Convergence of metrics under self - dual Weyl tensor and Scalar curvature bounds

TL;DR

This paper proves a compactness theorem for metrics with bounded self-dual Weyl tensor and scalar curvature, using harmonic radius estimates and blow-up analysis, with applications to the Calabi flow on complex surfaces.

Contribution

It introduces a $C^{1,eta}$ compactness theorem for such metrics, advancing understanding of geometric bounds and their implications.

Findings

Established $C^{1,eta}$ compactness under specified curvature bounds.

Used blow-up analysis to estimate harmonic radius.

Potential applications to Calabi flow on complex surfaces.

Abstract

We establish a $C^{1,\alpha}$ compactness theorem for the metrics with bounded self - dual Weyl tensor and Scalar curvature. The key step is to estimate the $C^{1,\alpha}$ harmonic radius, where we use the blow up analysis as in \cite{Anderson90}. The result is motivated by, and may be applied to the Calabi flow on complex surfaces \cite{Calabi82}.