On the exponential convergence of Kobayashi geodesics in strongly convex domains

2026/07/19 by Kingshook Biswas, Sanjoy Chatterjee, Amar Deep Sarkar
#math.CV

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Abstract

In this paper, we have proved a qualitative version of the approaching geodesic property for certain convex domains. We have proved that that if Ω⊂ ℂ^d is a bounded strongly convex domain with C³ boundary and γ₁, γ₂:[0, ∞) → Ω are two geodesics such that γ₁(∞)=γ₂(∞)=ξ∈ ∂ Ω. Then if the images of γ₁ and γ₂ are contained in the same complex geodesic, then there exists T∈ ℝ lim_{t → ∞} (1)/(t) log K_Ω(γ₁(t), γ₂(t+T)) = -2, otherwise lim_{t → ∞} (1)/(t) log K_Ω(γ₁(t), γ₂(t+T)) = -1. Furthermore, using this property we provided a characterization of strongly pseudoconvex domain via a biholomorphic invariant function namely generalized squeezing function. We have proved that: For every α>0 there exists ε(d,α)>0 such that the following holds: if Ω⊂ ℂ^d is a bounded convex domain with C^2,α-boundary and T_Ω^D(z)≥ 1-ε outside a compact subset of Ω, where D \Subset ℂ^d is a balanced strongly convex domain with C³ boundary and T_Ω^D is the squeezing function of Ω with respect to the domain D then Ω is strongly pseudoconvex.

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