Regularity and convergence of 4-dimensional extremal Kahler metrics

We establish a regularity result for the metric on any 4-dimensional extremal K\"ahler manifold, and a weak compactness theorem on the space of such metrics. Specifically, the sectional curvature at a point is bounded when the quantity $L^2(|\Riem|)$ in a surrounding ball is sufficiently small compared to the pointwise norm of its scalar curvature. Consequently sequences of 4-dimensional extremal K\"ahler metrics with uniformly bounded Calabi energies and scalar curvature have convergent subsequences in the Gromov-Hausdorff topology. Gromov-Hausdorff limits are length spaces with the structure of Riemannian orbifolds away from finitely many point-like singularities of unknown structure.