The behavior of uniform domains under quasisymmetric and quasi-Möbius maps between proper metric spaces is studied. Recall that a metric space is proper if closed balls are compact.
Let \(c\geq 1\), and let \(\Omega\) be a subdomain of a metric space \((X,d)\) with nonempty boundary. A path \(\gamma\colon [0,1]\to\Omega\) is called a \(c\)-uniform curve if (1) \(\ell(\gamma) \leq c\,d(\gamma(0),\gamma(1))\), and (2) \(c\,d(\gamma(t)) \geq \min\{\ell(\gamma| [0,t]),\ell(\gamma| [t,1])\}\) for all \(t\in[0,1]\), where for \(z\in\Omega\), \(d(z)\) is the distance of \(z\) the the boundary \(\partial \Omega\). The domain \(\Omega\) is called \(c\)-uniform if every two points \(x,y\in\Omega\) can be joined by a \(c\)-uniform curve. Let \(\lambda<0\leq 1\) and \(c\geq 1\). If for any \(x\in \Omega\), and for any two points \(y_1,y_2\in B(x,\lambda d(x,\partial \Omega))\), there is a path \(\gamma\) in \(\Omega\) connecting \(y_1,y_2\) with \(\ell(\gamma)\leq c\,d(y_1,y_2)\) then the domain \(\Omega\) is called \((\lambda,c)\)-quasiconvex.
The main result states that if a \(c_1\)-uniform domain \(\Omega_1\) is quasimöbius equivalent to a \((\lambda,c_2)\)-quasiconvex domain \(\Omega_2\), then \(\Omega_2\) is \(c\)-uniform for some constant \(c\). Quantitative estimates for the constant \(c\), with some additional assumptions, are also given.
The author also proves that the uniformity is preserved in quasisymmetric maps between proper metric spaces. This result, a slight generalization to the one presented by J. Väisälä [Banach Cent. Publ. 48, 55–118 (1999; Zbl 0934.30018)], is used for proving the main results.