Conformal metrics on the unit ball: The Gehring-Hayman property and the volume growth

We continue the study of conformal metrics on the unit ball in Euclidean space. We assume that the density $\rho$ associated with the metric satisfies a Harnack inequality and then consider how much we can relax the volume growth condition from that in [Proc. London Math. Soc. Vol. 77 (3) (1998), 635-664] so that the Gehring-Hayman property still holds along the radii, i.e., if a boundary point can be accessed via a path with $\rho$-length $M<\infty$, then the $\rho$-length of the corresponding radius is bounded by $CM$. It turns out that if the path is inside a Stolz cone, then this result holds irrespective of the volume growth condition. Moreover, even if the path is not inside a Stolz cone, we are able to relax the volume growth condition for large $r$, and still conclude that the corresponding radius is $\rho$-rectifiable. This observation leads to a new estimate on the size of the boundary set corresponding to the $\rho$-unrectifiable radii.