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Bi-Lipschitz Smoothing under Ricci and Injectivity Bounds

Maja Gwozdz

TL;DR

This paper proves that a complete Riemannian manifold with a positive uniform lower bound on injectivity radius and a positive uniform lower bound on Ricci curvature admits an $L^{\infty}$-close bi-Lipschitz smooth metric with two-sided Ricci bounds and a uniform positive lower bound on injectivity radius. The authors combine Petersen-Wei-Ye (PWY) controlled smoothing with Croke's universal local volume bound and the Cheeger-Gromov-Taylor injectivity-radius estimate, using scaling arguments to relate the original and smoothed metrics. The main result formalizes that for each dimension $n$ and bounds $(\ell,k)$ there exist constants $C,L,K$ depending only on these parameters such that $g$ can be smoothed to $h$ with $\frac{1}{C}g\le h\le Cg$, $\text{inj}(M,h)\ge L$, and $|\text{Ric}_h|_h\le K$. This addresses Morgan–Pansu’s Bandara Question 2 and provides a robust tool for geometric analysis by ensuring stable curvature bounds under smoothing while preserving a controlled metric distortion.

Abstract

We prove that a complete Riemannian manifold with a positive uniform lower bound on injectivity radius and a positive uniform lower bound on Ricci curvature admits an $L^\infty$-close (bi-Lipschitz) smooth metric with two-sided Ricci curvature bounds and a uniform positive lower bound on injectivity radius. This answers Question 2 in the Morgan--Pansu list of open problems from the conference Modern Trends in Differential Geometry (São Paulo, 2018), proposed by L. Bandara. In the proof, we rely on controlled smoothing with Croke's universal local volume lower bound and the Cheeger--Gromov--Taylor injectivity radius estimate.

Bi-Lipschitz Smoothing under Ricci and Injectivity Bounds

TL;DR

This paper proves that a complete Riemannian manifold with a positive uniform lower bound on injectivity radius and a positive uniform lower bound on Ricci curvature admits an -close bi-Lipschitz smooth metric with two-sided Ricci bounds and a uniform positive lower bound on injectivity radius. The authors combine Petersen-Wei-Ye (PWY) controlled smoothing with Croke's universal local volume bound and the Cheeger-Gromov-Taylor injectivity-radius estimate, using scaling arguments to relate the original and smoothed metrics. The main result formalizes that for each dimension and bounds there exist constants depending only on these parameters such that can be smoothed to with , , and . This addresses Morgan–Pansu’s Bandara Question 2 and provides a robust tool for geometric analysis by ensuring stable curvature bounds under smoothing while preserving a controlled metric distortion.

Abstract

We prove that a complete Riemannian manifold with a positive uniform lower bound on injectivity radius and a positive uniform lower bound on Ricci curvature admits an -close (bi-Lipschitz) smooth metric with two-sided Ricci curvature bounds and a uniform positive lower bound on injectivity radius. This answers Question 2 in the Morgan--Pansu list of open problems from the conference Modern Trends in Differential Geometry (São Paulo, 2018), proposed by L. Bandara. In the proof, we rely on controlled smoothing with Croke's universal local volume lower bound and the Cheeger--Gromov--Taylor injectivity radius estimate.
Paper Structure (2 sections, 8 theorems, 50 equations)

This paper contains 2 sections, 8 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Main result

Key Result

Theorem 1

We restate Question 2 in more detail. Let us fix an integer $n\ge 2$. For every $\ell>0$ and $k>0$, there exist constants $C,L,K>0$ (that depend only on $n,\ell,k$) with the following property. Let $(M^n,g)$ be a complete Riemannian manifold that satisfies: There exists a smooth complete Riemannian metric $h$ on $M$ with: Here $\operatorname{inj}(M,g):=\inf_{p\in M}\operatorname{inj}_g(p)$ and $

Theorems & Definitions (17)

  • Theorem 1: Smoothing under $\operatorname{inj}$ and $\operatorname{Ric}$ lower bounds
  • Lemma 1: Metric comparison
  • proof
  • Lemma 2: Constant rescaling
  • proof
  • Lemma 3: Morrey--Sobolev on a unit ball
  • proof
  • Lemma 4: Morrey estimate
  • proof
  • Lemma 5
  • ...and 7 more