International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 3, Pages 383-391
doi:10.1155/IJMMS.2005.383
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
Abstract
Let M be a complete Riemannian manifold and N a complete noncompact Riemannian manifold. Let ϕ:M→N 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(m≥3) 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.