Smoothing Riemannian Metrics with Bounded Ricci Curvatures in Dimension Four, II

TL;DR

This paper proves that 4-manifold metrics with bounded Ricci curvature and no local curvature concentration can be smoothed to metrics with bounded sectional curvature, even in collapsing scenarios, without requiring bounds on local Sobolev constants.

Contribution

It extends smoothing results for 4-manifold metrics by removing the need for local Sobolev constant bounds, enabling applications to collapsing cases.

Findings

Metrics with bounded Ricci curvature and no local concentration can be smoothed to bounded sectional curvature

The smoothing applies even in collapsing scenarios without Sobolev constant bounds

Provides a method to improve regularity of Riemannian metrics under specific curvature conditions

Abstract

This note is a continuation of the author's paper \cite{Li}. We prove that if the metric $g$ of a 4-manifold has bounded Ricci curvature and the curvature has no local concentration everywhere, then it can be smoothed to a metric with bounded sectional curvature. Here we don't assume the bound for local Sobolev constant of $g$ and hence this smoothing result can be applied to the collapsing case.