Existence of global Chebyshev nets on surfaces of absolute Gaussian curvature less than \({{2\pi}}\)

Abstract

We prove the existence of a global smooth Chebyshev net on complete, simply connected surfaces when the total absolute curvature is bounded by \({2\pi}\). Following Samelson and Dayawansa, we look at Chebyshev nets given by a dual curve, splitting the surface into two connected half-surfaces, and a distribution of angles along it. An analogue to the Hazzidakis’ formula is used to control the angles of the net on each half-surface with the integral of the Gaussian curvature of this half-surface and the Cauchy boundary conditions. We can then prove the main result using a theorem about splitting the Gaussian curvature with a geodesic, obtained by Bonk and Lang.

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Correspondence to Yannick Masson.

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Masson, Y., Monasse, L. Existence of global Chebyshev nets on surfaces of absolute Gaussian curvature less than \({{2\pi}}\) . J. Geom. 108, 25–32 (2017). https://doi.org/10.1007/s00022-016-0319-1

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Mathematics Subject Classification

  • 53C21
  • 58G30

Keywords

  • Chebyshev nets
  • curvature bounds