REALIZATION OF THE IDEAL BOUNDARY OF A RI EMANN SURFACE IN CONFORMAL MAPPING

Let a noncompact Riemann surface R of positive finite genus g be given . If f : R --> R' is a conformal mapping of R into a compact Riemann s urface R' of genus g, we have a realization of the ideal boundary of R on the surface R'. We consider (for the fixed R) all the possible R' and the associated conformal mappings, and study how large the realize d boundary can be. To this aim we pass to the (common) universal space C-g of the Jacobi variety of any R' and show that the image sets of t he ideal boundary of R in C-g are uniformly bounded.