Constant mean curvature cylinders
This thesis is concerned with the problem of constructing surfaces of constant mean curvature. More specifically, we are interested in obtaining immersions of the twice punctured Riemann sphere into three dimensional Euclidean space with constant mean curvature. Constant mean curvature surfaces can be described by a Weier-strass type representation, the DPW method, in terms of holomorphic loop Lie algebra valued 1-forms. For the construction of cylinders with the DPW method, the need arises to investigate holonomy problems and we derive explicit conditions on the holomorphic loop Lie algebra valued 1-forms which ensure periodicity of the resulting immersion. This allows us to construct three new families of constant mean curvature cylinders, each of which include surfaces that possess umbilic points. The first class consists of surfaces with a closed planar geodesic on which arbitrarily many umbilics may be positioned. In the second class each surface has a closed curve of points with a common tangent plane. The third class consists of cylinders with one end asymtotic to a Delaunay surface.