Constant Mean Curvature Surfaces in Homogeneous Manifolds

In this dissertation new constant mean curvature surfaces in homogeneous 3-manifolds are constructed. They arise as sister surfaces of Plateau solutions.

The first example, a two-parameter family of MC Hsurfaces in §igma( kappa) times mathbb{Rwith H in[0,1/2]and kappa+4H^2 leq0, has genus 0, 2kends and k-fold dihedral symmetry, k geq2. The existence of the minimal sister follows from the construction of a mean convex domain. The projection of the domain is non-convex.

The second example is an MC 1/2surface in mathbb{H^2 times mathbb{Rwith kends, genus 1and k-fold dihedral symmetry, k geq3. One has to solve two period problems in the construction. The first period guarantees that the surface has exactly one horizontal symmetry. For the second period the control of a horizontal mirror curve proves the dihedral symmetry.

For H=1/2all surfaces are Alexandrov-embedded.