Related papers: Hyperbolic Circle Packings and Total Geodesic Curv…
We wish to draw attention to an interesting and promising interaction of two theories. On the one hand, it is the theory of \textbf{pseudo-triangulations} which was useful for implicit solution of thecarpenter's rule problem and proved…
We show that for any closed surface of genus greater than one and for any finite weighted graph filling the surface, there exists a hyperbolic metric which realizes the least Dirichlet energy harmonic embedding of the graph among a fixed…
Certain topics on polygons are extended from Euclidean to hyperbolic geometry. This first part deals with uniqueness and existence of cocyclic polygons with prescribed sidelengths. The non-Euclidean versions are more difficult due to the…
We define discrete constant mean curvature (cmc) surfaces in the three-dimensional Euclidean and Lorentz spaces in terms of sphere packings with orthogonally intersecting circles. These discrete cmc surfaces can be constructed from…
Hyperbolic curvature flow is a geometric evolution equation that in the plane can be viewed as the natural hyperbolic analogue of curve shortening flow. It was proposed by Gurtin and Podio-Guidugli (1991) to model certain wave phenomena in…
The geodesic total curvature of rectifiable spherical curves is analyzed. We extend to the case of high dimension spheres the explicit formula that holds true for curves supported into the 2-sphere. For this purpose, we take advantage of…
Given a complex of groups $G(\mathcal{Y}) = (G_\sigma, \psi_a, g_{a,b})$ where all $G_\sigma$ are relatively hyperbolic, the $\psi_a$ are inclusions of full relatively quasiconvex subgroups, and the universal cover $X$ is CAT$(0)$ and…
We investigate the geometry of hyperbolic knots and links whose diagrams have a high amount of twisting of multiple strands. We find information on volume and certain isotopy classes of geodesics for the complements of these links, based…
We study in this article the curvature of complete maximal spacelike submanifolds in pseudo-hyperbolic spaces. We show that the scalar curvature of these submanifolds is nonpositive in every signature. This gives, together with a result of…
Periodic geodesics on the modular surface correspond to periodic orbits of the geodesic flow in its unit tangent bundle $\mathrm{PSL}_2(\mathbb{Z})\backslash\mathrm{PSL}_2(\mathbb{R})$. The complement of any finite number of orbits is a…
Thurston's sphere packing on a 3-dimensional manifold is a generalization of Thusrton's circle packing on a surface, the rigidity of which has been open for many years. In this paper, we prove that Thurston's Euclidean sphere packing is…
We prove a structure theorem for ergodic homological rotation sets of homeomorphisms isotopic to the identity on a closed orientable hyperbolic surface: this set is made of a finite number of pieces that are either one-dimensional or almost…
In \cite{G3}, Glickenstein introduced the discrete conformal structures on polyhedral surfaces in an axiomatic approach from Riemannian geometry perspective. Glickenstein's discrete conformal structures include Thurston's circle packings,…
We investigate the vertex curve, that is the set of points in the hyperbolic region of a smooth surface in real 3-space at which there is a circle in the tangent plane having at least 5-point contact with the surface. The vertex curve is…
We study rotational hypersurfaces with constant Gauss-Kronecker curvature. We solve the ODE for the generating curves of such hypersurfaces and analyze several geometric properties of such hypersurfaces. In particular, we discover a class…
We investigate criteria for circle packing(CP) types of disk triangulation graphs embedded into simply connected domains in $ \mathbb{C}$. In particular, by studying combinatorial curvature and the combinatorial Gauss-Bonnet theorem…
Moduli spaces of hyperbolic surfaces with geodesic boundary components of fixed lengths may be endowed with a symplectic structure via the Weil-Petersson form. We show that, as the boundary lengths are sent to infinity, the Weil-Petersson…
This article is a first step towards the understanding of the dynamics of the horocycle flow on foliated manifolds by hyperbolic surfaces. This is motivated by a question formulated by M. Martinez and A. Verjovsky on the minimality of this…
We study the boundaries of relatively hyperbolic HHGs. Using the simplicial structure on the hierarchically hyperbolic boundary, we characterize both relative hyperbolicity and being thick of order 1 among HHGs. In the case of relatively…
We prove that a 3-dimensional hyperbolic cusp with convex polyhedral boundary is uniquely determined by the metric induced on its boundary. Furthemore, any hyperbolic metric on the torus with cone singularities of positive curvature can be…