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:

• (i) If $\int _M|\tau (\phi )|^pdv_g<\infty ,$ where $2\le p<\infty$ is a real constant and there exists at least one hyperbolic point on $N$, then $\phi$ is harmonic.

• (ii) If $\phi$ is non-degenerate in at least two directions, $\int _{B_\rho (x_0)}|\tau (\phi )|^{p+2}dv_g$, $p\ge 0$, is of at most polynomial growth of $\rho$ and there exists a constant $\epsilon >0$ such that the sectional curvature of $(N, h)$ is smaller than $-\epsilon$ everywhere, then $\phi$ is harmonic.

We also prove Liouville-type theorems on complete Riemannian manifolds and give applications of our non-existence result to bi-harmonic submersions.