International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 3, Pages 383-391

Harmonic morphisms and subharmonic functions

Gundon Choi1 and Gabjin Yun2

1Global Analysis Research Center (GARC) and Department of Mathematical Sciences, Seoul National University, San 56-1, Shillim-Dong, Seoul 151-747, Korea
2Department of Mathematics, Myongji University, San 38-2, Namdong, Yongin Do 449-728, Kyunggi, Korea

Received 18 July 2004; Revised 22 November 2004

Copyright © 2005 Gundon Choi and Gabjin Yun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let M be a complete Riemannian manifold and N a complete noncompact Riemannian manifold. Let ϕ:MN be a surjective harmonic morphism. We prove that if N admits a subharmonic function with finite Dirichlet integral which is not harmonic, and ϕ has finite energy, then ϕ is a constant map. Similarly, if f is a subharmonic function on N which is not harmonic and such that |df| is bounded, and if M|dϕ|<, then ϕ is a constant map. We also show that if Nm(m3) has at least two ends of infinite volume satisfying the Sobolev inequality or positivity of the first eigenvalue of the Laplacian, then there are no nonconstant surjective harmonic morphisms with finite energy. For p-harmonic morphisms, similar results hold.