Harmonic morphisms and subharmonic functions

Gundon Choi,Gabjin Yun

doi:10.1155/ijmms.2005.383

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.

Highlights

Let (Mn, g) and (Nm, h) be complete Riemannian manifolds of dimension n and m, respectively, and let φ : M → N be a C1 map
A map φ : M → N is called harmonic if φ is a critical point of the energy functional defined by (1.1) on any compact domain D ⊂ M, or equivalently the tension field τ(φ) = trg ∇dφ ∈ Γ(φ−1TN) is identically zero, where trg and ∇ denote the trace with respect to the metric g and Levi-Civita connection on M, respectively
Yau generalized [19] the Liouville theorem to harmonic functions on Riemannian manifolds of nonnegative Ricci curvature

Summary

HARMONIC MORPHISMS AND SUBHARMONIC FUNCTIONS

GUNDON CHOI AND GABJIN YUN Received 18 July 2004 and in revised form 22 November 2004. 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, φ is a constant map. If f is a subharmonic function on N which is not harmonic and such that |df | is bounded, and if M |dφ| < ∞, φ is a constant map. We 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, there are no nonconstant surjective harmonic morphisms with finite energy.

Introduction

TpM and the horizontal space is

Thus u is a subharmonic function and

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