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Kobayashi hyperbolicity in Riemannian manifolds

Hervé Gaussier, Alexandre Sukhov

TL;DR

This work extends the Kobayashi geometry to domains in arbitrary Riemannian manifolds by using conformal harmonic discs to define and study the Kobayashi-Royden metric $F_D$. It establishes a localization principle and sharp boundary estimates that lead to complete hyperbolicity results for strictly pseudoconvex domains, and it proves a Fatou-type boundary limit theorem and a Riemannian Picard theorem for bounded or punctured-disc conformal harmonic maps. The methods fuse minimal surface theory via MPSH functions with Kobayashi metric techniques, yielding both structural results on hyperbolicity and concrete boundary-extension theorems with broad implications for geometric analysis on manifolds. Collectively, the results provide a robust framework for understanding boundary behavior, hyperbolicity, and extension phenomena of conformal harmonic maps in the Riemannian setting, along with illustrative examples of hyperbolic manifolds.

Abstract

We study the boundary behavior of the Kobayashi-Royden metric and the Kobayashi hyperbolicity of domains in Riemannian manifolds. As an application, we prove a Fatou type theorem on the existence, almost everywhere, of non tangential limits for bounded conformal harmonic immersed discs. We also prove a Picard theorem for conformal harmonic discs and give some examples of Kobayashi hyperbolic Riemannian manifolds.

Kobayashi hyperbolicity in Riemannian manifolds

TL;DR

This work extends the Kobayashi geometry to domains in arbitrary Riemannian manifolds by using conformal harmonic discs to define and study the Kobayashi-Royden metric . It establishes a localization principle and sharp boundary estimates that lead to complete hyperbolicity results for strictly pseudoconvex domains, and it proves a Fatou-type boundary limit theorem and a Riemannian Picard theorem for bounded or punctured-disc conformal harmonic maps. The methods fuse minimal surface theory via MPSH functions with Kobayashi metric techniques, yielding both structural results on hyperbolicity and concrete boundary-extension theorems with broad implications for geometric analysis on manifolds. Collectively, the results provide a robust framework for understanding boundary behavior, hyperbolicity, and extension phenomena of conformal harmonic maps in the Riemannian setting, along with illustrative examples of hyperbolic manifolds.

Abstract

We study the boundary behavior of the Kobayashi-Royden metric and the Kobayashi hyperbolicity of domains in Riemannian manifolds. As an application, we prove a Fatou type theorem on the existence, almost everywhere, of non tangential limits for bounded conformal harmonic immersed discs. We also prove a Picard theorem for conformal harmonic discs and give some examples of Kobayashi hyperbolic Riemannian manifolds.
Paper Structure (11 sections, 24 theorems, 71 equations)

This paper contains 11 sections, 24 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Minimal surfaces, conformal and harmonic maps
  4. MPSH functions
  5. Localization principle and preliminary boundary estimates
  6. Complete hyperbolicity of strictly pseudoconvex domains
  7. Theorem of Fatou
  8. Picard theorems and hyperbolicity
  9. Degeneracy locus of the Kobayashi pseudodistance
  10. The Picard theorem
  11. Some examples of hyperbolic manifolds

Key Result

Lemma 2.1

Let $p \in M$ and let $E$ be a 2-dimensional subspace of $T_pM$. Then there exists a conformal harmonic immersion $u: \mathbb D \to M$ with $u(0) = p$ and such that the tangent space $T_p u(\Delta)$ coincides with $E$. Furthermore, this immersion depends smoothly on $p$, $E$ and $g$.

Theorems & Definitions (37)

  • Lemma 2.1
  • proof
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • Theorem 3.1
  • ...and 27 more