Biharmonic Maps Between Conformally Compact Manifolds

TL;DR

This paper proves that biharmonic maps between conformally compact manifolds with non-positive curvature are harmonic if their boundary restriction is non-constant, generalizing results related to the Chen's Conjecture.

Contribution

It establishes a non-existence result for biharmonic maps under specific boundary and curvature conditions, extending the understanding of biharmonic maps in geometric analysis.

Findings

Biharmonic maps with non-constant boundary restriction are harmonic in non-positively curved conformally compact manifolds.

The result generalizes the Chen's Conjecture to a broader class of manifolds without energy integrability assumptions.

Properly embedded submanifolds with boundary in such manifolds are biharmonic if and only if they are minimal.

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.