On the exponential convergence of Kobayashi geodesics in strongly convex domains

In this paper, we have proved a qualitative version of the approaching geodesic property for certain convex domains. More precisely, we have proved that that if $\mathrm{\Omega }\subset {\mathbb{C}}^d$ is a bounded strongly convex domain with ${\mathcal{C}}^3$ boundary and ${\gamma }_1,{\gamma }_2:[0,\mathrm{\infty })\to \mathrm{\Omega }$ are two geodesics such that ${\gamma }_1(\mathrm{\infty })={\gamma }_2(\mathrm{\infty })=\xi \in \mathrm{\partial }\mathrm{\Omega }$. Then there exists $A({\gamma }_1(0),{\gamma }_2(0))>0$ and $T\in \mathbb{R}$ such that $$K_{\mathrm{\Omega }}({\gamma }_1(t),{\gamma }_2(t+T))\le A{e}^{-\frac{t}{2}}\mathrm{\forall }t\ge 0.$$ 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 $\alpha >2$ there exists $\epsilon (d,\alpha )>0$ such that the following holds: if $\mathrm{\Omega }\subset {\mathbb{C}}^d$ is a bounded convex domain with ${\mathcal{C}}^{2,\alpha }$-boundary and $$T_{\mathrm{\Omega }}^D(z)\ge 1-\epsilon$$ outside a compact subset of $\mathrm{\Omega }$, where $D⋐{\mathbb{C}}^d$ is a balanced strongly convex domain with ${\mathcal{C}}^3$ boundary and $T_{\mathrm{\Omega }}^D$ is the squeezing function of $\mathrm{\Omega }$ with respect to the domain $D$ then $\mathrm{\Omega }$ is strongly pseudoconvex.