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Roman Prosanov

Publications and source records attributed to Roman Prosanov.

13 recordsLinked to original sources
arXiv2026

Second-order rigidity of coned polytope frameworks and the stress-flex conjecture from a vector-valued Schl\"afli formula

A coned polytope framework (CPF) is the bar-joint framework obtained from the 1-skeleton of a convex polytope by coning over some interior point. It was recently shown that CPFs are rigid, though the exact order of rigidity remained open. In this paper we introduce the Wachspress stress and use it to show that CPFs are prestress stable, in particular, second-order rigid. To this end, we resolve the stress-flex conjecture in the case of the Wachspress stress by identifying its dual formulation as a corollary of a vector-valued Schl\"afli-type formula introduced by Schlenker and Souam. We give a new and purely discrete-geometric proof of this generalized Schl\"afli formula.

math.MG↗
arXiv2025

Polyhedral surfaces in anti-de Sitter (2+1)-spacetimes

We first prove that given a Fuchsian representation $\rho_\circ: \pi_1S \ra {\rm PSL}(2,\R)$, where $S$ is a closed oriented surface of genus $\geq 2$, any hyperbolic cone-metric on $S$ with cone-angles $>2\pi$ isometrically embeds as a future-convex bent Cauchy surface in a globally hyperbolic maximal Cauchy compact (GHMC) anti-de Sitter (2+1)-spacetime whose left representation is $\rho_\circ$. Second, we show that given any two such cone-metrics, there exists a GHMC anti-de Sitter (2+1)-spacetime in which the cone-metrics embed simultaneously, one as a future-convex bent Cauchy surface and one as a past-convex. Furthermore, in both cases we establish that such a spacetime and embeddings are unique provided that the cone-metrics are sufficiently small.

math.GT↗
arXiv2023

Polyhedral surfaces in flat (2+1)-spacetimes and balanced cellulations on hyperbolic surfaces

We first prove that given a hyperbolic metric $h$ on a closed surface $S$, any flat metric on $S$ with negative singular curvatures isometrically embeds as a convex polyhedral Cauchy surface in a unique future-complete flat globally hyperbolic maximal (2+1)-spacetime whose linear part of the holonomy is given by $h$. The Gauss map allows to translate this statement to a purely 2-dimensional problem of finding a balanced geodesic cellulation on the hyperbolic surface, from which the flat metric can be easily recovered. We show next that given two such flat metrics on the surface, there exists a unique pair of future- and past-complete flat globally hyperbolic maximal (2+1)-spacetimes with the same holonomy, in which the flat metrics embed respectively as convex polyhedral Cauchy surfaces. The proof follows from convexity properties of the total length of the associated balanced geodesic cellulations over Teichm\"uller space.

math.MG↗
arXiv2023

Prescribed curvature problem for discrete conformality on convex spherical cone-metrics

Let $S$ be the 2-sphere and $V \subset S$ be a finite set of at least three points. We show that for each function $κ: V \rightarrow (0, 2π)$ satisfying elementary necessary conditions, in each discrete conformal class of spherical cone-metrics there exists a unique metric realizing $κ$ as its discrete curvature. This can be seen as a discrete version of a result of Luo and Tian.

math.MG↗
arXiv2022

Hyperbolic 3-manifolds with boundary of polyhedral type

Let $M$ be a compact orientable 3-manifold with hyperbolizable interior and non-empty boundary such that all boundary components have genii at least 2. We study an Alexandrov-Weyl-type problem for convex hyperbolic cone-metrics on $\partial M$. We consider a class of hyperbolic metrics on M with convex boundary, which we call bent metrics, and which naturally generalize hyperbolic metrics on $M$ with convex polyhedral boundary. We show that for each convex hyperbolic cone-metric $d$ on $\partial M$, with few simple exceptions, there exists a bent metric on $M$ such that the induced intrinsic metric on $\partial M$ is $d$. Next, we prove that if a bent realization is what we call controllably polyhedral, then it is unique up to isotopy. We exhibit a large subclass of hyperbolic cone-metrics on $\partial M,$ called balanced, which is open and dense among all convex hyperbolic cone-metrics in the sense of Lipschitz topology, and for which we show that their bent realizations are controllably polyhedral. We additionally prove that any convex realization of a convex hyperbolic cone-metric on $\partial M$ is bent. Finally, we deduce that there exists an open subset of the space of convex cocompact metrics on the interior of $M$, including all metrics with polyhedral convex cores, such that the metrics in this subset are (1) globally rigid with respect to the induced intrinsic metrics on the boundaries of their convex cores; (2) infinitesimally rigid with respect to their bending laminations. This gives partial progress towards conjectures of W. Thurston.

math.GT↗
arXiv2022

New invariants of Gromov-Hausdorff limits of Riemannian surfaces with curvature bounded below

Let $\{X_i\}$ be a sequence of compact $n$-dimensional Alexandrov spaces (e.g. Riemannian manifolds) with curvature uniformly bounded below which converges in the Gromov-Hausdorff sense to a compact Alexandrov space $X$. In an earlier paper by the first author there was described (without a proof) a construction of an integer valued function on $X$; this function carries additional geometric information on the sequence such as the limit of intrinsic volumes of $X_i$'s. In this paper we consider sequences of closed 2-surfaces and (1) prove the existence of such a function in this situation; and (2) classify the functions which may arise from the construction.

math.DG↗
arXiv2022

Dual metrics on the boundary of strictly polyhedral hyperbolic 3-manifolds

Let $M$ be a compact oriented 3-manifold with non-empty boundary consisting of surfaces of genii $>1$ such that the interior of $M$ is hyperbolizable. We show that for each spherical cone-metric $d$ on $\partial M$ such that all cone-angles are greater than $2\pi$ and the lengths of all closed geodesics that are contractible in $M$ are greater than $2\pi$ there exists a unique strictly polyhedral hyperbolic metric on $M$ such that $d$ is the induced dual metric on $\partial M$.

math.MG↗
arXiv2021

Rigidity of compact Fuchsian manifolds with convex boundary

A compact Fuchsian manifold with boundary is a hyperbolic 3-manifold homeomorphic to $S_g \times [0; 1]$ such that the boundary component $S_g \times \{ 0\}$ is geodesic. We prove that a compact Fuchsian manifold with convex boundary is uniquely determined by the induced path metric on $S_g \times \{1\}$. We do not put further restrictions on the boundary except convexity.

math.GT↗
arXiv2019

Ideal polyhedral surfaces in Fuchsian manifolds

Let $S_{g,n}$ be a surface of genus $g > 1$ with $n>0$ punctures equipped with a complete hyperbolic cusp metric. Then it can be uniquely realized as the boundary metric of an ideal Fuchsian polyhedron. In the present paper we give a new variational proof of this result. We also give an alternative proof of the existence and uniqueness of a hyperbolic polyhedral metric with prescribed curvature in a given conformal class.

math.GT↗
arXiv2018

Chromatic numbers of spheres

The chromatic number of a subset of Euclidean space is the minimal number of colors sufficient for coloring all points of this subset in such a way that any two points at the distance 1 have different colors. We give new upper bounds for chromatic numbers of spheres.

math.CO↗