Biharmonic Maps Between Conformally Compact Manifolds
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917197208616960 |
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| author | Usula, Marco |
| author_facet | Usula, Marco |
| contents | 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 $Σ\hookrightarrow N$ is a properly embedded submanifold with boundary meeting $\partial N$ transversely, then $Σ$ is biharmonic if and only if it is minimal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_13580 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Biharmonic Maps Between Conformally Compact Manifolds Usula, Marco Differential Geometry 53-XX, 58E20, 58-XX, 58J05, 35J70 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 $Σ\hookrightarrow N$ is a properly embedded submanifold with boundary meeting $\partial N$ transversely, then $Σ$ is biharmonic if and only if it is minimal. |
| title | Biharmonic Maps Between Conformally Compact Manifolds |
| topic | Differential Geometry 53-XX, 58E20, 58-XX, 58J05, 35J70 |
| url | https://arxiv.org/abs/2502.13580 |