On the Gromov hyperbolicity of the minimal metric (Q6593028)

The paper discusses the Gromov hyperbolicity of domains in the Euclidean space \(\mathbb R^n\) for the minimal metric introduced by \textit{F. Forstnerič} and \textit{D. Kalaj} in [Anal. PDE 17, 981--1003 (2024; Zbl 1544.53071)]. The minimal metric for domains in \(\mathbb R^n\) is an analogue of the Kobayashi metric for domains in \(\mathbb C^n\); instead of using holomorphic maps from the unit disc, it is defined by using conformal harmonic maps. The author proves that every bounded strongly minimal convex domain (the equivalent of the pseudoconvex domain in the complex case) is Gromov hyperbolic, and its Gromov compactification is equivalent to its Euclidean closure. Furthermore, there are no nonconstant conformal harmonic discs in the boundary of a Gromov hyperbolic convex domain. The proofs are inspired by the proofs of the corresponding results for the Kobayashi metric.