On the stability of $\phi$-uniform domains

TL;DR

This paper investigates the properties and stability of $$-uniform domains, showing that removing certain points preserves their $$-uniformity, which advances understanding of their geometric structure.

Contribution

It introduces the class of $$-uniform domains defined by metric comparability and proves their stability under point removal.

Findings

$$-uniform domains are stable under removal of geometric sequences of points.

The class of $$-uniform domains remains within the same category after certain modifications.

The paper establishes foundational properties of $$-uniform domains in geometric analysis.

Abstract

We study two metrics, the quasihyperbolic metric and the distance ratio metric of a subdomain $G \subset {\mathbb R}^n$. In the sequel, we investigate a class of domains, so called $\varphi$-uniform domains, defined by the property that these two metrics are comparable with respect to a homeomorphism $\varphi$ from $[0,\infty)$ to itself. Finally, we discuss a number of stability properties of $\varphi$-uniform domains. In particular, we show that the class of $\varphi$-uniform domains is stable in the sense that removal of a geometric sequence of points from a $\varphi$-uniform domain yields a $\varphi_1$-uniform domain.