On the Gromov hyperbolicity of the minimal metric

Abstract

In this paper we study the hyperbolicity in the sense of Gromov of domains in ${\mathbb{R}}^d$ $(d\ge 3)$ with respect to the minimal metric introduced by Forstnerič and Kalaj. 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 non-trivial conformal harmonic disks. Finally, we study the relation between the minimal metric and the Hilbert metric in convex domains.

References in corpus (5)

• Schwarz-Pick lemma for harmonic maps which are conformal at a point
• The horofunction boundary of a Gromov hyperbolic space
• Hyperbolic domains in real Euclidean spaces
• Backward dynamics of non-expanding maps in Gromov hyperbolic metric spaces
• Worm domains are not Gromov hyperbolic