Moduli Spaces of Embedded Constant Mean Curvature Surfaces with Few Ends   and Special Symmetry
  We give necessary conditions on complete embedded \cmc surfaces with three or four ends subject to reflection symmetries. The respective submoduli spaces are two-dimensional varieties in the moduli spaces of general \cmc surfaces. We characterize fundamental domains of our \cmc surfaces by associated great circle polygons in the three-sphere. 
K. Brauckmann and R. Kusner