W^{1,2}_{loc}$ Riemannian metric on $M$ and $\Sigma \subseteq M$ is a compact set so that the following holds: Then there are $T,L>0$ depending only on $n, \Lambda_0, L_0, p, \delta, r_0,h$ and a smooth solution $g(t)$ to the Ricci-DeTurck $h$-flow on $M\times(0,T]$

Theorems & Definitions (21)

  • Theorem 1.1
  • Lemma 2.2
  • proof : Sketch of Proof
  • Proposition 2.4
  • proof
  • proof : Proof of Existence in Theorem \ref{['thm:intro-application-1-statement']} when $\Sigma=\emptyset$
  • Lemma 3.1
  • proof
  • proof : Proof of Theorem \ref{['thm:intro-application-1-statement']} when $\Sigma\neq \emptyset$
  • Definition 4.1
  • ...and 11 more