Constant Mean Curvature Surfaces in Homogeneous Manifolds

Julia Plehnert

ISBN 978-3-8325-3206-2
93 pages, year of publication: 2012
price: 35.00 €
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 H surfaces in ∑(κ) × R with H ∈ [0,1/2] and κ + 4H² ≤ 0, has genus 0,2k ends and k-fold dihedral symmetry, k ≥ 2. 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 1/2 surface in H² ∈ R with k ends, genus 1 and k-fold dihedral symmetry, k ≥ 3. 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=1/2 all surfaces are Alexandrov-embedded.