Ideal boundaries of a Riemann surface for the equation
Author:
J. L. Schiff
Journal:
Proc. Amer. Math. Soc. 66 (1977), 57-61
MSC:
Primary 30A50
DOI:
https://doi.org/10.1090/S0002-9939-1977-0450544-9
MathSciNet review:
0450544
Full-text PDF
Abstract | References | Similar Articles | Additional Information
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 P-harmonic boundary
of the P-compactification of R, have been employed in the study of energy-finite solutions of
on R. We show that
is homeomorphic to
, where
is the P-singular 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
.
- [1]
M. Glasner and R. Katz, On the behavior of
at the Royden boundary, J. Analyse Math. 22 (1969), 345-354. MR 0257344 (41:1995)
- [2] M. Glasner and M. Nakai, The roles of nondensity points, Duke Math. J. 43 (1976), 579-595. MR 0412447 (54:573)
- [3]
Y. K. Kwon and L. Sario, The P-singular point of the P-compactification for
, Bull. Amer. Math. Soc. 77 (1971), 128-133. MR 0267119 (42:2021)
- [4]
Y. K. Kwon, L. Sario and J. Schiff, Bounded energy-finite solutions of
on a Riemannian manifold, Nagoya Math. J. 42 (1971), 95-108. MR 0287485 (44:4689)
- [5]
-, The P-harmonic boundary and energy-finite solutions of
, Nagoya Math. J. 42 (1971), 31-41. MR 0288696 (44:5892)
- [6]
M. Nakai and L. Sario, A new operator for elliptic equations and the P-compactification for
, Math. Ann. 189 (1970), 242-256. MR 0279326 (43:5049)
- [7] L. Sario and M. Nakai, Classification theory of Riemann surfaces, Springer-Verlag, Berlin, Heidelberg and New York, 1970. MR 0264064 (41:8660)
- [8] J. L. Schiff, Relations between boundaries of a Riemannian manifold, Bull. Austral. Math. Soc. 6 (1972), 25-30. MR 0293541 (45:2618)
- [9] C. Wang, Quasibounded P-harmonic functions, Doctoral Dissertation, Univ. of California, Los Angeles, 1970.
Retrieve articles in Proceedings of the American Mathematical Society with MSC: 30A50
Retrieve articles in all journals with MSC: 30A50
Additional Information
DOI:
https://doi.org/10.1090/S0002-9939-1977-0450544-9
Keywords:
Hyperbolic Riemann surface,
Royden harmonic boundary,
nondensity points,
P-harmonic boundary,
P-singular point,
canonical isomorphism,
energy-finite solutions of
Article copyright:
© Copyright 1977
American Mathematical Society