Variations of meromorphic differentials under quasiconformal deformations (Q801439)

Let R be an arbitrary open Riemann surface, let \(\Lambda\) be the Hilbert space of square integrable complex differentials on R with inner norm \[ <\lambda_ 1,\lambda_ 2>=Re\iint_{R}\lambda_ 1\wedge^*{\bar \lambda}_ 2\quad (\lambda_ i\in \Lambda,\quad i=1,2), \] let \(\Lambda_{e0}\) be the subspace of \(\Lambda\) which consists of closed \(\lambda\in \Lambda\) such that \(<\lambda,\omega >=0\) for any harmonic \(\omega\in \Lambda\), and let \(\Lambda_ x\) be the behavior space of Shiba's type [\textit{M. Shiba}, ibid. 11, 495-525 (1971; Zbl 0227.30022)]. The author proves the first and second variational formulas of differentials of \(\Lambda_ x+\Lambda_{e0}\) under quasiconformal deformations of R. If we take \(\Lambda_ x=\Lambda_{-1}=i\Gamma_ h\), then we obtain the first and the second variational formulas for Green's functions which gives an extension of Guerrero's result, and Robin's constants and some others.