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Relations between boundaries of a riemannian manifold

Published online by Cambridge University Press:  17 April 2009

J.L. Schiff
J.L. Schiff
Affiliation:
University of Auckland, Auckland, New Zealand.
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For a noncompact riemannian manifold R, let MP (R) be the P-algebra, and R*P the P-compactification, with the assumption that ∫RPdV = ∞. If s is the P-singular point of the P-harmonic boundary ΔP, and Δ is the harmonic boundary of Royden's compactification R*, we construct a continuous mapping π: R* → R*P such that π(Δ) = Δp or π(Δ) = ΔP – s.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 6 , Issue 1 , February 1972 , pp. 25 - 30
Copyright
Copyright © Australian Mathematical Society 1972

References

[1]Chow, Kwang-nan, “Representing measures on the Royden boundary for solutions of Δu = Pu on a Riemannian manifold”, (Doctoral dissertation, California Institute of Technology, Pasadena, 1970).CrossRefGoogle Scholar
[2]Constantinescu, Corneliu und Cornea, Aurel, Ideale Ränder Riemannscher Flächen (Ergebnisse der Mathematik und ihrer Grenzgebiete, N.F., Band 32. Springer-Verlag, Berlin, Göttingen, Heidelberg, 1963).CrossRefGoogle Scholar
[3]Glasner, Moses and Katz, Richard, “On the behavior of solutions of Δu = Pu at the Royden boundary”, J. Analyse Math. 22 (1969), 343–354.CrossRefGoogle Scholar
[4]Kwon, Y.K. and Sario, L., “The P-singular point of the P-compactification for Δu = Pu”, Bull. Amer. Math. Soc. 77 (1971), 128–133.CrossRefGoogle Scholar
[5]Kwon, Y.K., Sario, L., and Schiff, J., “The P-harmonic boundary and energy-finite solutions of Δu = Pu”, Nagoya Math. J. 42 (1971), 31–41.CrossRefGoogle Scholar
[6]Kwon, Y.K., Sario, L., and Schiff, J., “Bounded energy-finite solutions of Δu = Pu on a Riemannian manifold”, Nagoya Math. J. 42 (1971), 95–108.CrossRefGoogle Scholar
[7]Nakai, Mitsuru, “Dirichlet finite solutions of Δu = Pu, and classification of Riemann surfaces”, Bull. Amer. Math. Soc. 77 (1971), 381–385.CrossRefGoogle Scholar
[8]Nakai, Mitsuru and Sario, Leo, “A new operator for elliptic equations and the P-compactification for Δu = Pu”, Math. Ann. 189 (1970), 242–256.CrossRefGoogle Scholar
[9]Sario, L.; Nakai, M., Classification theory of Riemann surfaces (Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Band 164. Springer-Verlag, Berlin, Heidelberg, New York, 1970).CrossRefGoogle Scholar
[10]Wang, C., “Quasibounded P-harmonic functions”, (Doctoral dissertation, University of California, Los Angeles, 1970).Google Scholar