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Hyperbolic domains in real Euclidean spaces

Barbara Drinovec Drnovsek, Franc Forstneric

Abstract

The second named author and David Kalaj introduced a pseudometric on any domain in the real Euclidean space $\mathbb R^n$, $n\ge 3$, defined in terms of conformal harmonic discs, by analogy with Kobayashi's pseudometric on complex manifolds, which is defined in terms of holomorphic discs. They showed that on the unit ball of $\mathbb R^n$, the minimal metric coincides with the classical Beltrami-Cayley-Klein metric, one of the models of hyperbolic geometry. In the present paper we investigate properties of the minimal pseudometric and give sufficient conditions for a domain to be (complete) hyperbolic, meaning that the minimal pseudometric is a (complete) metric. We show in particular that a convex domain is complete hyperbolic if and only if it does not contain any affine 2-planes. One of our main results is that a domain with a negative minimal plurisubharmonic exhaustion function is hyperbolic, and a bounded strongly minimally convex domain is complete hyperbolic. We also prove a localization theorem for the minimal pseudometric.

Hyperbolic domains in real Euclidean spaces

Abstract

The second named author and David Kalaj introduced a pseudometric on any domain in the real Euclidean space , , defined in terms of conformal harmonic discs, by analogy with Kobayashi's pseudometric on complex manifolds, which is defined in terms of holomorphic discs. They showed that on the unit ball of , the minimal metric coincides with the classical Beltrami-Cayley-Klein metric, one of the models of hyperbolic geometry. In the present paper we investigate properties of the minimal pseudometric and give sufficient conditions for a domain to be (complete) hyperbolic, meaning that the minimal pseudometric is a (complete) metric. We show in particular that a convex domain is complete hyperbolic if and only if it does not contain any affine 2-planes. One of our main results is that a domain with a negative minimal plurisubharmonic exhaustion function is hyperbolic, and a bounded strongly minimally convex domain is complete hyperbolic. We also prove a localization theorem for the minimal pseudometric.

Paper Structure

This paper contains 12 sections, 33 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Definition and basic properties of the minimal metric
  3. The minimal distance is given by chains of conformal harmonic discs
  4. Characterizations of hyperbolic domains
  5. Hyperbolicity of convex domains
  6. The minimal metric on $\Omega\subset\mathbb{R}^n$ and the Kobayashi metric on $\Omega\times \mathfrak{i}\mathbb{R}^n\subset\mathbb{C}^n$
  7. A pseudometric defined by minimal log-plurisubharmonic functions
  8. Localization theorems for the minimal pseudometric
  9. Bounded strongly minimally convex domains are complete hyperbolic
  10. Complete hyperbolicity of some unbounded strongly minimally convex domains
  11. Extending conformal minimal surfaces across punctures
  12. Open problems

Key Result

Proposition \oldthetheorem

A hyperbolic domain $\Omega\subset \mathbb{R}^n$ is complete hyperbolic if and only if for every point $\mathbf{p}\in \Omega$ and number $r>0$ the closed ball $\{\mathbf{x}\in \Omega: \rho_\Omega(\mathbf{p},\mathbf{x})\le r\}$ is compact.

Theorems & Definitions (73)

  • Definition \oldthetheorem
  • Example \oldthetheorem: Minimal metric on the ball
  • Proposition \oldthetheorem
  • Corollary \oldthetheorem
  • Example \oldthetheorem
  • Example \oldthetheorem
  • Theorem \oldthetheorem
  • proof
  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • ...and 63 more