Ideal boundaries of a Riemann surface for the equation
Author:
J. L. Schiff
Journal:
Proc. Amer. Math. Soc. 66 (1977), 5761
MSC:
Primary 30A50
MathSciNet review:
0450544
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Abstract: For a nonnegative density P on a hyperbolic Riemann surfaces R, let be the subset of the Royden harmonic boundary consisting of the nondensity points of P. This ideal boundary, as well as the Pharmonic boundary of the Pcompactification of R, have been employed in the study of energyfinite solutions of on R. We show that is homeomorphic to , where is the Psingular point. It follows that fails to characterize the space in the sense that it is possible for to be homeomorphic to , but is not canonically isomorphic to .
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 M. Glasner and R. Katz, On the behavior of at the Royden boundary, J. Analyse Math. 22 (1969), 345354. MR 0257344 (41:1995)
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 M. Glasner and M. Nakai, The roles of nondensity points, Duke Math. J. 43 (1976), 579595. MR 0412447 (54:573)
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 Y. K. Kwon and L. Sario, The Psingular point of the Pcompactification for , Bull. Amer. Math. Soc. 77 (1971), 128133. MR 0267119 (42:2021)
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 Y. K. Kwon, L. Sario and J. Schiff, Bounded energyfinite solutions of on a Riemannian manifold, Nagoya Math. J. 42 (1971), 95108. MR 0287485 (44:4689)
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 , The Pharmonic boundary and energyfinite solutions of , Nagoya Math. J. 42 (1971), 3141. MR 0288696 (44:5892)
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 M. Nakai and L. Sario, A new operator for elliptic equations and the Pcompactification for , Math. Ann. 189 (1970), 242256. MR 0279326 (43:5049)
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 L. Sario and M. Nakai, Classification theory of Riemann surfaces, SpringerVerlag, Berlin, Heidelberg and New York, 1970. MR 0264064 (41:8660)
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 J. L. Schiff, Relations between boundaries of a Riemannian manifold, Bull. Austral. Math. Soc. 6 (1972), 2530. MR 0293541 (45:2618)
 [9]
 C. Wang, Quasibounded Pharmonic functions, Doctoral Dissertation, Univ. of California, Los Angeles, 1970.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939197704505449
PII:
S 00029939(1977)04505449
Keywords:
Hyperbolic Riemann surface,
Royden harmonic boundary,
nondensity points,
Pharmonic boundary,
Psingular point,
canonical isomorphism,
energyfinite solutions of
Article copyright:
© Copyright 1977
American Mathematical Society
