Some remarks on the Gehring-Hayman theorem

Abstract

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.

12 pages. Results improved from the first version and one more coauthor added