Ivan Izmestiev

Three-Dimensional Manifolds Defined by Coloring a Simple Polytope

In the present paper we introduce and study a class of three-dimensional manifolds endowed with the action of the group ℤ23 whose orbit space is a simple convex polytope. These manifolds originate from three-dimensional polytopes whose faces allow a coloring into three colors with the help of the construction used for studying quasitoric manifolds. For such manifolds we prove the existence of an equivariant embedding into Euclidean space ℝ4. We also describe the action on the set of operations of the equivariant connected sum and the equivariant Dehn surgery. We prove that any such manifold can be obtained from a finitely many three-dimensional tori with the canonical action of the group ℤ23 by using these operations.

A Variational Proof of Alexandrov’s Convex Cap Theorem

We give a variational proof of the existence and uniqueness of a convex cap with the given metric on the boundary. The proof uses the concavity of the total scalar curvature functional (also called Hilbert-Einstein functional) on the space of generalized convex caps. As a by-product, we prove that generalized convex caps with the fixed metric on the boundary are globally rigid, that is uniquely determined by their curvatures.

INFINITESIMAL RIGIDITY OF POLYHEDRA WITH VERTICES IN CONVEX POSITION

Let P C R 3 be a polyhedron. It was conjectured that if P is weakly convex (that is, its vertices lie on the boundary of a strictly convex domain) and decomposable (that is, P can be triangulated without adding new vertices), then it is infinitesimally rigid. We prove this conjecture under a weak additional assumption of codecomposability. The proof relies on a result of independent interest about the Hilbert― Einstein function of a triangulated convex polyhedron. We determine the signature of the Hessian of that function with respect to deformations of the interior edges. In particular, if there are no interior vertices, then the Hessian is negative definite.

There is no triangulation of the torus with vertex degrees 5, 6, . . ., 6, 7 and related results: Geometric proofs for combinatorial theorems

There is no 5,7-triangulation of the torus, that is, no triangulation with exactly two exceptional vertices, of degree 5 and 7. Similarly, there is no 3,5-quadrangulation. The vertices of a 2,4-hexangulation of the torus cannot be bicolored. Similar statements hold for 4,8-triangulations and 2,6-quadrangulations. We prove these results, of which the first two are known and the others seem to be new, as corollaries of a theorem on the holonomy group of a euclidean cone metric on the torus with just two cone points. We provide two proofs of this theorem: One argument is metric in nature, the other relies on the induced conformal structure and proceeds by invoking the residue theorem. Similar methods can be used to prove a theorem of Dress on infinite triangulations of the plane with exactly two irregular vertices. The non-existence results for torus decompositions provide infinite families of graphs which cannot be embedded in the torus.

Gauss images of hyperbolic cusps with convex polyhedral boundary

We prove that a 3-dimensional hyperbolic cusp with convex poly- hedral boundary is uniquely determined by its Gauss image. Furthermore, any spherical metric on the torus with cone singularities of negative curvature and all closed contractible geodesics of length greater than 2π is the metric of the Gauss image of some convex polyhedral cusp. This result is an analog of the Rivin-Hodgson theorem characterizing compact convex hyperbolic polyhedra in terms of their Gauss images. The proof uses a variational method. Namely, a cusp with a given Gauss image is identified with a critical point of a functional on the space of cusps with cone-type singularities along a family of half-lines. The functional is shown to be concave and to attain maximum at an interior point of its domain. As a byproduct, we prove rigidity statements with respect to the Gauss image for cusps with or without cone-type singularities. In a special case, our theorem is equivalent to existence of a circle pattern on the torus, with prescribed combinatorics and intersection angles. This is the genus one case of a theorem by Thurston. In fact, our theorem extends Thurstons theorem in the same way as Rivin-Hodgsons theorem extends An- dreevs theorem on compact convex polyhedra with non-obtuse dihedral angles. The functional used in the proof is the sum of a volume term and curvature term. We show that, in the situation of Thurstons theorem, it is the potential for the combinatorial Ricci flow considered by Chow and Luo. Our theorem represents the last special case of a general statement about isometric immersions of compact surfaces.

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

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. 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.

Extension of colorings

Let K be a combinatorial (d - 1)-sphere with vertices colored in n colors, n ≥ d + 1. We prove that K bounds an n-colored combinatorial ball. This theorem generalizes previously known facts for d = 2 and 3. A further generalization is obtained. Namely, let L be a simplicial complex of dimension d and K be a subcomplex of L. Then any vertex coloring of K in n ≥ d + 1 colors extends to some subdivision of L relative to K. Besides, in both cases the extension can be required to use only d + 1 of n colors in the complement to K.

Derived subdivisions make every PL sphere polytopal

We give a simple proof that some iterated derived subdivision of every PL sphere is combinatorially equivalent to the boundary of a simplicial polytope, thereby resolving a problem of Billera (personal communication).

SHAPES OF POLYHEDRA, MIXED VOLUMES AND HYPERBOLIC GEOMETRY

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 \pi/2.

Iterating Evolutes and Involutes

This paper concerns iterations of two classical geometric constructions, the evolutes and involutes of plane curves, and their discretizations: evolutes and involutes of plane polygons. In the continuous case, our main result is that the iterated involutes of closed locally convex curves with rotation number one (possibly, with cusps) converge to their curvature centers (Steiner points), and their limit shapes are hypocycloids, generically, astroids. As a consequence, among such curves only the hypocycloids are homothetic to their evolutes. The bulk of the paper concerns two kinds of discretizations of these constructions: the curves are replaced by polygons, and the evolutes are formed by the circumcenters of the triples of consecutive vertices (