arXiv research

Locally finite complexes with polyhedral $CAT(0)$ metrics are arborescent.

Tree complex linked to polyhedral shapes like associahedra and cyclohedra.

In this article we introduce the notion of Polyhedral Kahler manifolds, even dimensional polyhedral manifolds with unitary holonomy. We concentrate on the 4-dimensional case, prove that such manifolds are smooth complex surfaces, and classify the singularities of the metric. The singularities form a divisor and the res…

Study circle patterns and polyhedral surfaces in hyperbolic ends, proving manifold properties.

The ramification of a polyhedral space is defined as the metric completion of the universal cover of its regular locus. We consider mainly polyhedral spaces of two origins: quotients of Euclidean space by a discrete group of isometries and polyhedral metrics on the complex projective plane with singularities at a colle…

Conditions for polyhedral Kähler metrics on CP^n with specific singularities.

New criterion for almost-complex 4-manifolds using polyhedral decompositions.

The study finds PK cone metrics on complex manifolds near hyperplane arrangements.

Let $M$ be a smooth closed orientable surface. Let $F$ be the space of Morse functions on $M$ having fixed number of critical points of each index, moreover at least $χ(M)+1$ critical points are labeled by different labels (enumerated). A notion of a skew cylindric-polyhedral complex, which generalizes the notion of a …

Given a finite simplicial complex L and a collection of pairs of spaces indexed by its vertex set, one can define their polyhedral product. We record a simple formula for its Euler characteristic. In special cases the formula simplifies further to one involving the h-polynomial of L.

It is proved that every discrete Morse function in the sense of Forman on a finite regular CW complex can be represented by a polyhedral Morse function in the sense of Banchoff on an appropriate embedding in Euclidean space of the barycentric subdivision of the CW complex; such a representation preserves critical point…

3-manifold triangulations are Golod and tight, proven through a topological characterization.

The "polyhedral product functor" produces a space from a simplicial complex L and a collection of pairs of spaces, {(A(i),B(i))}, where i ranges over the vertex set of L. We give necessary and sufficient conditions for the resulting space to be aspherical. There are two similar constructions, each of which starts with …

Study on discrete Gaussian curvature for polyhedral surfaces.

Polyhedral surfaces are fundamental objects in architectural geometry and industrial design. Whereas closeness of a given mesh to a smooth reference surface and its suitability for numerical simulations were already studied extensively, the aim of our work is to find and to discuss suitable assessments of smoothness of…

A discrete conformality for hyperbolic polyhedral surfaces is introduced in this paper. This discrete conformality is shown to be computable. It is proved that each hyperbolic polyhedral metric on a closed surface is discrete conformal to a unique hyperbolic polyhedral metric with a given discrete curvature satisfying …

Link condition for simplicial complexes to be CUB spaces.

Polyhedral semantics for intermediate logics; Nerve Criterion ensures completeness.

The paper proves rigidity of bordered polyhedral surfaces using variational principles.

We develop a method to find a set of diminimal polyhedral maps on the torus from which all other polyhedral maps on the torus may be generated by face splitting and vertex splitting. We employ this method, though not to its completion, to find 53 diminimal polyhedral maps on the Torus.

Characterizes metrics on triangulated surfaces using glued Euclidean triangles.

We study hyperbolic polyhedral surfaces with faces isometric to regular hyperbolic polygons satisfying that the total angles at vertices are at least $2π.$ The combinatorial information of these surfaces is shown to be identified with that of Euclidean polyhedral surfaces with negative combinatorial curvature everywher…

A discrete conformality for polyhedral metrics on surfaces is introduced in this paper which generalizes earlier work on the subject. It is shown that each polyhedral metric on a surface is discrete conformal to a constant curvature polyhedral metric which is unique up to scaling. Furthermore, the constant curvature me…

A polyhedral map is called $\{p, q\}$-equivelar if each face has $p$ edges and each vertex belongs to $q$ faces. In 1983, it was shown that there exist infinitely many geometrically realizable $\{p, q\}$-equivelar polyhedral maps if $q > p = 4$, $p > q = 4$ or $q - 3 > p = 3$. It was shown in 2001 that there exist infi…

Smooth 2-tori in R^4 can be approximated by polyhedral Lagrangian or isotropic tori.

Survey on discrete curvature concepts for polygons and polyhedral surfaces.

Deep learning has received much attention lately due to the impressive empirical performance achieved by training algorithms. Consequently, a need for a better theoretical understanding of these problems has become more evident in recent years. In this work, using a unified framework, we show that there exists a polyhe…

We show that area minimizing polyhedral surfaces are saddle.

The study provides conditions for approximating Riemannian manifolds with polyhedral metrics.

Study calculates Floer homology for binary polyhedral spaces.

New method for identifying best designs in vector optimization with uncertain feedback.

We study rigidity of polyhedral surfaces and the moduli space of polyhedral surfaces using variational principles. Curvature like quantities for polyhedral surfaces are introduced. Many of them are shown to determine the polyhedral metric up to isometry. The action functionals in the variational approaches are derived …

This paper investigates several global rigidity issues for polyhedral surfaces including inversive distance circle packings. Inversive distance circle packings are polyhedral surfaces introduced by P. Bowers and K. Stephenson as a generalization of Andreev-Thurston's circle packing. They conjectured that inversive dist…

New bounds show polyhedral surrogates are optimal for generalization.

New index theory proves Gromov's dihedral conjectures.

Constructs a moment map flow for isotropic maps on surfaces.

This work describes compactifications of metric spaces and vector spaces using asymmetric norms.

We investigate the rigidity of hyperbolic cone metrics on $3$-manifolds which are isometric gluing of ideal and hyper-ideal tetrahedra in hyperbolic spaces. These metrics will be called ideal and hyper-ideal hyperbolic polyhedral metrics. It is shown that a hyper-ideal hyperbolic polyhedral metric is determined up to i…

Researchers decompose hyperbolic n-manifolds with totally geodesic boundaries into polyhedral cells.

The study connects polyhedral manifolds to Riemannian ones with geometric bounds.

We show that a compact length space is polyhedral if a small spherical neighborhood of any point is conic.

Softens tilings in 3D space, proving conjectures about polyhedral tilings.

Study shows non-polyhedral structure in moduli spaces for n≥8.

The Cartesian subgroup in graph products of groups is studied with bounds and algorithms.

Flat metrics on hyperbolic surfaces embed as polyhedral surfaces in (2+1)-spacetimes.

Study shows Bergman metric is non-Einstein for certain domains.

We study the rigidity of polyhedral surfaces using variational principle. The action functionals are derived from the cosine laws. The main focus of this paper is on the cosine law for a non-triangular region bounded by three possibly disjoint geodesics. Several of these cosine laws were first discovered and used by Fe…

We present and apply a method for disproving the existence of polyhedral immersions in $\mathbb{R}^3$ of certain triangulations on non-orientable surfaces. In particular, it is proved that neither of the two vertex-minimal, neighborly 9-vertex triangulations of the non-orientable surface of genus 5 are realizable as im…