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Constant Mean Curvature Surfaces Based on Fundamental Quadrilaterals


We describe the construction of CMC surfaces with symmetries in \(\mathbb {S}^{3}\) and \(\mathbb {R}^{3}\) using a CMC quadrilateral in a fundamental tetrahedron of a tessellation of the space. The fundamental piece is constructed by the generalized Weierstrass representation using a geometric flow on the space of potentials.


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The first author is partially supported by the DFG Collaborative Research Center TRR 109 Discretization in Geometry and Dynamics. The second author is supported by the DFG grant HE 6829/3-1 of the DFG priority program SPP 2026 Geometry at Infinity. The third author is supported by the DFG Collaborative Research Center TRR 109 Discretization in Geometry and Dynamics.


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Correspondence to Sebastian Heller.

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Communicated by:Alexander P. Veselov

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Bobenko, A.I., Heller, S. & Schmitt, N. Constant Mean Curvature Surfaces Based on Fundamental Quadrilaterals. Math Phys Anal Geom 24, 37 (2021).

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  • CMC surface
  • Flat connections
  • DPW method
  • Tesselations

Mathematics Subject Classification (2010)

  • 53A10
  • 53C42
  • 53C43