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Local mollification of metrics with small curvature concentration

Man-Chun Lee, Tang-Kai Lee

TL;DR

This paper proves a local smoothing result for metrics with small curvature concentration that does not rely on Ricci curvature lower bounds and is fully local. By developing localized a priori estimates for Ricci flow and constructing a local smoothing flow, the authors show that metrics with small $||\mathrm{Rm}||_{L^{n/2}}$ can be regularized on small scales, while preserving key Sobolev and entropy constants. They apply this to obtain compactness results for manifolds with bounded curvature concentration under Ahlfors $n$-regularity and bounded Sobolev constant, and they prove a topological gap: if a complete non-compact manifold satisfies a Euclidean-type Sobolev inequality, Euclidean volume growth, and small curvature concentration, then it is diffeomorphic to $\mathbb{R}^n$. The framework unifies local smoothing, compactness, and topological rigidity in a curvature-concentration setting without relying on Ricci bounds, with potential extensions to Euclidean submanifolds and broader geometric contexts.

Abstract

In this work, we establish a local smoothing result on metrics with small curvature concentration with respect to Sobolev constants and volume growth. In contrast with all previous works, we remove the Ricci curvature condition and completely localize the smoothing. As an application, we prove the compactness of the space of compact manifolds with bounded curvature concentration under Ahlfors $n$-regularity and bounded Sobolev constant. In the complete non-compact case, we show that manifolds with Euclidean type Sobolev inequality, Euclidean volume growth, and small curvature concentration are necessarily diffeomorphic to Euclidean spaces.

Local mollification of metrics with small curvature concentration

TL;DR

This paper proves a local smoothing result for metrics with small curvature concentration that does not rely on Ricci curvature lower bounds and is fully local. By developing localized a priori estimates for Ricci flow and constructing a local smoothing flow, the authors show that metrics with small can be regularized on small scales, while preserving key Sobolev and entropy constants. They apply this to obtain compactness results for manifolds with bounded curvature concentration under Ahlfors -regularity and bounded Sobolev constant, and they prove a topological gap: if a complete non-compact manifold satisfies a Euclidean-type Sobolev inequality, Euclidean volume growth, and small curvature concentration, then it is diffeomorphic to . The framework unifies local smoothing, compactness, and topological rigidity in a curvature-concentration setting without relying on Ricci bounds, with potential extensions to Euclidean submanifolds and broader geometric contexts.

Abstract

In this work, we establish a local smoothing result on metrics with small curvature concentration with respect to Sobolev constants and volume growth. In contrast with all previous works, we remove the Ricci curvature condition and completely localize the smoothing. As an application, we prove the compactness of the space of compact manifolds with bounded curvature concentration under Ahlfors -regularity and bounded Sobolev constant. In the complete non-compact case, we show that manifolds with Euclidean type Sobolev inequality, Euclidean volume growth, and small curvature concentration are necessarily diffeomorphic to Euclidean spaces.
Paper Structure (5 sections, 13 theorems, 69 equations)

This paper contains 5 sections, 13 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. A priori estimates of Ricci flow
  3. Local smoothing using Ricci flow
  4. Compactness of manifolds with bounded curvature concentration
  5. Application to topological gap theorems

Key Result

Theorem 1.1

For $n\geq 3$ and $v_0,C_s>0$, there exist $\delta_1(n,v_0,C_s),\Lambda_1(n,v_0,C_s),L(n,v_0,C_s)$, $r_1(n,v_0,C_s),$ and $\hat{T}(n,v_0,C_s)>0$, such that the following holds. Let $(M^n,g_0)$ be a manifold such that $B_{g_0}(x_0,3+\bar{r})\Subset M$ for some $(x_0,\bar{r})\in M\times [0,+\infty)$. Then there exists a smooth Ricci flow solution $g(t)$ defined on $B_{g_0}(x_0,1+\bar{r})\times [0,\

Theorems & Definitions (41)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Claim 2.6
  • ...and 31 more