Gromov hyperbolicity of pseudoconvex finite type domains in $${\mathbb {C}}^2$$

Abstract

In this paper we study the hyperbolicity in the sense of Gromov of domains in R d (d ≥ 3) with respect to the minimal metric introduced by Forstnerič and Kalaj in [13].In particular, we prove that every bounded strongly minimally convex domain is Gromov hyperbolic and its Gromov compactification is equivalent to its Euclindean closure. Moreover, we prove that the boundary of a Gromov hyperbolic convex domain does not contain nontrivial conformal harmonic disks. Finally, we study the relation between the minimal metric and the Hilbert metric in convex domains.

Local and global visibility and Gromov hyperbolicity of domains with respect to the Kobayashi distance

“…In [6,7,8] this point of view has been used to prove extension of biholomorphisms between Gromov hyperbolic convex domains, proving, for instance, that every convex map from the ball whose image is convex extends as a homeomorphism up to the boundary regardless the regularity of the image. In [20,21] it has been proved that Gromov hyperbolicity of convex smooth bounded domains is related to D'Angelo type finiteness of the boundary, while in [15] the same result has been proved in C 2 for pseudoconvex domains. In [9] Gromov hyperbolicity of convex domains is shown to be equivalent to the existence of a negatively pinched metric close to the boundary, giving the idea that Gromov hyperbolicity should be read only by local properties near the boundary.…”

“…(1) bounded smooth strongly pseudoconvex domains (see [1]), (2) bounded smooth convex domains of finite D'Angelo type (see [20]), (3) Gromov hyperbolic (with respect to the Kobayashi distance) convex domains (see [8]), (4) bounded smooth pseudoconvex domains of finite D'Angelo type in C 2 (see [15]), (5) bounded Gromov hyperbolic (with respect to the Kobayashi distance) C-convex domains with Lipschitz boundary (see [21] for the C 1 -smooth case and Section 4 for the Lipschitz case) (6) any domain biholomorphic to a Gromov model domain such that the biholomorphism extends as a homeomorphism up to the boundary. Theorem 1.6 allows to "localize" the previous list as follows:…”

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

“…In particular, the bounded convex domains endowed the Kobayashi metric are also geodesic spaces [2]. Ω is Gromov hyperbolic with Kobayashi metric by Fiacchi [10,Theorem 1.1].…”

“…Later Zimmer [22] and Fiacchi [10] proved the Gromov hyperbolicity for bounded convex domains in C n or pseudoconvex domain in C 2 with smooth boundary of finite type. They also showed that the Gromov boundary is homeomorphic to the Euclidean boundary which could be used to show the homeomorphic extensions for the quasi-isometric mappings.…”

Backward dynamics of non-expanding maps in Gromov hyperbolic metric spaces

“…Notice that Theorems 1.5 and 1.6 can be applied to holomorphic self-maps of bounded strongly pseudoconvex domains in C q , to smoothly bounded convex domains of finite D'Angelo type in C q , and to smoothly bounded pseudoconvex domains of finite D'Angelo type in C 2 . The Gromov compactification is equivalent to the Euclidean compactifications in all those cases (see respectively [10], [34] and [20]).…”