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.