Smoothing metrics on closed Riemannian manifolds through the Ricci flow

TL;DR

This paper demonstrates that under certain conditions, Riemannian metrics on closed manifolds can be smoothed to achieve sectional curvature bounds, extending previous estimates in geometric analysis.

Contribution

It introduces a new smoothing technique for metrics with Ricci curvature bounds, improving curvature regularity under Sobolev inequality assumptions.

Findings

Metrics with Ricci bounds can be smoothed to sectional bounds.

The smoothing process relies on the uniform local Sobolev inequality.

This extends previous a priori estimates in Riemannian geometry.

Abstract

Under the assumption of the uniform local Sobolev inequality, it is proved that Riemannian metrics with an absolute Ricci curvature bound and a small Riemannian curvature integral bound can be smoothed to having a sectional curvature bound. This partly extends previous a priori estimates of Ye Li (J. Geom. Anal. 17 (2007) 495-511; Advances in Mathematics 223 (2010) 1924-1957).