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

Abstract

Let D ⊂ C n be a smoothly bounded pseudoconvex Levi corank one domain with defining function r, i.e., the Levi form ∂∂r of the boundary ∂D has at least (n − 2) positive eigenvalues everywhere on ∂D. The main goal of this article is to obtain bounds for the Carathéodory, Kobayashi and the Bergman distance between a given pair of points p, q ∈ D in terms of parameters that reflect the Levi geometry of ∂D and the distance of these points to the boundary. Applications include an understanding of Fridman's invariant for the Kobayashi metric on Levi corank one domains, a description of the balls in the Kobayashi metric on such domains that are centered at points close to the boundary in terms of Euclidean data and the boundary behaviour of Kobayashi isometries from such domains.

Remarks on the higher dimensional Suita conjecture

“…This includes the class of all smoothly bounded pseudoconvex domains of finite type in C 2 . For Levi corank one domains, all invariant metrics are uniformly comparable -see for example [11], [12] and [2]. Combining this with the general lower bound F a D ≥ 1 obtained by B locki -Zwonek already means that for any Levi corank one domain, F k D is bounded below by a positive constant (in fact, this holds even when the Levi corank one assumption does not hold globally; see Lemma 1.3 below).…”

The Kobayashi balls of ( $${\mathbb {C}}$$ C -)convex domains

“…The weakly pseudoconvex finite type case in C 2 , as well as the convex finite type case in C n , are treated in [6,Propositions 8.8 and 8.9] as byproducts of long considerations. The strongly pseudoconvex case in C n and the weakly pseudoconvex finite type in C 2 are particular cases of the pseudoconvex Levi corank one case which is considered in [3,Theorem 1.3]. The behavior of the Kobayashi balls in all the mentioned results is given in terms of the Levi geometry of the boundary which is assumed smooth and bounded.…”

“…So there exist constants c ′ = c ′ (r, n) and c ′′ = c ′′ (r) such that Lemma 3.10], the sizes of these polydiscs are comparable (in terms of small/big constant depending on D) with the sizes of polydiscs in [3,6] arising from the Levi geometry of the boundary. Thus Theorem 1 extends [6,Propositions 8.9].…”

Analysing the Wu metric on a class of eggs in ℂ n – I

“…Now, recall from Proposition 5.1 that, on M − , the contact point of the square transforms of the Kobayashi indicatrix and the Wu ellipsoid, lies on the upper K-curve, where the Kobayashi metric is described as follows (cf. equation (7.29) of [3]):…”