On rigidity of the ideal boundary of an infinite riemann surface

Abstract

An end D of an infinite Riemann surface R is called to be rigid if every quasiconformal self-mapping of R, homotopic to the identity on R and conformal on D, is actually equal to the identity on D. If there is a determining sequence of rigid ends of R, we say that R has a rigid ideal boundary point. The Teichmüller space of a Riemann surface having a rigid ideal boundary point contrasts with that of a finite Riemann surface. Lei R be a Riemann surface whose Fuchsian model is of the first kind. Then we show that R has a rigid ideal boundary point, either if R contains an infinite planar end, or if R is an infinitely branched two-sheeted covering of a planar region, or if R admits a Green's function.

AMS No:

• 30F25