Bi-Lipschitz Smoothing under Ricci and Injectivity Bounds

TL;DR

This paper demonstrates that under certain bounds on Ricci curvature and injectivity radius, a Riemannian manifold can be smoothly approximated by a bi-Lipschitz equivalent metric with controlled curvature and injectivity radius.

Contribution

It provides a positive answer to an open problem by establishing the existence of a smooth bi-Lipschitz metric with bounded Ricci curvature on manifolds with given geometric bounds.

Findings

Existence of smooth bi-Lipschitz metrics with Ricci bounds

Application of controlled smoothing techniques

Use of volume and injectivity radius estimates

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 (S\~ao 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.