On the deformation of Riemann surfaces and differentials by quasiconformal mappings

Abstract

On the Teichmtiller space of a compact Riemann surface, Ahlfors [2] first showed the continuity of Dirichlet norms o f Abelian differentials with prescribed A-periods. Recently, this result has been extended to some classes of open Riemann surfaces (cf. Kusunoki-Taniguchi [8], Shiga [16]). O n the other hand, M inda [12] proved that a quasiconformal mapping of Riemann surfaces induces isomorphisms between the H ilbert of square integrable differentials with specific properties, and these isomorphisms are quasiisometric (cf. Proposition 2.2). To generalize these results, we shall define here the notion of the f am ily of Hilbert spaces and investigate the variation of reproducing kernels for bounded linear functionals (Sec. 1). The subspaces of square integrable harmonic differentials and the isomorphisms induced by quasiconformal mappings whose maximal dilatations converge to one are typical examples of our deformation family. In Sec. 2, we shall prove the variational formulae of the period reproducing differentials for subspaces of square integrable harmonic differentials by using the results of Sec. 1 (e.g. Theorem 2.3). Further we shall show the continuity of norms of reproducing differentials fo r a fixed Jordan arc on a surface, which gives an extension of the author's previous result [17]. We shall use freely the concepts in Ahlfors-Sario [4] (or Kusunoki [7]), especially notations and basic facts for the square integrable differentials on Riemann surfaces. Finally, the author wishes to thank Professors S. Mori and Y. Kusunoki for helpful conversations with valuable suggestions.