Visibility of Kobayashi geodesics in convex domains and related properties

Filippo Bracci (1) , Nikolai Nikolov (2) , Pascal J. Thomas (3)

1 Università degli Studi di Roma Tor Vergata [Roma] = University of Rome Tor Vergata
2 BAS - Bulgarian Academy of Sciences
3 IMT - Institut de Mathématiques de Toulouse UMR5219

Let $D$ be a bounded convex domain in $\mathbb C^n$. A pair of distinct boundary points $\{p,q\}$ of $D$ has the visibility property provided there exist a compact subset $K_{p,q}\subset D$ 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 enjoy 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.

Domaines: Variables complexes [math.CV]

Filippo Bracci, Nikolai Nikolov, Pascal J. Thomas. Visibility of Kobayashi geodesics in convex domains and related properties. 2024. ⟨hal-03120855⟩