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Register a new userThe Deformation Space of Geodesic Triangulations and Generalized Tuttes Embedding Theorem
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We proved the contractibility of the deformation space of the geodesic triangulations on a closed surface of negative curvature. This solves an open problem proposed by Connelly et al. in 1983, in the case of hyperbolic surfaces. The main part of the proof is a generalization of Tuttes embedding theorem for closed surfaces of negative curvature.
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We prove that the deformation space of geodesic triangulations of a flat torus is homotopy equivalent to a torus. This solves an open problem proposed by Connelly et al. in 1983, in the case of flat tori. A key tool of the proof is a generalization of Tuttes embedding theorem for flat tori. When this paper is under preparation, Erickson and Lin proved a similar result, which works for all convex drawings.
The lengths of geodesics on hyperbolic surfaces satisfy intriguing equations, known as identities, relating these lengths to geometric quantities of the surface. This paper is about a large family of identities that relate lengths of closed geodesics and orthogeodesics to boundary lengths or number of cusps. These include, as particular cases, identities due to Basmajian, to McShane and to Mirzakhani and Tan-Wong-Zhang. In stark contrast to previous identities, the identities presented here include the lengths taken among all closed geodesics.
Let n>2 and let M be an orientable complete finite volume hyperbolic n-manifold with (possibly empty) geodesic boundary having Riemannian volume vol(M) and simplicial volume ||M||. A celebrated result by Gromov and Thurston states that if M has empty boundary then the ratio between vol(M) and ||M|| is equal to v_n, where v_n is the volume of the regular ideal geodesic n-simplex in hyperbolic n-space. On the contrary, Jungreis and Kuessner proved that if the boundary of M is non-empty, then such a ratio is strictly less than v_n. We prove here that for every a>0 there exists k>0 (only depending on a and n) such that if the ratio between the volume of the boundary of M and the volume of M is less than k, then the ratio between vol(M) and ||M|| is greater than v_n-a. As a consequence we show that for every a>0 there exists a compact orientable hyperbolic n-manifold M with non-empty geodesic boundary such that the ratio between vol(M) and ||M|| is greater than v_n-a. Our argument also works in the case of empty boundary, thus providing a somewhat new proof of the proportionality principle for non-compact finite-volume hyperbolic n-manifolds without boundary.
In the following article we discuss Delaunay triangulations for a point cloud on an embedded surface in $mathbb{R}^3$. We give sufficient conditions on the point cloud to show that the diagonal switch algorithm finds an embedded Delaunay triangulation.
We prove that any arithmetic hyperbolic $n$-manifold of simplest type can either be geodesically embedded into an arithmetic hyperbolic $(n+1)$-manifold or its universal $mathrm{mod}~2$ Abelian cover can.
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