Dubois, J. (2020). Discretization of flat fronts in hyperbolic space [Diploma Thesis, Technische Universität Wien]. reposiTUm. http://hdl.handle.net/20.500.12708/79939
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Number of Pages:
43
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Abstract:
The aim of this thesis is to investigate the discretization of flat fronts in the 3-dimensional hyperbolic space. We follow the methods and results obtained in the field of discrete differential geometry, and we also take advantage of the representation of the hyperbolic space in Lie sphere geometry. In the first part, we present the two Poincaré models and the hyperboloid model of the hyperbolic space, and introduce the fundamentals of Lie sphere geometry. In the second part, we define fronts in the hyperbolic space. For that we define the Legendrian immersions that produce these fronts. Then we focus on the flat fronts, for which we define the curvature and the hyperbolic Gauss maps. We also prove, using the Weierstrass representation, that the hyperbolic Gauss maps form a Darboux pair and that they are sufficient data to construct a flat front. In the third and last part, we discretize the flat fronts. First we introduce the discrete surfaces and discrete Legendrian immersions, and we also define a discrete Gaussian curvature. With all these elements we are able to study the discrete flat fronts. Thanks to the discrete Weierstrass representation, we prove a discrete equivalent of the above result: the hyperbolic Gauss maps of a discrete flat front form a discrete Darboux pair.