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Biharmonic Maps Between Conformally Compact Manifolds

Marco Usula

TL;DR

This work investigates biharmonic maps between conformally compact manifolds, proving a rigidity result: a simple $b$-map $u:M\to N$ with $(N,h)$ non-positively curved is biharmonic only if it is harmonic, provided $u|_{\partial M}$ is non-constant on boundary components. The proof develops a boundary-normal analysis via the $0$-calculus, introducing model simple $b$-maps $u_{p}:M_{p}\to N_{u(p)}$ and computing their tension fields and Jacobi operators, including indicial roots, to control the asymptotics. With Mazzeo’s Fredholm theory for $0$-elliptic operators and curvature-induced decay, the tension field must vanish, leading to harmonicity and, for immersions, minimality; this yields a generalized Chen conjecture for conformally compact, non-positively curved targets. The paper demonstrates that, even without energy finiteness, the boundary behavior in the conformally compact setting enforces strong rigidity for biharmonic maps, showcasing the power of $0$-calculus in nonlinear geometric PDEs. The results provide a robust rigidity framework for biharmonic submanifolds in asymptotically hyperbolic geometries with non-positive curvature.

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 $Σ\hookrightarrow N$ is a properly embedded submanifold with boundary meeting $\partial N$ transversely, then $Σ$ is biharmonic if and only if it is minimal.

Biharmonic Maps Between Conformally Compact Manifolds

TL;DR

This work investigates biharmonic maps between conformally compact manifolds, proving a rigidity result: a simple -map with non-positively curved is biharmonic only if it is harmonic, provided is non-constant on boundary components. The proof develops a boundary-normal analysis via the -calculus, introducing model simple -maps and computing their tension fields and Jacobi operators, including indicial roots, to control the asymptotics. With Mazzeo’s Fredholm theory for -elliptic operators and curvature-induced decay, the tension field must vanish, leading to harmonicity and, for immersions, minimality; this yields a generalized Chen conjecture for conformally compact, non-positively curved targets. The paper demonstrates that, even without energy finiteness, the boundary behavior in the conformally compact setting enforces strong rigidity for biharmonic maps, showcasing the power of -calculus in nonlinear geometric PDEs. The results provide a robust rigidity framework for biharmonic submanifolds in asymptotically hyperbolic geometries with non-positive curvature.

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 -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 -map between conformally compact manifolds is biharmonic, its restriction to the boundary is non-constant, and moreover is non-positively curved, then is harmonic. We do not assume any integrability condition on : in particular, is not required to have finite energy, nor is its tension field required to be in for any . Our result implies the following version of the Generalized Chen's Conjecture: if is a non-positively curved conformally compact manifold, and is a properly embedded submanifold with boundary meeting transversely, then is biharmonic if and only if it is minimal.

Paper Structure

This paper contains 14 sections, 16 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. The $0$-world
  3. A preliminary remark on polyhomogeneity
  4. Conformally compact manifolds
  5. $0$-connections
  6. $0$-differential operators
  7. Harmonic and biharmonic $b$-maps
  8. Simple $b$-maps
  9. Harmonic and biharmonic simple $b$-maps
  10. Model simple $b$-maps for $u$
  11. The model problem
  12. The model tension field
  13. The model Jacobi operator and its indicial roots
  14. Proof of the main theorem

Key Result

Theorem 1

Let $\left(M^{m+1},g\right)$ and $\left(N^{n+1},h\right)$ be conformally compact manifolds, with $\left(N,h\right)$ of non-positive sectional curvature. Let $u:M\to N$ be a simple $b$-map. If $m\geq1$, assume that $u_{|\partial M}$ is not constant on any connected component of $\partial M$. Then $u$

Theorems & Definitions (43)

  • Theorem 1
  • Corollary 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Remark 6
  • Remark 7
  • Definition 8
  • Definition 9
  • Definition 10
  • ...and 33 more