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Constant Mean Curvature Surfaces in Homogeneous Manifolds
Julia Plehnert
ISBN 978-3-8325-3206-2
93 Seiten, Erscheinungsjahr: 2012
Preis: 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.
