Curvature bounds for regularized riemannian metrics

TL;DR

This paper studies how smoothing Riemannian metrics affects their curvature, providing bounds under certain geometric conditions, which helps understand the stability of curvature properties during regularization.

Contribution

It establishes uniform curvature bounds for regularized metrics assuming Ricci bounds and injectivity radius, extending to metrics with bounded harmonic radius.

Findings

Uniform curvature estimates under Ricci and injectivity bounds

Extension to metrics with bounded harmonic radius

Quantitative control of curvature change during mollification

Abstract

We investigate regularization of riemannian metrics by mollification. Assuming both-sided bounds on the Ricci tensor and a lower injectivity radius bound we obtain a uniform estimate on the change of the sectional curvature. Actually, our result holds for any metric with a uniform bound on the $W^{2,p}$-harmonic radius.