Abstract

In this paper, we develop some new tools and theory that are useful in describing the geometry of properly embedded, constant mean curvature surfaces in R^3 with bounded second fundamental form. More precisely, we prove dynamics type results for the space of translational limits of such a surface. As a consequence of our main theorems, in subsequent papers we obtain rigidity results for certain properly embedded, constant mean curvature surfaces in R^3, as well as derive curvature estimates for complete, embedded, constant mean curvature surfaces in complete locally homogeneous three- manifolds.