Harmonic measure and hyperbolic distance in John disks

Decomposition and removability properties of John domains

“…In particular, a decomposition theorem follows as we see below in Theorem A. A Jordan domain D is a quasidisk if and only if both D and D * := C\D are John disks, and every quasidisk is a John disk (see [8]). Hence John disks can be thought of as 'one-sided quasidisks'.…”

The maximal entropy measure detects non-uniform hyperbolicity

“…This result should be compared with a result of Kim and Langmeyer [8,Theorem 2.3], that a bounded Jordan domain is a John domain if and only if its harmonic measure is doubling on the boundary. Note however that there are planar simply-connected John domains for which the harmonic measure is not doubling on the boundary [1].…”

Inner Uniform Domains, the Quasihyperbolic Metric and Weak Bloch Functions

“…The proofs are similar and here we use the inner length metric instead of the inner diameter metric. For a simply connected domain D ⊂ R 2 , the equivalence of (i) and (ii) was proved in [11,Theorem 4.1]. For a domain D ⊂ R n , n > 2, the proof of Theorem 3.6 in [12] shows that (ii) implies that every quasihyperbolic geodesic in D is a double b-cone arc and thus (ii) implies that D is John.…”