Quasiconformal Variations of Green's and Neumann's Functions for Arbitrary Hyperbolic Riemann Surfaces

2009-04-07T06:57:46Z

2009-04-07T06:57:46Z

1984-03

The purpose of this paper is to show the variational formulas of potential theoretic quantities on arbitrary Riemann surfaces under quasiconformal deformations. We have already given the variational formulas for Riemann's period matrices and certain abelian differentials. There we used a notion of behavior spaces in the Hilbert space over the complex number field. In this paper, we shall use behavior spaces (Shiba's type) in the Hilbert space of first order complex differentials over the real number field. Applying the previous consideration to them, we can get a variational formula of Green's functions on arbitrary hyperbolic Riemann surfaces, which is an extension of the one by Guerrero on finite Riemann surfaces. Further we shall obtain the variational formulas for period reproducing differentials, slits mappings, Neumann's functions, Robin's constants and Bergmann kernels.

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Reprinted from the Memoirs of the Faculty of Industrial Arts, Kyoto Technical University, Science and Technology, Vol.32 (1983) pp.61-71

http://hdl.handle.net/10212/1905

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