  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 $\mathbb{M}^{\kappa}_n$ of the
sectional curvature $\kappa<0$ with the dimension $n \geq 2$ and hyperbolic
fillings.
