Quasim\"obius invariance of uniform domains

TL;DR

This paper explores the invariance of uniform domains under quasim"obius maps in Banach spaces, establishing equivalences among geometric properties and addressing open questions in the field.

Contribution

It demonstrates the equivalence of various geometric properties of uniform domains under certain conditions and provides new methods to prove related recent results.

Findings

Equivalence of geometric properties in $ ext{psi}$-natural domains.

Partial answers to open questions by V"ais"al"a.

A new proof technique for recent quasim"obius map results.

Abstract

In this paper, we study quasim\"obius invariance of uniform domains in Banach spaces. We first investigate implications of certain geometric properties of domains in Banach spaces, such as the (diameter) uniformity, the $\delta$-uniformity and the min-max property. Then we show that all of these conditions are equivalent if the domain is $\psi$-natural. As applications, we answer partially to an open question proposed by V\"ais\"al\"a, and provide a new method to prove a recent result of M. Huang, Y. Li, M. Vuorinen, and X. Wang in [On quasim\"obius maps in real Banach spaces, Israel J. Math. 198 (2013), 467-486], which also gives an answer to another question raised by V\"ais\"al\"a.