On convergence of holomorphic abelian differentials on the Teichmüller spaces of arbitrary Riemann surfaces (Q1068261)

In an earlier paper [J. Math. Kyoto Univ. 22, 293-305 (1982; Zbl 0495.30036)] the author investigated three types of convergence for holomorphic abelian differentials on the Teichmüller space of a compact Riemann surface. In the present paper the author extends this investigation to the Teichmüller space of an arbitrary Riemann surface. The three types of convergence are geometrical, metrical and strongly metrical. The two main results of the author are that (i) geometrical convergence implies metrical convergence for square integrable differentials and (ii) metrical or strongly metrical convergence implies geometrical convergence when additional technical hypotheses are satisfied. As applications of these main theorems the author discusses the convergence differentials associated with Green's function or a period reproducing differential.