We study biharmonic maps between conformally compact manifolds, a large class of complete manifolds with bounded geometry, asymptotically negative curvature, and smooth compactification. These manifolds 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 transverse to the boundary of the target manifold. The main result of this paper is a non-existence result: if a simple b-map <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$u:\left( M,g\right) \rightarrow \left( N,h\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo>:</mo> <mfenced close=")" open="("> <mi>M</mi> <mo>,</mo> <mi>g</mi> </mfenced> <mo stretchy="false">→</mo> <mfenced close=")" open="("> <mi>N</mi> <mo>,</mo> <mi>h</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> between conformally compact manifolds is biharmonic, its restriction to the boundary is non-constant, and if <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\left( N,h\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <mi>N</mi> <mo>,</mo> <mi>h</mi> </mfenced> </math></EquationSource> </InlineEquation> is non-positively curved, then u is harmonic. We do not assume any integrability condition on u: in particular, u is not required to have finite energy, nor is its tension field required to be in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$L^{p}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation> for any p. Our result implies the following version of the Generalized Chen’s Conjecture: if <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\left( N,h\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <mi>N</mi> <mo>,</mo> <mi>h</mi> </mfenced> </math></EquationSource> </InlineEquation> is a non-positively curved conformally compact manifold, and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Sigma \hookrightarrow N$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Σ</mi> <mo stretchy="false">↪</mo> <mi>N</mi> </mrow> </math></EquationSource> </InlineEquation> is a properly embedded submanifold with boundary meeting <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\partial N$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mi>N</mi> </mrow> </math></EquationSource> </InlineEquation> transversely, then <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\Sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Σ</mi> </math></EquationSource> </InlineEquation> is biharmonic if and only if it is minimal.

Biharmonic maps between conformally compact manifolds

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发表日期：

2026年1月12日

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Marco Usula

摘要

We study biharmonic maps between conformally compact manifolds, a large class of complete manifolds with bounded geometry, asymptotically negative curvature, and smooth compactification. These manifolds 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 transverse 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) \rightarrow \left( N,h\right) $ $u : (M, g) \to (N, h)$ between conformally compact manifolds is biharmonic, its restriction to the boundary is non-constant, and if $\left( N,h\right) $ $(N, h)$ 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 $L^{p}$ $L^{p}$ for any p. Our result implies the following version of the Generalized Chen’s Conjecture: if $\left( N,h\right) $ $(N, h)$ is a non-positively curved conformally compact manifold, and $\Sigma \hookrightarrow N$ $Σ ↪ N$ is a properly embedded submanifold with boundary meeting $\partial N$ $\partial N$ transversely, then $\Sigma $ $Σ$ is biharmonic if and only if it is minimal.