Visibility of Kobayashi geodesics in convex domains and related properties

Abstract

Let D ⊂ C n be a bounded convex domain. A pair of distinct boundary points {p, q} of D has the visibility property provided there exist a compact subset K p,q ⊂ D and open neighborhoods U p of p and U q of q, such that the real geodesics for the Kobayashi metric of D which join points in U p and U q intersect K p,q . Every Gromov hyperbolic convex domain enjoys the visibility property for any couple of boundary points. The Goldilocks domains introduced by Bharali and Zimmer and the log-type domains of Liu and Wang also enjoys the visibility property.In this paper we prove that a certain estimate on the growth of the Kobayashi distance near the boundary points is a necessary condition for visibility and provide new cases where this estimate and the visibility property hold.We also exploit visibility for studying the boundary behavior of biholomorphic maps.

Local and global visibility and Gromov hyperbolicity of domains with respect to the Kobayashi distance

“…A pair {p, q} is not visible iff p, q ∈ {0} × B d (and so the conclusion of Proposition 4.1 does not hold in this example by [10,Theorem 3.3]). Indeed, if p belongs to ∂ Ω \ {z 0 = 0}, then we can choose W a neighborhood of p such that (Ω ∩W ) ∩ {z 0 = 0} = / 0, so that Ω ∩W is Gromov hyperbolic and has Lipschitz boundary, so it has the visibility property by Proposition 4.1, then it is easy to show that no geodesics from p to q can escape from all compacta of Ω.…”

“…Roughly speaking this means that geodesic lines that converges to different points in the Gromov boundary bend inside the space. However, visibility (with respect to the Euclidan boundary) has been exhibited for domains which are not Gromov hyperbolic in [3,2,10,17], and turns out to be a key notion for continuous extension of biholomorphisms and Denjoy-Wolff type theorems. In [13], this notion has been extended to embedded submanifolds of C d .…”

“…With this definition at hand, our initial discussion can be summarized (cfr. [10]) as follows: There is not a complete characterization of Gromov model domains, however, it is known that the following are Gromov model domains:…”

“…Assume Ω 1 is a Gromov model domain (which implies that (Ω 2 , K Ω 2 ) is Gromov hyperbolic). the identity map id Ω 2 : Ω 2 → Ω 2 extends continuously as a surjective continuous mapΩ 2 G → Ω 2 ifand only if F extends as a continuous surjective map fromΩ 1 to Ω 2 .If D ⊂⊂ C d is a domain, following[10], we say that a geodesic line γ : (−∞, +∞) → D is a geodesic loop in D if γ has the same cluster set Γ in D at +∞ and −∞. In such a case we say that Γ is the vertex of the geodesic loop γ.Rephrasing[10, Thm.…”

“…the identity map id Ω 2 : Ω 2 → Ω 2 extends continuously as a surjective continuous mapΩ 2 G → Ω 2 ifand only if F extends as a continuous surjective map fromΩ 1 to Ω 2 .If D ⊂⊂ C d is a domain, following[10], we say that a geodesic line γ : (−∞, +∞) → D is a geodesic loop in D if γ has the same cluster set Γ in D at +∞ and −∞. In such a case we say that Γ is the vertex of the geodesic loop γ.Rephrasing[10, Thm. 3.3] we have that a Gromov hyperbolic, visible domain is a Gromov model domain if and only if it has no geodesic loops.…”