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Constant Mean Curvature Surfaces Based on Fundamental Quadrilaterals

Abstract

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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Acknowledgements

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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Open Access funding enabled and organized by Projekt DEAL.

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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). https://doi.org/10.1007/s11040-021-09397-z

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  • DOI: https://doi.org/10.1007/s11040-021-09397-z

Keywords

  • CMC surface
  • Flat connections
  • DPW method
  • Tesselations

Mathematics Subject Classification (2010)

  • 53A10
  • 53C42
  • 53C43