The Gehring-Hayman type theorems on complex domains

In this paper we establish Gehring-Hayman type theorems for some complex domains. Suppose that $\mathrm{\Omega }\subset {\mathbb{C}}^n$ 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 $\mathrm{\Omega }$ is less than $c_1\mathrm{\mid }x-y{\mathrm{\mid }}^{c_2}$. Furthermore, if $\mathrm{\Omega }$ 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.