Remarks on the nonexistence of biharmonic maps

TL;DR

This paper proves that under certain boundedness and integrability conditions, biharmonic maps from complete Riemannian manifolds into nonpositively curved manifolds must be harmonic, extending previous nonexistence results.

Contribution

It establishes new nonexistence conditions for biharmonic maps into nonpositively curved manifolds, improving upon prior theorems by relaxing assumptions.

Findings

Biharmonic maps are harmonic under boundedness and integrability conditions.

Results extend previous nonexistence theorems for biharmonic maps.

Conditions include volume, curvature, and rank assumptions.

Abstract

In this short note we study nonexistence result of biharmonic maps from a complete Riemannian manifold into a Riemannian manifold with nonpositive sectional curvature. Assume that $\phi:(M,g)\to (N, h)$ is a biharmonic map, where $(M, g)$ is a complete Riemannian manifold and $(N,h)$ a Riemannian manifold with nonpositive sectional curvature, we will prove that $\phi$ is a harmonic map if one of the following conditions holds: (i) $|d\phi|$ is bounded in $L^q(M)$ and $ \int_M|\tau(\phi)|^pdv_g<\infty, $ for some $1\leq q\leq\infty$, $1< p<\infty$; or (ii) $Vol(M)=\infty$ and $ \int_M|\tau(\phi)|^pdv_g<\infty, $ for some $1< p<\infty$. In addition if $N$ has negative sectional curvature, we assume that $rank\phi(q)\geq2$ for some $q\in M$ and $\int_M|\tau(\phi)|^pdv_g<\infty, $ for some $1< p<\infty$. These results improve the related theorems due to Baird et al.(cf. \cite{BFO}), Nakauchi et al.(cf. \cite{NUG}), Maeta(cf. \cite{Ma}) and Luo(cf. \cite{Luo}).