Existence of global Chebyshev nets on surfaces of absolute Gaussian curvature less than 2π

Abstract : We prove the existence of a global smooth Chebyshev net on complete, simply connected surfaces when the total absolute curvature is bounded by 2π. 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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Yannick Masson, Laurent Monasse. Existence of global Chebyshev nets on surfaces of absolute Gaussian curvature less than 2π. Journal of Geometry, Springer Verlag, 2016, ⟨http://link.springer.com/article/10.1007/s00022-016-0319-1⟩. ⟨hal-01233113⟩

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