Conformally invariant metrics and lack of Hölder continuity

Abstract

The modulus metric between two points in a subdomain of $\mathbb{R}^n, n\ge 2,$ is defined in terms of moduli of curve families joining the boundary of the domain with a continuum connecting the two points. This metric is one of the conformally invariant hyperbolic type metrics, which have become a standard tool in geometric function theory. We prove that the modulus metric is not Hölder continuous with respect to the hyperbolic metric.

Figures (2)

• Figure 1: \ref{['Fig01']}: The graph of $\mu_{\mathbb{B}^2}(x,0)$, its lower bound in Corollary \ref{['cor_muLowBound']} (dashed), and its lower bound in \ref{['twoDimsMubounds']} (dotted) when $\rho_{\mathbb{B}^2}(x,0)<2$, where $0<x\leq 0.75$. \ref{['Fig02']}: The graph of $\mu_{\mathbb{B}^2}(x,0)$, its lower bound in \ref{['twoDimsMubounds']} (dotted), and its lower bound in Corollary \ref{['cor_muLowBound']} (dashed) when $\rho_{\mathbb{B}^2}(x,0)\geq 2$, where $0.75<x<1$.
• Figure 2: In the Euclidean midpoint rotation, two distinct points $x,y$ in the unit disk $\mathbb{B}^2$ are rotated around their midpoint $(x+y)/2$ so that the smaller angle $\nu$ between the lines $L(x,y)$ and $L(0,(x+y)/2)$ increases from $0$ to $\pi/2$.

Theorems & Definitions (16)

• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Lemma 3.4
• proof
• Corollary 3.7
• proof
• Remark 3.8
• Theorem 3.9