Abelian differentials with normal behavior and complex pinching deformation (Q1824725)

L. Ahlfors (1960) obtained variational formulas for the periods of normalized abelian differentials on compact Riemann surfaces. His method has been applied to a number of other classes of square integrable abelian differentials on arbitrary Riemann surfaces. The key ingredients of the method are the isomorphism induced by a quasiconformal mapping between various classes of square integrable harmonic differentials and orthogonality relations between certain classes of harmonic differentials. In a series of papers, the author has used extensions of this method to derive variational formulas for period reproducers and Green's functions on arbitrary Riemann surfaces under pinching deformations. In this paper the author employs similar techniques to derive variational formulas for the period matrices of normalized differentials with specified behavior (modeled on the behavior of certain families of square integrable harmonic differentials) outside of a compact set under a complex pinching deformation. A special instance is a deformation formula of Fay and Yamada for compact Riemann surfaces.