Hilbert metric and quasiconformal mappings

TL;DR

This work establishes a precise link between the Hilbert metric and the visual angle metric on the unit disk by representing both through the Poincaré hyperbolic metric, yielding the fundamental identity $\tan\frac{v}{2}=\sqrt{\frac{1+m}{1-m}}\,\tanh\frac{h}{4}$. Utilizing this identity, it derives sharp distortion bounds for $K$-quasiregular mappings (and analytic functions) in terms of the Hilbert metric, with a bound of the form $\tanh\frac{h(f(a),f(b))}{4} \le D\left(\tanh\frac{h(a,b)}{4}\right)^{1/K}$. The paper also proves that Hilbert circles in the unit disk are Euclidean ellipses, computed via Gröbner bases, and explores their relation to hyperbolic circles, providing both theoretical results and practical computational tools. Collectively, these results deepen the understanding of metric distortion in planar geometric function theory and offer new tools for analyzing quasiconformal and analytic maps through the Hilbert metric.

Abstract

We prove a functional identity between the Hilbert metric and the visual angle metric in the unit disk. The proof utilizes the Poincaré hyperbolic metric in terms of which both metrics can be expressed. This identity then yields sharp distortion results for quasiregular mappings and analytic functions, expressed in terms of the Hilbert metric. We also prove that Hilbert circles are, in fact, Euclidean ellipses. The proof makes use of computer algebra methods. In particular, Gröbner bases are used.

Figures (7)

• Figure 1: $\rho_{\mathbb{B}^2}(h,k) = \rho_{\mathbb{B}^2}(l,j)$
• Figure 2:
• Figure 3:
• Figure 4: Under the inversion $\tau,$$v_{{\mathbb{B}^2}}(a,b) = v_{{\mathbb{B}^2}}(\tau(a),\tau(b))$ and $\rho_{{\mathbb{B}^2}}(a,b) = \rho_{{\mathbb{B}^2}}(\tau(a),\tau(b)).$ Observe that $\measuredangle(a, q_2, b)= \measuredangle(\tau(a),q,\tau(b))$$=v_{{\mathbb{B}^2}}(a,b)\,$ and $\tau(q_2)=q.$
• Figure 5: A Hilbert circle and a maximal inscribed hyperbolic circle and a minimal circumscribed hyperbolic circle with the same center, cf. Theorem \ref{['hilhyp0115']} for details.

Theorems & Definitions (27)

• Theorem 1.1
• Theorem 1.2
• Theorem 3.2
• proof
• Remark 3.4
• Theorem 4.2
• Theorem 4.3
• proof
• Theorem 4.5
• Theorem 4.7