Liouville-Type Theorems on Complete Manifolds and Non-Existence of Bi-Harmonic Maps

Abstract

In this note we study a non-existence result of bi-harmonic maps from a complete Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. Assume that $$\phi {:}\,(M,g)\rightarrow (N, h)$$ is a bi-harmonic map, where $$(M, g)$$ is a complete Riemannian manifold and $$(N,h)$$ a Riemannian manifold with non-positive sectional curvature. We prove: We also prove Liouville-type theorems on complete Riemannian manifolds and give applications of our non-existence result to bi-harmonic submersions.