arXiv research

Polyhedra rigidity theorem in hyperbolic space proved.

The paper proves rigidity and uniformization theorems for infinite circle patterns and convex polyhedra in hyperbolic 3-space.

Spinors prove rigidity for polyhedral spacetime data.

Study on polyhedra rigidity, finding non-existence of flexible weakly convex decomposable polyhedra.

Polyhedra's structure is uniquely defined by edge lengths and dihedral angles, even nonconvex.

Weakly convex polyhedra which are star-shaped with respect to one of their vertices are infinitesimally rigid. This is a partial answer to the question whether every decomposable weakly convex polyhedron is infinitesimally rigid. The proof uses a recent result of Izmestiev on the geometry of convex caps.

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 Hilbert-Einstein functional on the space of "warped polyhedra" with a fixed metric on the boundary. This approach is in a sense dual to using derivatives of the volume i…

Characterizes infinite ideal polyhedra in hyperbolic 3-space and proves their existence and rigidity.

The main motivation here is a question: whether any polyhedron which can be subdivided into convex pieces without adding a vertex, and which has the same vertices as a convex polyhedron, is infinitesimally rigid. We prove that it is indeed the case for two classes of polyhedra: those obtained from a convex polyhedron b…

We show the rigidity of the hexagonal Delaunay triangulated plane under Luo's PL conformality. As a consequence, we obtain a rigidity theorem for a particular type of locally finite convex ideal hyperbolic polyhedra.

Proves Gromov's conjecture and answers Stoker's polyhedron conjecture.

The paper proves a spacetime version of dihedral rigidity for cubes in 3D spacetime.

Let $P$ be a (non necessarily convex) embedded polyhedron in $\R^3$, with its vertices on an ellipsoid. Suppose that the interior of $P$ can be decomposed into convex polytopes without adding any vertex. Then $P$ is infinitesimally rigid. More generally, let $P$ be a polyhedron bounding a domain which is the union of p…

The Stoker problem, first formulated in 1968, consists in understanding to what extent a convex polyhedron is determined by its dihedral angles. By means of the double construction, this problem is intimately related to rigidity issues for 3-dimensional cone-manifolds. In a former paper, two such rigidity results were …

The paper proves rigidity for submanifolds in warped product manifolds.

The study of comparison theorems in geometry has a rich history. In this paper, we establish a comparison theorem for polyhedra in 3-manifolds with nonnegative scalar curvature, answering affirmatively a dihedral rigidity conjecture by Gromov. For a large collections of polyhedra with interior non-negative scalar curva…

Polyhedra can mimic constant curvature surfaces, even with self-intersections.

Study rigidity and volume optimization of hyperbolic polyhedra.

New proof for global rigidity of vertex scaling on polyhedral surfaces.

The deformation theory of hyperbolic and Euclidean cone-manifolds with all cone angles less then 2π plays an important role in many problems in low dimensional topology and in the geometrization of 3-manifolds. Furthermore, various old conjectures dating back to Stoker about the moduli of convex hyperbolic and Euclidea…

Let $P \subset \R^3$ be a polyhedron. It was conjectured that if $P$ is weakly convex (i. e. its vertices lie on the boundary of a strictly convex domain) and decomposable (i. e. $P$ can be triangulated without adding new vertices), then it is infinitesimally rigid. We prove this conjecture under a weak additional assu…

Löbell polyhedra have small systoles and are quasi-arithmetic.

Introduces Coxeter polyhedra in various geometries.

New hyperbolic polyhedra with $π/3$ angles and volumes calculated.

The paper sets new limits on hyperbolic polyhedra volumes.

The study sets limits on dihedral angles of large hyperbolic polyhedra.

Improved volume estimates for right-angled polyhedra in hyperbolic space.

We study convex polyhedra in $\mathbb{R}\mathbb{P}^3$ with all their vertices on a sphere. We do not require, in particular, that the polyhedra lie in the interior of the sphere, hence the term "weakly inscribed". Such polyhedra can be interpreted as ideal polyhedra, if we regard $\mathbb{R}\mathbb{P}^3$ as a combinati…

We review several results related to the characterization of polyhedra in hyperbolic 3-space. In particular we present Rivin's theorem that gives a characterization of compact convex hyperbolic polyhedra, and Hodgson's proof of the Adreev's theorem. We also review the analogous characterization of ideal polyhedra, and …

Improved bounds on ideal vertices in right-angled hyperbolic polyhedra.

Study calculates mass of special polyhedra in hyperbolic space.

Mass in relativity linked to polyhedra geometry.

Polyhedra volume conjecture supports Stoker conjecture weakly.

In this paper we consider a class of right-angled polyhedra in three-dimensional Lobachevsky space, all vertices of which lie on the absolute. New upper bounds on volumes in terms the number of faces of the polyhedron are obtained. Volumes of polyhedra with at most 23 faces are computed. It is shown that the minimum vo…

This article defines a pair of combinatorial operations on the combinatorial structure of compact right-angled hyperbolic polyhedra in dimension three called decomposition and edge surgery. It is shown that these operations simplify the combinatorics of such a polyhedron, while keeping it within the class of right-angl…

Intermediate logic of all convex polyhedra is axiomatized.

Study approximates Riemannian manifolds using polyhedra.

We present a notion of mutation of hyperbolic polyhedra, analogous to mutation in knot theory, and then present a general question about commensurability of mutant pairs of polyhedra. We motivate that question with several concrete examples of mutant pairs for which commensurability is unknown. The polyhedra we conside…

New families of hyperbolic polyhedra yield infinitely many unique reflection groups.

The paper solves three problems related to monostable polyhedra.

In this article we establish the relation between the spines of 3-manifolds and the polyhedra with identified faces. We do this by showing that the spines of the closed, connected, orientable 3-manifolds can be presented through polyhedra with identified faces in a very natural way. We also prove the equivalence betwee…

Paper solves long-standing problem of infinite ideal polyhedra in hyperbolic space.

Mass in relativity linked to polyhedra geometry.

We consider a natural question: "Is it true that each homotopy domination of a polyhedron over itself is a homotopy equivalence?" and a strongly related problem of K. Borsuk (1967): "Is it true that two ANR's homotopy dominating each other have the same homotopy type?" The answer was earlier known to be positive for ma…

An algorithm for determining the list of smallest volume right-angled hyperbolic polyhedra in dimension 3 is described. This algorithm has been implemented on computer using the program Orb to compute volumes, and the first 825 polyhedra in the list have been determined.

New periodic polyhedra found in curved spaces.

Study geodesics on spherical polyhedra, estimating their number.

Software finds ideal polyhedra with rational dihedral angles and volume maxima.