Curvature Estimates for Critical 4-Manifolds with a Lower Ricci Curvature Bound

TL;DR

This paper establishes curvature estimates for 4-manifolds with a lower Ricci curvature bound, using geometric and topological methods to achieve fixed-scale control without relying on Sobolev constants.

Contribution

It introduces a novel approach to curvature estimation on 4-manifolds that bypasses traditional elliptic regularity techniques and Sobolev constant dependence.

Findings

Curvature bounds are achieved on a fixed scale for 4-manifolds with Ricci lower bounds.

The approach avoids Sobolev constant control, relying instead on geometric and topological arguments.

Results extend understanding of regularity in critical 4-manifold geometries.

Abstract

We draw elliptic regularity results for 4-manifolds with an elliptic system, without Sobolev constant control. Direct use of analysis is circumvented; the results come mainly through geometric and topological arguments. In contrast to our previous paper, which worked predominantly on the scale of the curvature radius, the results here provide curvature controls on a fixed scale.