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Hilbert metric and Hölder continuity

Şahsene Altınkaya, Masayo Fujimura, Marcelina Mocanu, Matti Vuorinen

TL;DR

This work develops explicit formulas and relations for Hilbert's metric in the unit disk and leverages them to analyze Hölder-type continuity of quasiregular mappings from the unit disk to bounded convex domains. It connects the Hilbert metric to hyperbolic and Euclidean metrics, describes the geometry of Hilbert circles and higher-dimensional Hilbert spheres (as ellipsoids of revolution with explicit centers and axes), and establishes a sharp-looking Hölder bound for $K$-quasiregular maps in terms of the Hilbert metric. Key contributions include a precise Hölder-continuity inequality with explicit constants depending on $K$ and a geometric framework relating Hilbert geometry to classical hyperbolic geometry. The results are complemented by open problems and observations about the structure of Hilbert disks and spheres in various domains, suggesting directions for further refinement and generalization.

Abstract

We prove several formulas for the Hilbert metric in the unit disk and apply these results to study quasiregular mappings of the unit disk $\mathbb{B}^2$ onto a bounded convex domain $D$. The main result deals with the Hölder continuity of these mappings with respect to Hilbert metrics of $\mathbb{B}^2$ and $D$. Also several open problems are formulated.

Hilbert metric and Hölder continuity

TL;DR

This work develops explicit formulas and relations for Hilbert's metric in the unit disk and leverages them to analyze Hölder-type continuity of quasiregular mappings from the unit disk to bounded convex domains. It connects the Hilbert metric to hyperbolic and Euclidean metrics, describes the geometry of Hilbert circles and higher-dimensional Hilbert spheres (as ellipsoids of revolution with explicit centers and axes), and establishes a sharp-looking Hölder bound for -quasiregular maps in terms of the Hilbert metric. Key contributions include a precise Hölder-continuity inequality with explicit constants depending on and a geometric framework relating Hilbert geometry to classical hyperbolic geometry. The results are complemented by open problems and observations about the structure of Hilbert disks and spheres in various domains, suggesting directions for further refinement and generalization.

Abstract

We prove several formulas for the Hilbert metric in the unit disk and apply these results to study quasiregular mappings of the unit disk onto a bounded convex domain . The main result deals with the Hölder continuity of these mappings with respect to Hilbert metrics of and . Also several open problems are formulated.

Paper Structure

This paper contains 5 sections, 17 theorems, 130 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminary facts
  3. Hilbert metric in the unit disk
  4. Hilbert circles and spheres
  5. Hölder continuity

Key Result

Theorem 1.3

Let $f: \mathbb{B}^2 \to f(\mathbb{B}^2) = D$ be a $K$-quasiregular mapping onto a convex bounded domain $D$ and $a, b \in \mathbb{B}^2$. Then where $m$ denotes the Euclidean distance from the origin to the line $L[a, b]$, and $c(K),$$c(K) \to 1$ when $K \to 1,$ is as given in Theorem 2.12.

Figures (8)

  • Figure 1: The points $a,b, {\rm cen},p, m_{ab}$ in Lemma \ref{['lemFuji']}.
  • Figure 2: Lemma \ref{['lem:cross-ratio']} (1).
  • Figure 3: Lemma \ref{['newLem']}(2): $L[a,b]$ and $L[c,d]$ are parallel.
  • Figure 4: The point $w$ in Theorem \ref{['lemBisect']} (2) is chosen so that the circle $C(a, b,w)$ through the points $a, b,w,$ is tangent to the unit circle, see Remark \ref{['vamrmk']}. In both cases (A) and (B) $|c-u|=|d-v|$ and $|c-m|=|d-m|$ where $m=(c+d)/|c+d|\,.$
  • Figure 5: (A) Corollary \ref{['LittleThm']} (1): If $L[a,b]$ and $L[c,d]$ are parallel, then the circle $C(b_2,a_2,u)$ is tangent to the unit circle at $u$ and $\exp h_{{\mathbb B}^2}(a_2, b_2)=|a,a_2,b_2, b|= \frac{|a-c|^2}{|a-d|^2} \,.$ (B) Corollary \ref{['LittleThm']} (2): If $L[a,b]$ and $L[c,d]$ are parallel and $p \in [a,b]\,,$ then the circle $C(p, c_2,d_2)$ is tangent to the chord $[a,b]$ at the point $p$ and $|a,c_2,d_2,b|=|a,d,c,b| \,.$
  • ...and 3 more figures

Theorems & Definitions (32)

  • Theorem 1.3
  • Lemma 2.9
  • proof
  • Lemma 2.15
  • proof
  • Theorem 3.3
  • Proposition 3.5
  • proof
  • Corollary 3.6
  • proof
  • ...and 22 more