On the Gromov hyperbolicity of the minimal metric

Abstract

AbstractIn this paper, we study the hyperbolicity in the sense of Gromov of domains in $$\mathbb {R}^d$$ R d $$(d\ge 3)$$ ( d ≥ 3 ) with respect to the minimal metric introduced by Forstnerič and Kalaj (Anal PDE 17(3):981–1003, 2024). In particular, we prove that every bounded strongly minimally convex domain is Gromov hyperbolic and its Gromov compactification is equivalent to its Euclidean 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.