Authors prove a formula relating the Gaussian curvature of polyhedral vertex stars to their Gauss images.
problem Proving a formula connecting discrete Gaussian curvature to the algebraic area of Gauss images.
method Comparing winding numbers and critical point index of a normal vector to deduce the formula.
result Formula significantly limits possible shapes of Gauss images of polyhedral vertex stars.
Locally finite complexes with polyhedral CAT(0) metrics are arborescent.
problem Characterizing locally finite complexes with CAT(0) metrics. method Proving arborescence for complexes with polyhedral CAT(0) metrics. result Locally finite complexes with polyhedral CAT(0) metrics are arborescent. New proof for global rigidity of vertex scaling on polyhedral surfaces.
problem Global rigidity of vertex scaling on polyhedral surfaces.
method Elementary variational proof based on continuity of eigenvalues and extension of convex functions.
result Global rigidity of vertex scaling proved without involving 3D hyperbolic geometry.
We present and apply a method for disproving the existence of polyhedral immersions in R3 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…
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…
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.
The paper develops algorithms for finding metrics with prescribed combinatorial curvature on polyhedral surfaces.
problem Finding metrics with prescribed combinatorial curvature on polyhedral surfaces.
method Discrete uniformization theorem, combinatorial α-Yamabe flow, combinatorial α-Calabi flow, edge flipping surgery.
result Longtime existence and convergence of combinatorial α-Yamabe flow and combinatorial α-Calabi flow with surgery.
Proves a generalized Whitehead cut vertex lemma for tree groups.
problem Extending Whitehead's cut vertex lemma to tree group conjugacy classes.
method Proves a version of Whitehead's lemma for tree groups.
result Establishes a cut vertex in star graphs for tree group conjugacy classes.
The paper tackles prescribing discrete Gaussian curvature on polyhedral surfaces.
problem Prescribing discrete Gaussian curvature on polyhedral surfaces.
method Discrete conformal theory and variational principles with constraints.
result Proves Kazdan-Warner type theorems for polyhedral surfaces.
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.
New maps on the plane with specific symmetry properties identified.
problem Characterizing maps with quasi-vertex-transitive properties.
method Analyzing the automorphism groups and vertex orbits of maps on the plane.
result Existence of quasi-vertex-transitive maps of certain types, but not vertex-transitive.
Paper proves rigidity of discrete conformal structures on polyhedral surfaces.
problem Rigidity of discrete conformal structures on polyhedral surfaces.
method Variational principles.
result Proves Glickenstein's conjecture on the rigidity of discrete conformal structures.
Paper analyzes Birkhoff relaxation for graph alignment, providing theoretical guarantees.
problem Finding vertex correspondence between two graphs to maximize edge overlap.
method Birkhoff relaxation as a convex relaxation of the quadratic assignment problem (QAP).
result Theoretical guarantees on the performance of Birkhoff relaxation under specific conditions.
We extend Jendrol' and Skupień's results about the local structure of maps on the 2-sphere: In this paper we show that if a polyhedral map G on a surface $\M$ of Euler characteristic $χ(\M) \le 0$ has more than $126|χ(\M)|$ vertices, then G has a vertex with "nearly" non-negative combinatorial curvature. As a corol…
Study rigidity and volume optimization of hyperbolic polyhedra.
problem Rigidity and volume optimization of hyperbolic polyhedra.
method Analyzing decorated 1-3 type hyperbolic polyhedra and their metrics.
result Decorated 1-3 type hyperbolic polyhedra are rigid up to isometry and change of decorations.
Tropical geometry and weighted lattices improve curve and surface fitting.
problem Fitting max-⋆ tropical curves and surfaces to data. method Max-⋆ algebra, weighted lattices, morphological adjunctions. result Optimal piecewise-linear regression for max-⋆ curves and surfaces. The paper introduces combinatorial curvature and flow for polyhedral surfaces, proving rigidity and solving the Yamabe problem.
problem Discrete conformal structures on polyhedral surfaces and their rigidity.
method Parameterized combinatorial curvature, combinatorial α-Ricci flow, and flow extension through singularities.
result Existence and convergence of combinatorial α-Ricci flow for solving the Yamabe problem.
Polyhedra can mimic constant curvature surfaces, even with self-intersections.
problem Understanding curvature constraints in discrete vs. smooth settings.
method Constructive proof showing any surface can be realized as a polyhedral surface with uniform angular defect.
result Closed surfaces can be realized as polyhedral surfaces with constant angular defect.
The study proves that in normal tilings, at least two vertices are required per cell.
problem Understanding the minimum number of vertices required in normal tilings.
method The research examines both periodic and monohedral tilings in 2D, proving the minimum number of non-smooth vertices required.
result The study confirms that for normal tilings, at least two vertices are necessary per cell.
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 classifies Halin graphs with positive curvature.
problem Classifying Halin graphs with specific curvature.
method Analyzing generalized Halin graphs formed by connecting tree leaves.
result Identified all generalized Halin graphs with positive Lin-Lu-Yau curvature.
Computing uniformization maps for surfaces has been a challenging problem and has many practical applications. In this paper, we provide a theoretically rigorous algorithm to compute such maps via combinatorial Calabi flow for vertex scaling of polyhedral metrics on surfaces, which is an analogue of the combinatorial Y…
This work reveals a new scaling law for optimal design of multirotor aerial vehicles.
problem Designing optimal configurations for fully-actuated multirotor aerial vehicles.
method Formulated on the product manifold of Projective Lines \RP^2^N, minimizing a coordinate-invariant Log-Volume isotropy metric.
result The topology of the global optima is governed by the symmetry of the chassis, leading to a N-5 Scaling Law.
Graph alignment problem solved with convex relaxations for correlated matrices.
problem Recovering hidden vertex permutations from correlated Gaussian matrices.
method Convex relaxations of the quadratic assignment problem over doubly stochastic matrices.
result The solution of the convex relaxation concentrates around the ground-truth permutation matrix for certain correlation parameters.
Discrete conformal maps on surfaces with vertex decorations are studied.
problem Discrete conformal equivalence for decorated piecewise Euclidean surfaces.
method Intimate relationship between decorated PE-surfaces, canonical tessellations of hyperbolic surfaces, and convex hyperbolic polyhedra; concave variational principle.
result Proof of discrete uniformization theorem for decorated PE-surfaces.
Differential chains are a proper subspace of de Rham currents given as an inductive limit of Banach spaces endowed with a geometrically defined strong topology. Boundary is a continuous operator, as are operators that dualize to Hodge star, Lie derivative, pullback and interior product. Partitions of unity exist in thi…
Sharp stability of Alexandrov's theorem for C1 domains in the small-excess regime
problem Stability of Alexandrov's theorem for C1 domains in the small-excess regime method Combines a BV version of Fuglede's spectral-gap argument, a star-shaped rearrangement for sets of finite perimeter, quantitative estimates for the part of the boundary contained in the tentacles, and a polyhedral approximation argument for the non-graphical region result Sharp stability estimate in a genuinely non-parametric regime
A connected combinatorial 2-manifold is called degree-regular if each of its vertices have the same degree. A connected combinatorial 2-manifold is called weakly regular if it has a vertex-transitive automorphism group. Clearly, a weakly regular combinatorial 2-manifold is degree-regular and a degree-regular combinator…
It is known that the (2k−1)-sphere has at most 2O(nklogn) combinatorially distinct triangulations with n vertices, for every k≥2. Here we construct at least 2Ω(nk) such triangulations, improving on the previous constructions which gave 2Ω(nk−1) in the general case (Kalai) and $2^{Ω(n^{5/…
Fractional combinatorial flow improves surface conformal structures.
problem Improving discrete conformal structures on surfaces.
method Introducing a fractional combinatorial Calabi flow for discrete conformal structures on surfaces.
result Longtime existence and global convergence of the fractional combinatorial Calabi flow for various surface types.
Paper shows minimum 10 vertices for hyperbolic origami 2-torus.
problem Finding minimum vertices for hyperbolic origami 2-torus.
method Geodesic triangulation and isometric polyhedral embedding.
result 10 vertices are the minimum required for a hyperbolic origami 2-torus.
Collapsibility is a combinatorial strengthening of contractibility. We relate this property to metric geometry by proving the collapsibility of any complex that is CAT(0) with a metric for which all vertex stars are convex. This strengthens and generalizes a result by Crowley. Further consequences of our work are: (1) …
The study proves non-orientable surfaces can map to a torus.
problem Embedding non-orientable surfaces in 4D space.
method Proving mapping to a 2D torus for 2-convex surfaces.
result Projective plane and Klein bottle cannot be 2-convex in 4D space.
All Higman groups on 5 or more generators are uniquely measure equivalent.
problem Proving uniqueness of measure equivalence for Higman groups.
method Using measured group theory and polyhedral complexes.
result Superrigidity for measure equivalence of Higman groups.
The paper consists of two parts. In the first one we show that a relatively hyperbolic group G splits as a star graph of groups whose central vertex group is finitely generated and the other vertex groups are maximal parabolic subgroups. As a corollary we obtain that every group which admits 3-discontinuous and 2-coc…
Study on discrete Gaussian curvature for polyhedral surfaces.
problem Discretization of Gaussian curvature for polyhedral surfaces.
method Generalization of discrete conformal equivalence to define discrete Gaussian curvature and classify polyhedral surfaces.
result Existence of polyhedral surfaces with constant discrete Gaussian curvature in every discrete conformal class.
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 …
Conjecturally, the only knots in S3 with non-integer surgeries producing Seifert fibered spaces are torus knots and cables of torus knots. In this paper, we make progress on the associated realization problem. Let Y be a small Seifert fibered space arising by p/q-surgery on a knot in S3, where p/q is positi…
Polyhedral semantics for intermediate logics; Nerve Criterion ensures completeness.
problem Characterize polyhedrally-complete intermediate logics.
method Developed Nerve Criterion to characterize polyhedrally-complete logics combinatorially.
result Nerve Criterion provides a necessary and sufficient condition for polyhedrally-completeness.
The paper proves rigidity of bordered polyhedral surfaces using variational principles.
problem Determining the rigidity of bordered polyhedral surfaces.
method Using the variational principle, the paper shows that bordered polyhedral surfaces are determined by boundary values and discrete curvatures on interior edges.
result The paper re-proves the classical result that two Euclidean or hyperbolic cyclic polygons are congruent if their side lengths are equal.
We show that the torsion of any simple closed curve Γ in Euclidean 3-space changes sign at least 4 times provided that it is star-shaped and locally convex with respect to a point o in the interior of its convex hull. The latter condition means that through each point p of Γ there passes a plane H, not cont…
The paper studies deformation of discrete conformal structures on surfaces using combinatorial curvature flows.
problem Finding piecewise constant curvature metrics on surfaces with prescribed combinatorial curvatures.
method Combinatorial curvature flows, including Ricci flow and Calabi flow, are applied to deform Glickenstein's discrete conformal structures.
result The solution of the combinatorial Ricci flow can be uniquely extended and converges exponentially fast for any initial value under certain conditions.
The paper proves a theorem for discretizing Gaussian curvature on surfaces.
problem Discretizing Gaussian curvature on surfaces with nonpositive Euler number.
method Discrete conformal theory and variational principles with constraints.
result Each decorated piecewise Euclidean metric on surfaces with nonpositive Euler number is discrete conformal to a metric with a specific discrete curvature constant.
Characterizes metrics on triangulated surfaces using glued Euclidean triangles.
problem Describing metrics on triangulated surfaces constructed from glued Euclidean triangles.
method Carefully constructing polyhedral metrics and proving their uniqueness.
result Polyhedral metrics are the only intrinsic metrics preserving Euclidean triangle lengths.
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…
The paper finds and visualizes unique geometric polyhedra and tori with few vertices.
problem Finding and visualizing geometric polyhedra and tori with specific vertex configurations.
method Using Schlegel diagrams and geometric realization in 3D and 4D space.
result Identifies and visualizes 12 triangulations of the 2-torus and 12 triangulations of the 2D projective plane.