  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.

Complex Variables