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N-manifolds carrying bounded but no Dirichlet finite harmonic functions

Published online by Cambridge University Press:  22 January 2016

Dennis Hada ,
Leo Sario  and
Cecilia Wang
Dennis Hada
Affiliation:
University of California, Los Angeles
Leo Sario
Affiliation:
University of California, Los Angeles
Cecilia Wang
Affiliation:
University of California, Los Angeles
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Among the most remarkable results in the theory of harmonic functions on Riemann surfaces is the strictness of the inclusion relations OG < OHB < OHD, established by Ahlfors [1,2], Royden [2,4], and Tôki [8] two decades ago. Subsequently the strictness of the relations OG < OHP < OHB was shown and a somewhat simpler proof of OHB < OHD given by Sario [5] and Tôki [9]. Here OG is the class of parabolic surfaces, and OHP, OHB, OHD stand for the classes of surfaces which do not carry nonconstant harmonic functions which are positive, bounded, or Dirichlet finite, respectively.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 54 , July 1974 , pp. 1 - 6
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1974

References

[1] Ahlfors, L., Remarks on the classification of open Riemann surfaces, Ann. Acad. Sci. Fenn. Ser. A. I. No. 87 (1951), 8 pp.Google Scholar
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[3] Hada, D., Sario, L., and Wang, C., Dirichlet finite biharmonic functions on the Poincarè N-ball, J. Reine Angew. Math, (to appear).Google Scholar
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