On symmetries of constant mean curvature surfaces

Abstract

We start the investigation of immersions $\Psi$ of a simply connected domain $D$ into three dimensional Euclidean space $R^3$, which have constant mean curvature (CMC-immersions), and allow for a group of automorphisms of $D$ which leave the image $\Psi (D)$ invariant. On one hand, this leads to a detailed description of symmetric CMC-surfaces and the associated symmetry groups. On the other hand, it allows us to start the classification of CMC-immersions of an arbitrary, compact or noncompact Riemann surface $M$ into $R^3$ in terms of Weierstrass-type data, as introduced by Pedit, Wu, and one of the authors [D]. We use our general results to prove, that there are no CMC-tori or Delaunay surfaces in the dressing orbit of the cylinder. As an example, we apply the discussion to Smyth surfaces and to a CMC-surface with a branchpoint.

42 pages, LaTeX, no figures, revision correcting several misprints

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• Deformations of constant mean curvature surfaces preserving symmetries and the Hopf differential
• Classification of Homogeneous Willmore Surfaces in $S^N$
• Discrete surfaces of constant mean curvature via dressing
• Minimal surfaces with non-trivial topology in the three-dimensional Heisenberg group
• Investigation and application of the dressing action on surfaces of constant mean curvature