Common fixed points in hyperbolic Riemann surfaces and convex domains

In this paper we prove that a commuting family of continuous self-maps of a bounded convex domain in ${\mathbb{C}^n}$ which are holomorphic in the interior has a common fixed point. The proof makes use of three basic ingredients: iteration theory of holomorphic maps, a precise description of the structure of the boundary of a convex domain, and a similar result for commuting families of self-maps of a hyperbolic domain of a compact Riemann surface.