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Weak Limits of Riemannian Metrics in Surfaces with integral Curvature Bound

In this paper, we study the weak compactness of the set of conformal metrics in any Riemann surface without boundary whose Calabi energy and area are uniformly bounded. We prove that for any sequence of such metrics, there alwasy exists a subsequence which converges in H\sp{2,2}_\sb{loc} everywhere except a finite number of bubble points. Blowup analysis near bubble shows that the bubble on bubble phenomenon occurs. The limit metric gives rise to a tree structure decomposition, where each node in the tree represents a limit metric of a subsequence at that stage while the edge of the tree structure represents the neck on the process of blowing up. We also show that the number of the nodes which have more than three edges attached is finite.