Behavior of the poincaré metric near a fractal boundary

Abstract

Given a simply connected domain ω in the complex plane with boundary of Minkowski dimension α, we show that the area with respect to the Poincare metric divided by the area with respect to the quasihyperbolic metric of the super level sets of the distance function to ∂ω is asymptotically bounded by α2 as the distance decreases to zero. These bounds are established using a symmetrization argument, showing that the Poincare area of a subdomain of ω can be estimated in terms of a conformal modulus of its complement. We find a generalization of this result to locally simply connected domains and also show that the curvature of the quasi-hyperbolic metric tends to -α on the average, as we approach ∂ω. This behavior of the curvature leads us to conjecture that the bound α2 can be replaced by α.