Geometry of horospheres in Kobayashi hyperbolic domains

For a Kobayashi hyperbolic domain, Abate introduced the notion of small and big horospheres of a given radius at a boundary point with a pole. In this article, we investigate which domains have the property that closed big horospheres and closed small horospheres centered at a given point and of a given radius intersect the boundary only at that point? We prove that any model-Gromov-hyperbolic domain have this property. To provide examples of non-Gromov-hyperbolic domains, we show that unbounded locally model-Gromov-hyperbolic domains and bounded, Dini-smooth, locally convex domains, that are locally visible, also have this property. Finally, using the geometry of the horospheres, we present a result about the homeomorphic extension of biholomorphisms and give an application of it.

💡 Research Summary

The paper studies the geometry of horospheres in Kobayashi hyperbolic domains, focusing on the question of when the closed small and big horospheres centered at a boundary point intersect the boundary only at that point. For a Kobayashi hyperbolic domain Ω, Abate defined the small horosphere H⁽ˢ⁾ₒ(x,R) and the big horosphere H⁽ᵇ⁾ₒ(x,R) with pole o∈Ω, center x∈∂Ω and radius R>0. The central problem (∗) asks whether for every choice of x, o and R one has
 H⁽ˢ⁾ₒ(x,R)∩∂Ω = H⁽ᵇ⁾ₒ(x,R)∩∂Ω = {x}.

The authors first prove that any model‑Gromov‑hyperbolic domain satisfies (∗). A model‑Gromov‑hyperbolic domain is a Kobayashi hyperbolic space (Ω,kΩ) that is Gromov hyperbolic and for which the identity map extends to a homeomorphism from the Gromov compactification Ωᴳ onto the end‑compactification Ωᴱ. Bharali–Zimmer showed that such domains are complete and possess the visibility property with respect to Kobayashi geodesics. Visibility means that any two distinct boundary points can be joined by geodesics that must pass through a fixed compact set. Using the uniform separation of geodesic rays converging to the same Gromov boundary point, the authors demonstrate that both small and big horospheres can meet the boundary only at their center, establishing Theorem 1.3.

The paper then investigates non‑Gromov‑hyperbolic examples. The first class consists of locally model‑Gromov‑hyperbolic domains: each boundary point has a neighbourhood that, after restriction, is a model‑Gromov‑hyperbolic domain. By a localization argument (Results 2.4–2.5) and the fact that local visibility propagates to global visibility, Theorem 1.5 shows that (∗) holds for any complete Kobayashi hyperbolic domain that is hyperbolically embedded in ℂᵈ and locally model‑Gromov‑hyperbolic.

The second class comprises bounded domains with Dini‑smooth boundary that are locally convex and visible. A domain Ω is locally convex if near each boundary point x there exist neighbourhoods Uₓ⊃Vₓ and a biholomorphism Φₓ mapping Vₓ∩Ω onto a convex set. If each such convex piece enjoys the visibility property for Kobayashi geodesics, Theorem 1.6 proves that (∗) holds. This includes many convex domains of finite D’Angelo type, as well as the non‑Gromov‑hyperbolic convex domains constructed by Zimmer and by Bharali–Zimmer.

Having identified a broad family of domains where horospheres intersect the boundary uniquely, the authors turn to an application: continuous extension of biholomorphisms. They introduce the notion of a metrically‑regular domain: there exists a pole o such that the limit
 lim_{w→ξ}