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}.

21 pages. arXiv admin note: text overlap with arXiv:math/0612491, arXiv:math/0312318