Estimates of invariant distances on “convex” domains

Abstract

Estimates for invariant distances of convexifiable, Cconvexifiable and planar domains are given.

Estimates of the Bergman distance on Dini-smooth bounded planar domains

“…By [6,Proposition 8], the same result holds for √ 2c D and √ 2k D instead of b D . So, we have the following Corollary 2.…”

“…Combining the argument in the case n ∈ A from the previous proof and the estimates from Proposition 1, we may find an r 2 ∈ (0, r 1 ) such that if z, w ∈ E r 2 and γ : [0, 1] → D is a smooth curve for which γ(0) = 1, γ(1) = w and On the other hand, [6,Proposition 8] gives the estimates from Proposition 1 for 2k D instead of √ 2b D . Then we obtain as above lim z,w→1 z =w κ Er (z, w) κ D (z, w) = 1.…”

“…(b) One of the missing properties of b D in comparison with c D and l D is monotonicity under inclusion of (planar) domains. However, the invariants M D and K D share this property which allows us to modify the approach from [6]. (c) Results in C n in the spirit of Proposition 1 and Corollary 3 can be found in [1] and [7], respectively, where the strictly pseudoconvex domains are treated.…”

“…Recall now another comparison result between c D and k D (see [6,Proposition 9]): if D is a finitely connected bounded planar domain without isolated boundary points, 1 then…”

Bounds for invariant distances on pseudoconvex Levi corank one domains and applications

“…Partial answers to this question may be found in [1], [3] and [23], all of which deal with strongly pseudoconvex domains in C n . Optimal estimates of the boundary behaviour of invariant distances for domains with C 1+ǫ -smooth boundary in dimension one, may be found in the recent work [43] where estimates for convexifiable domains are also dealt with. A more complete treatment for strongly pseudoconvex domains in C n was given by Balogh-Bonk in [6] using the Carnot-Carathéodory metric that exists on the boundary of these domains.…”