On the exponential convergence of Kobayashi geodesics in strongly convex domains

Abstract

In this paper, we have proved a quantitative version of the approaching geodesic property for certain convex domains. We have proved that that if $Ω\subset \mathbb{C}^{d}$ is a bounded strongly convex domain with $\mathcal{C}^3$ boundary and $γ_{1}, γ_{2}:[0, \infty) \to Ω$ are two geodesic rays such that $γ_{1}(\infty)=γ_{2}(\infty)=ξ\in \partial Ω$. Then if the images of $γ_{1}$ and $γ_{2}$ are contained in the same complex geodesic, then there exists $T\in \mathbb{R}$ \[ \lim_{t \to \infty} \frac{1}{t} \log K_Ω\big(γ_{1}(t), γ_{2}(t+T)\big) = -2, \] otherwise \[ \lim_{t \to \infty} \frac{1}{t} \log K_Ω\big(γ_{1}(t), γ_{2}(t+T)\big) = -1. \] 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 $Ω\subset \mathbb{C}^d$ is a bounded convex domain with $\mathcal{C}^{2,α}$-boundary and \[ T_Ω^{D}(z)\geq 1-ε\] outside a compact subset of $Ω$, where $D \Subset \mathbb{C}^{d}$ is a balanced strongly convex domain with $\mathcal{C}^{3}$ boundary and $T_Ω^{D}$ is the squeezing function of $Ω$ with respect to the domain $D$ then $Ω$ is strongly pseudoconvex. We also establish exponential convergence of a certain family of quasi-geodesics in the unit ball of $\mathbb{C}^{d}$. We further show that the study of this family of quasi-geodesics provides a useful tool that allows the exponential convergence property of geodesics to be transferred from local subdomains to the ambient domain, as well as in the reverse direction.