On a relation between harmonic measure and hyperbolic distance on planar domains

Abstract

We prove that a domain in the Riemann sphere is Gromov hyperbolic if and only if it is conformally equivalent to a uniform circle domain. This resolves a conjecture of Bonk-Heinonen-Koskela and also verifies Koebe's conjecture (Kreisnormierungsproblem) for the class of Gromov hyperbolic domains. Moreover, the uniformizing conformal map from a Gromov hyperbolic domain onto a circle domain is unique up to Möbius transformations. We also undertake a careful study of the geometry of inner uniform domains in the plane and prove the above uniformization and rigidity results for such domains.

On a Property of Harmonic Measure on Simply Connected Domains

“…Proof. Property (i) is immediate by the construction of D. So, we prove properties (ii) and (iii) (for a similar calculation see [15]). The Riemann mapping theorem implies that there exists a conformal mapping ψ from D onto D such that ψ (0) = 0.…”

“…Since 0, R n and r n lie, in this order, along a hyperbolic geodesic (for more details see [15]), we have that [4, p. 14]). Combining this with (3.2) and (3.4), we deduce that Rn,rn) .…”

“…where k > 0 is a constant independent of n (see [15]). Now, taking limits in (3.7) as n → +∞, we obtain…”

On the finiteness of moments of the exit time of planar Brownian motion from comb domains

“…If µ is a discrete distribution, then Gross' construction yields a comb domain. Other examples include [26], in which a similar domain was used in order to construct a stopping time related to the winding of Brownian motion, and [17,19,18], in which similar domains were used as counterexamples to several conjectures concerning harmonic measure posed in [2,3]. Note that in Gross' paper in particular (see also [5,6,24]) the moments of the exit time are of importance, and yet it is not simple to show that they are finite for a given comb-like domain.…”

On the Hardy number of a domain in terms of harmonic measure and hyperbolic distance

“…In [19, p. 10] Poggi-Corradini proved that the Beurling-Nevanlinna projection theorem [1, p. 43-44] implies that for every α>0, Fα) and he stated the question [19, p. 36] whether the opposite inequality is also true for some positive constant. In [13] we proved that the answer is negative and only under additional assumptions involving the geometry of the domain ψ (D) it can be positive. However, the situation changes when we study integrals of the quantities stated above.…”