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

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}.