Multi-valued graphs in embedded constant mean curvature disks

In this thesis, we discuss several results concerning constant mean curvature surfaces. We prove that an embedded constant mean curvature disk with Gaussian curvature large at a point contains a multi-valued graph around that point on the scale of |A|2. This generalizes Colding and Minicozzi's result for minimal surfaces. Moreover, we provide examples of CMC surfaces containing arbitrary large multivalued graphs.