Harnack Inequalities for Critical 4-manifolds with a Ricci Curvature Bound

TL;DR

This paper establishes Harnack inequalities for critical 4-manifolds with Ricci curvature bounds, providing new elliptic estimates for curvature radius without relying on Sobolev constants.

Contribution

It introduces novel elliptic estimates for curvature radius in critical 4-manifolds with Ricci bounds, using blow-up techniques and geometric triviality results.

Findings

Derived elliptic estimates controlling sectional curvature

Established Harnack inequalities for critical 4-manifolds

Connected curvature radius bounds to geometric degenerations

Abstract

We study critical Riemannian 4-manifolds with a lower bound on Ricci curvature, but no a priori analytic constraints such as on Sobolev constants. We derive elliptic-type estimates for the local curvature radius, which itself controls sectional curvature. The primary method is construction of blow-ups of degenerating metrics, followed by a geometric/topological triviality result from a previous work.