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A Maximum Principle for Bounded Harmonic Functions on Riemannian Spaces

Published online by Cambridge University Press:  20 November 2018

Y. K. Kwon  and
L. Sario
Y. K. Kwon
Affiliation:
University of California, Los Angeles, California
L. Sario
Affiliation:
University of California, Los Angeles, California
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Harmonic functions with certain boundedness properties on a given open Riemann surface R attain their maxima and minima on the harmonic boundary ΔB of R. The significance of such maximum principles lies in the fact that the classification theory of Riemann surfaces related to harmonic functions reduces to a study of topological properties of Δ(cf. [11; 8; 3; 12].

For the corresponding problem in higher dimensions we shall first show that the complement of ΔR with respect to the Royden boundary ΓR of a Riemannian N-space R is harmonically negligible: given any non-empty compact subset E of ΓR – ΔR there exists an Evans superharmonic function v, i.e., a positive continuous function on R* = R ∪ ΓR, superharmonic on R, with v = 0 on ΔR, v ≡ ∞ on E, and with a finite Dirichlet integral over R.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 4 , 01 August 1970 , pp. 847 - 854
Copyright
Copyright © Canadian Mathematical Society 1970

References

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