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 $T\in \mathbb{R}$ \[ \lim_{t \to \infty} \frac{1}{t} \log K_Ω\big(γ_{1}(t), γ_{2}(t+T)\big) = -2, \] 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 $\alpha >0$ 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 \[ T_Ω^{D}(z)\geq 1-ε\] outside a compact subset of $\Omega$, where $D⋐{\mathbb{C}}^d$ 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.

Theorem 1.10 and Theorem 1.11 are added