Bi-Hölder extensions of quasi-isometries on pseudoconvex domains of finite type in ${\mathbb{C}}^2$

Abstract

In this paper, we prove that the identity map for the smoothly bounded pseudoconvex domain of finite type in ${\mathbb{C}}^2$ extends to a bi-Hölder map between the Euclidean boundary and Gromov boundary. As an application, we show the bi-Hölder boundary extensions for quasi-isometries between these domains. Moreover, we get a more accurate index of the Gehring-Hayman type theorem for the bounded $m$-convex domains with Dini-smooth boundary.