Gauss images of hyperbolic cusps with convex polyhedral boundary

Fillastre, François; Izmestiev, Ivan

doi:10.1090/s0002-9947-2011-05325-0

Cited by 14 publications

(16 citation statements)

References 33 publications

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“…Delaunay subdivisions can be constructed for (Euclidean, spherical or hyperbolic) surfaces with cone singularities. (In the spherical case, there is a restriction on the metric; see [15, Lemma 2.11].) As a vertex set

V

, one can choose any finite set containing the cone points.…”

Section: Related Workmentioning

confidence: 99%

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Shapes of Polyhedra, Mixed Volumes and Hyperbolic Geometry

Fillastre

Izmestiev

2016

Mathematika

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Abstract. We are generalizing to higher dimensions the Bavard-Ghys construction of the hyperbolic metric on the space of polygons with fixed directions of edges.The space of convex d-dimensional polyhedra with fixed directions of facet normals has a decomposition into type cones that correspond to different combinatorial types of polyhedra. This decomposition is a subfan of the secondary fan of a vector configuration and can be analyzed with the help of Gale diagrams.We construct a family of quadratic forms on each of the type cones using the theory of mixed volumes. The Alexandrov-Fenchel inequalities ensure that these forms have exactly one positive eigenvalue. This introduces a piecewise hyperbolic structure on the space of similarity classes of polyhedra with fixed directions of facet normals. We show that some of the dihedral angles on the boundary of the resulting cone-manifold are equal to π 2 .

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V

, one can choose any finite set containing the cone points.…”

Section: Related Workmentioning

confidence: 99%

“…Weighted Delaunay triangulations of Euclidean cone‐surfaces were introduced in [7] and those of hyperbolic and spherical cone‐surfaces were introduced in [14, 15]. In the non‐Euclidean case, instead of piecewise linear extensions one uses functions of a different sort.…”

Section: Related Workmentioning

confidence: 99%

Shapes of Polyhedra, Mixed Volumes and Hyperbolic Geometry

Fillastre

Izmestiev

2016

Mathematika

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“…In some theorems of Alexandrov or Minkowski type, the functional HE, respectively Vol, happens to be concave. This is the case with the Alexandrov convex cap theorem [24] and with the Alexandrov and Minkowski-type theorems for convex hyperbolic cusps [17,18]. However, in all these cases proofs of existence of a maximum point are quite difficult.…”

Section: Existence Theoremsmentioning

confidence: 99%

Infinitesimal Rigidity of Convex Polyhedra through the Second Derivative of the Hilbert–Einstein Functional

Izmestiev¹

2014

Can. j. math.

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Abstract. The paper is centered around a new proof of the infinitesimal rigidity of convex polyhedra. The proof is based on studying derivatives of the discrete HilbertEinstein functional on the space of "warped polyhedra" with a fixed metric on the boundary.The situation is in a sense dual to using derivatives of the volume in order to prove the Gauss infinitesimal rigidity of convex polyhedra. This latter kind of rigidity is related to the Minkowski theorem on the existence and uniqueness of a polyhedron with prescribed face normals and face areas.In the spherical and in the hyperbolic-de Sitter space, there is a perfect duality between the Hilbert-Einstein functional and the volume, as well as between both kinds of rigidity.We review some of the related work and discuss directions for future research.

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“…Discrete differential geometry was used to find an embedding of a piecewise Euclidean 2-sphere with positive vertex curvature into R 3 by Bobenko and Izmestiev [2], giving an iterative proof of Alexandrov's theorem. Hyperbolic embedding theorems from dihedral angle data had been treated by Hodgson and Rivin [39] and more recently [24,25]. Sphere packing metrics have been studied by a number of authors [15,[26][27][28], and there is some general theory on angle variations in three-dimensional piecewise Euclidean manifolds in [31] and further work on hyperbolic manifolds in [60].…”

Section: Three Dimensionsmentioning

confidence: 99%

Book Review: Ricci flow for shape analysis and surface registration: theories, algorithms and applications

Chow¹,

Glickenstein²,

Luo³

2016

Bull. Amer. Math. Soc.

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One of the goals in the field of modern geometric analysis is to study canonical structures on manifolds and vector bundles-existence, uniqueness, and moduli. One of the pioneers of analytical approaches in this field is Yau, who, with Schoen and other coauthors solved a number of outstanding geometric problems, many of which are related to elliptic PDEs. A personal account of these developments is in the two-volume set [65]. Around the same time, there were remarkable developments in other areas of geometry. A more geometric theory, including the fundamental theories of compactness and collapse, was developed by Cheeger, Gromov, and others. Thurston revolutionized low-dimensional topology by developing a geometric theory aimed at understanding 3-manifolds via the geometrization conjecture and related ideas. Soon after, inspired by the earlier work of Eells and Sampson on harmonic maps, Hamilton invented the Ricci flow, a parabolic PDE, and through a number of very original works nearly single-handedly developed it into a compelling program to approach the Poincaré and geometrization conjectures. Two decades later, Perelman solved the Poincaré and geometrization conjectures by introducing a plethora of deep and powerful ideas and methods into Ricci flow to complete Hamilton's program. At the present time, more than another decade later, geometric analysis is a thriving field with many active subfields, including Kähler geometry, influenced by the work of Tian, Donaldson, Chen, and others, the study of Ricci curvature by Cheeger, Colding, Minicozzi, and others, geometric flows such as the mean curvature flow by Huisken, White, and others, and the work of Brendle to solve long-standing conjectures, to just name a few of the many directions this field has taken.The idea that a discrete set of values determines geometry is the main tenet of discrete differential geometry. These ideas come from a number of disparate directions. Whitney used numerical analysis to connect de Rham theory and simplicial homology [64], and these ideas form the central tenets of discrete exterior calculus [20]. In a separate intellectual stream, Regge introduced a notion of discrete geometry that he considered a way of developing computational and quantum gravity from first principles that were motivated by, but not dependent on, the theory of smooth manifolds (see [34,51]). Thurston introduced the idea of circle packings for approximating Riemann mappings, and this developed into a whole field of circle packing and related topics (see Stephenson's book [58], reviewed in [6]). A discrete form of integrable systems was developed by Bobenko and Suris [5] into a geometric theory, while other groups [1,23] have developed similar or competing theories arising from numerical analysis and mathematical physics. All of these directions may be characterized as structure-preserving discretizations.

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Gauss images of hyperbolic cusps with convex polyhedral boundary

Cited by 14 publications

References 33 publications

Shapes of Polyhedra, Mixed Volumes and Hyperbolic Geometry

Shapes of Polyhedra, Mixed Volumes and Hyperbolic Geometry

Infinitesimal Rigidity of Convex Polyhedra through the Second Derivative of the Hilbert–Einstein Functional

Book Review: Ricci flow for shape analysis and surface registration: theories, algorithms and applications

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