Some remarks on the Gehring–Hayman theorem

Abstrakti

In this paper we provide new characterizations of the Gehring–Hayman theorem from the point of view of Gromov boundary and uniformity. We also determine the critical exponents for the uniformized space to be a uniform space in the case of the hyperbolic spaces, the model spaces ${\mathbb{M}}_n^{\kappa }$ of the sectional curvature $\kappa <0$ with the dimension $n\ge 2$ and hyperbolic fillings.