The Gehring-Hayman type theorems on complex domains

In this paper we establish Gehring-Hayman type theorems for some complex domains. Suppose that $\Omega\subset \mathbb{C}^n$ is a bounded $m$-convex domain with Dini-smooth boundary, or a bounded strongly pseudoconvex domain with $C^2$-smooth boundary. Then we prove that the Euclidean length of Kobayashi geodesic $[x,y]$ in $\Omega$ is less than $c_1|x-y|^{c_2}$. Furthermore, if $\Omega$ endowed with the Kobayashi metric is Gromov hyperbolic, then we can generalize this result to quasi-geodesics with respect to Bergman metric, Carath\'{e}odory metric or K\"{a}hler-Einstein metric. As applications, we prove the bi-H\"{o}lder equivalence between the Euclidean boundary and the Gromov boundary. Moreover, by using this boundary correspondence, we can show some extension results for biholomorphisms, and more general rough quasi-isometries with respect to the Kobayashi metrics between the domains.