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 $\Omega \subset \mathbb{C}^{d}$ is a bounded strongly convex domain with $\mathcal{C}^3$ boundary and $\gamma_{1}, \gamma_{2}:[0, \infty) \to \Omega$ are two geodesic rays such that $\gamma_{1}(\infty)=\gamma_{2}(\infty)=\xi \in \partial \Omega$. Then if the images of $\gamma_{1}$ and $\gamma_{2}$ are contained in the same complex geodesic, then there exists $T\in \mathbb{R}$ \[ \lim_{t \to \infty} \frac{1}{t} \log K_{\Omega}\big(\gamma_{1}(t), \gamma_{2}(t+T)\big) = -2, \] otherwise \[ \lim_{t \to \infty} \frac{1}{t} \log K_{\Omega}\big(\gamma_{1}(t), \gamma_{2}(t+T)\big) = -1. \] 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 $\alpha>0$ there exists $\epsilon(d,\alpha)>0$ such that the following holds: if $\Omega \subset \mathbb{C}^d$ is a bounded convex domain with $\mathcal{C}^{2,\alpha}$-boundary and \[ T_{\Omega}^{D}(z)\geq 1-\epsilon \] outside a compact subset of $\Omega$, where $D \Subset \mathbb{C}^{d}$ is a balanced strongly convex domain with $\mathcal{C}^{3}$ boundary and $T_{\Omega}^{D}$ is the squeezing function of $\Omega$ with respect to the domain $D$ then $\Omega$ 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.