Biharmonic Maps Between Conformally Compact Manifolds

Abstract

We study biharmonic maps between conformally compact manifolds, a large class of complete manifolds with bounded geometry, asymptotically negative curvature, and smooth compactification. These metrics provide a far-reaching generalization of hyperbolic space. We work on the class of simple $b$-maps, i.e. maps which send interior to interior, boundary to boundary, and are transversal to the boundary of the target manifold. The main result of this paper is a non-existence result: if a simple $b$-map $u:\left(M,g\right)\to\left(N,h\right)$ between conformally compact manifolds is biharmonic, its restriction to the boundary is non-constant, and moreover $\left(N,h\right)$ is non-positively curved, then $u$ is harmonic. We do not assume any integrability condition on $u$: in particular, $u$ is not required to have finite energy, nor is its tension field required to be in $L^{p}$ for any $p$. Our result implies the following version of the Generalized Chen's Conjecture: if $\left(N,h\right)$ is a non-positively curved conformally compact manifold, and $\Sigma\hookrightarrow N$ is a properly embedded submanifold with boundary meeting $\partial N$ transversely, then $\Sigma$ is biharmonic if and only if it is minimal.