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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


Ideal boundaries of a Riemann surface for the equation $ \Delta u=Pu$

Author: J. L. Schiff
Journal: Proc. Amer. Math. Soc. 66 (1977), 57-61
MSC: Primary 30A50
MathSciNet review: 0450544
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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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PII: S 0002-9939(1977)0450544-9
Keywords: Hyperbolic Riemann surface, Royden harmonic boundary, nondensity points, P-harmonic boundary, P-singular point, canonical isomorphism, energy-finite solutions of $ \Delta u = Pu$
Article copyright: © Copyright 1977 American Mathematical Society