Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Asymptotic Dirichlet problem for harmonic maps and conformal geodesics

Yoshihiko Matsumoto

TL;DR

This work formulates a holographic link between boundary conformal geodesics and harmonic maps into conformally compact Einstein fillings by solving the asymptotic Dirichlet problem for maps from the hyperbolic plane. The author constructs formal polyhomogeneous extensions of a boundary curve gamma and analyzes their tension field and second fundamental form to arbitrarily high order, showing that gamma is a conformal geodesic if and only if the ambient map achieves asymptotic total geodesicness. The analysis proceeds through precise first-, second-, and third-order coefficient computations, linking the vanishing of specific higher-order terms to the conformal geodesic equations on the boundary, and relating asymptotic isometry to the tangential component of these equations. This approach recovers, in a harmonic-map setting, the holographic content of the Fine–Herfray program while offering a streamlined route and potential generalizations to indefinite signatures and higher dimensions. The work also situates the discussion within the broader context of renormalized energy and its geometric interpretations, with renormalization details reserved to the appendix.

Abstract

The asymptotic Dirichlet problem for harmonic maps from the hyperbolic plane into conformally compact Einstein manifolds is used to give a holographic characterization of conformal geodesics on the boundary at infinity, in a way deeply inspired by a work of Fine and Herfray on renormalized area minimization.

Asymptotic Dirichlet problem for harmonic maps and conformal geodesics

TL;DR

This work formulates a holographic link between boundary conformal geodesics and harmonic maps into conformally compact Einstein fillings by solving the asymptotic Dirichlet problem for maps from the hyperbolic plane. The author constructs formal polyhomogeneous extensions of a boundary curve gamma and analyzes their tension field and second fundamental form to arbitrarily high order, showing that gamma is a conformal geodesic if and only if the ambient map achieves asymptotic total geodesicness. The analysis proceeds through precise first-, second-, and third-order coefficient computations, linking the vanishing of specific higher-order terms to the conformal geodesic equations on the boundary, and relating asymptotic isometry to the tangential component of these equations. This approach recovers, in a harmonic-map setting, the holographic content of the Fine–Herfray program while offering a streamlined route and potential generalizations to indefinite signatures and higher dimensions. The work also situates the discussion within the broader context of renormalized energy and its geometric interpretations, with renormalization details reserved to the appendix.

Abstract

The asymptotic Dirichlet problem for harmonic maps from the hyperbolic plane into conformally compact Einstein manifolds is used to give a holographic characterization of conformal geodesics on the boundary at infinity, in a way deeply inspired by a work of Fine and Herfray on renormalized area minimization.
Paper Structure (13 sections, 10 theorems, 125 equations)

This paper contains 13 sections, 10 theorems, 125 equations.

Table of Contents

  1. Conformal geodesics
  2. Harmonic maps and conformal geodesics
  3. Preliminaries on conformally compact Einstein manifolds
  4. Preliminaries on harmonic maps and outline of the proof
  5. First-order coefficients
  6. Second-order coefficients
  7. Third-order coefficients
  8. Second fundamental form
  9. Asymptotic isometricity
  10. Renormalized energy
  11. Renormalized volume and area
  12. Renormalized energy
  13. Critical points of the renormalized energy

Key Result

Theorem 1

Let $(X,g_+)$ be a conformally compact Einstein manifold of dimension $n+1\geqq 3$ with smooth conformal infinity $\partial_\infty X$. Suppose $\gamma\colon\partial_\infty\mathbb{H}^2\to\partial_\infty X$ is a smooth map having nowhere vanishing differential. (1) There is always a formal polyhomogen Also, such a formal extension $u$ of $\gamma$ always satisfies $\lvert\nabla du\rvert=O(e^{-2r})$.

Theorems & Definitions (21)

  • Theorem 1
  • Theorem 2: Fine--Herfray Fine-Herfray-22
  • Proposition 1.1
  • proof
  • Definition 1.2
  • proof : Proof of Theorem \ref{['thm:main']} (1)
  • Theorem 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • ...and 11 more