International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 3, Pages 383-391
Harmonic morphisms and subharmonic functions
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 be a complete Riemannian manifold and a complete noncompact Riemannian manifold. Let be a surjective harmonic morphism. We prove that if admits a subharmonic
function with finite Dirichlet integral which is not harmonic, and has finite energy, then is a constant map. Similarly, if is a subharmonic function on which is not harmonic and such that is bounded, and if , then is a constant map. We also show that if 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 -harmonic morphisms, similar results hold.