Ideal boundaries of a Riemann surface for the equation Ξπ’=ππ’
Joel L. Schiff Β· Proceedings of the American Mathematical Society Β· 1977
For a nonnegative density P on a hyperbolic Riemann surfaces R , let Ξ P {\Delta ^P} be the subset of the Royden harmonic boundary consisting of the nondensity points of P . This ideal boundary, as well as the P -harmonic boundary Ξ΄ P {\delta _P} of the P -compactification of R , have been employed in the study of energy-finite solutions of Ξ u = P u \Delta u = Pu on R . We show that Ξ P {\Delta ^P} is homeomorphic to Ξ΄ P β { s P } {\delta _P} - \{ {s_P}\} , where s P {s_P} is the P -singular point. It follows that Ξ΄ P {\delta _P} fails to characterize the space P B E ( R ) PBE(R) in the sense that it is possible for Ξ΄ P {\delta _P} to be homeomorphic to Ξ΄ Q {\delta _Q}