Variational formulas on arbitrary Riemann surfaces under pinching deformation (Q1100586)

In earlier work variational formulas on Riemann surfaces for certain Abelian differentials under quasiconformal deformation have been derived using the concept of orthogonal decomposition. The process involves two steps: first showing continuity in the Dirichlet norm using orthogonality in a family of differentials and then deriving variational formulas using the orthogonality of the family to the appropriate linear operator. In an earlier paper [J. Math. Kyoto Univ. 25, 597-617 (1985; Zbl 0597.30056)] the author extended the first step to the case of deformation by pinching a finite number of loops. The purpose of the present paper is to extend the second step to this case to obtain variational formulas for entities such as period reproducers and Green's functions.