We prove that a family of meromorphic mappings from a bidisc to a compact complex surface, which are equicontinuous in a neighborhood of the boundary of the bidisc, has the volumes of its graphs locally uniformly bounded.

Our primary goal in this paper is to understand wether the sets of normality of families of meromorphic mappings between general complex manifolds are pseudoconvex or not. It turns out that the answer crucially depends on the type of convergence one is interested in. We examine three natural types of convergence introduced by one of us earlier and prove pseudoconvexity of sets of normality for a large class of target manifolds for the so called weak and gamma convergencies. Furthermore we determine the structure of the exceptional components of the limit of a weakly/gamma but not strongly converging sequence, they turn to be rationally connected. This observation allows to determine effectively when a weakly/gamma converging sequence fails to converge strongly. An application to the Fatou sets of meromorphic self-maps of compact complex surfaces is given. Contents 1. Introduction 1 2. Topologies on the space of meromorphic mappings 6 3. Pseudoconvexity of sets of normality 10 4. Convergence of mappings with values in projective space 13 5. Bloch-Montel type normality criterion 17 6. Behavior of volumes of graphs under weak and gamma convergence 22 7. Rational connectivity of the exceptional components of the limit 24 8. Fatou components 27 References 32

Our primary goal in this paper is to understand whether the sets of normality of families of meromorphic mappings between general complex manifolds are pseudoconvex or not. It turns out that the answer crucially depends on the type of convergence one is interested in. We examine three natural types of convergence introduced by one of us earlier and prove pseudoconvexity of sets of normality for a large class of target manifolds for the so called weak and gamma convergencies. Furthermore we determine the structure of the exceptional components of the limit of a weakly/gamma but not strongly converging sequence, they turn to be rationally connected. This observation allows to determine effectively when a weakly/gamma converging sequence fails to converge strongly. An application to the Fatou sets of meromorphic self-maps of compact complex surfaces is given.

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