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A discrete analogue of the harmonic morphism and green kernelcomparison theorems

Published online by Cambridge University Press:  08 November 2000

Hajime Urakawa
Hajime Urakawa
Affiliation:
Mathematics Laboratories, Graduate School of Information Sciences, Tohoku University, Katahira, 2-1-1, Sendai, 980-8577, Japan. E-mail: urakawa@math.is.tokoku.ac.jp
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Abstract

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We give a discrete analogue of the harmonic morphism between two Riemannian manifolds. Roughly speaking, this is a mapping between two graphs preserving local harmonic functions. We characterize harmonic morphisms in terms of horizontal conformality. Many examples including coverings, non-complete extended p-sums and collapsings are given. Introducing the horizontal and vertical Laplacians, the Green kernel estimates are obtained for the harmonic morphism. As applications, a general and sharp estimate of the Green kernel for an infinite tree is obtained.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 42 , Issue 3 , September 2000 , pp. 319 - 334
Copyright
2000 Glasgow Mathematical Journal Trust