Ideal boundaries of a Riemann surface for the equation Δ𝑢=𝑃𝑢

Schiff, J. L.

doi:10.1090/S0002-9939-1977-0450544-9

Ideal boundaries of a Riemann surface for the equation $\Delta u=Pu$
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by J. L. Schiff PDF

Proc. Amer. Math. Soc. 66 (1977), 57-61 Request permission

Abstract:

For a nonnegative density P on a hyperbolic Riemann surfaces R, let ${\Delta ^P}$ 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 ${\delta _P}$ of the P-compactification of R, have been employed in the study of energy-finite solutions of $\Delta u = Pu$ on R. We show that ${\Delta ^P}$ is homeomorphic to ${\delta _P} - \{ {s_P}\}$, where ${s_P}$ is the P-singular point. It follows that ${\delta _P}$ fails to characterize the space $PBE(R)$ in the sense that it is possible for ${\delta _P}$ to be homeomorphic to ${\delta _Q}$, but $PBE(R)$ is not canonically isomorphic to $QBE(R)$.

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