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

Abstract

We continue the study of conformal metrics on the unit ball in Euclidean space. We assume that the density ρ \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 ρ \rho -length M > ∞ M>\infty , then the ρ \rho -length of the corresponding radius is bounded by C M 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 r , and still conclude that the corresponding radius is ρ \rho -rectifiable. This observation leads to a new estimate on the size of the boundary set corresponding to the ρ \rho -unrectifiable radii.