Harmonic morphisms and subharmonic functions. (21st March 2005)

• Record Type: Journal Article
• Authors: Choi, Gundon; Yun, Gabjin
• Is Part Of: International journal of mathematics and mathematical sciences. Number 3(2005)
• Page Start: 383
• Page End: 391
• Publication Date: 2005-03-21
• Subjects: Mathematics -- Periodicals; 510.5
• DOI: 10.1155/IJMMS.2005.383
• Languages: English
• ISSNs: 0161-1712

Abstract

Abstract : LetM be a complete Riemannian manifold andN a complete noncompact Riemannian manifold. Letϕ : M → N be a surjective harmonic morphism. We prove that ifN admits a subharmonic function with finite Dirichlet integral which is not harmonic, andϕ has finite energy, thenϕ is a constant map. Similarly, iff is a subharmonic function onN which is not harmonic and such that| d f | is bounded, and if∫ M | d ϕ | < ∞, thenϕ is a constant map. We also show that ifN m ( 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. Forp -harmonic morphisms, similar results hold.