Thurston's Circle Pattern Theorem studies existence and rigidity of circle patterns of a given combinatorial type and the given non-obtuse exterior intersection angles. Using topological degree theory, variational principle, Teichmuller theory, and Sard's Theorem, this paper generalizes Circle Pattern Theorem to the ca…
On-device research index
arXiv research
A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
169,181 papers · 148 categories
Trend · papers per month
48 results for “exterior angles”
Paper proves how to deform circle patterns with interstices.
problem Deforming circle patterns with interstices.
method Combining combinatorial type, exterior angles, and conformal structures.
result Complete description of circle patterns with interstices.
Paper extends circle pattern theory to obtuse angles.
problem Circle patterns with obtuse angles not previously covered.
method Using topological degree theory, extends Koebe-Andreev-Thurston Theorem.
result Generalized Andreev's Theorem for obtuse dihedral angles.
For Bezier curves, subdivision algorithms create control polygons as piecewise linear (PL) approximations that converge in terms of Hausdorff distance. We prove that the exterior angles of control polygons under subdivision converge to 0 at the rate of , where is the number of subdivisions.…
This note generalizes the visual angle to convex sets in 3D space.
problem Analyzing geometric properties of convex sets in 3D space.
method Generalizing the visual angle to convex sets in Euclidean space and expressing geometric quantities in terms of integrals of functions related to the solid angle.
result Invariant quantities of the original convex set can be expressed by integrals of functions related to the solid angle.
The paper studies circle patterns on surfaces with specific angles and curvature maps.
problem Investigating circle patterns with obtuse angles on surfaces of finite type.
method Characterizing curvature maps and establishing combinatorial Ricci flow conditions.
result Generalizations of circle pattern theorem and a computational method to find patterns.
We prove that the control polygon of a Bezier curve B becomes homeomorphic and ambient isotopic to B via subdivision, and we provide closed-form formulas to compute the number of iterations to ensure these topological characteristics. We first show that the exterior angles of control polygons converge exponentially to …
Characterizes infinite ideal polyhedra in hyperbolic 3-space and proves their existence and rigidity.
problem Characterize infinite ideal polyhedra in hyperbolic 3-space.
method Study ideal circle patterns (ICPs) and develop a uniform Ring Lemma via pointed Gromov-Hausdorff convergence.
result Establish existence and rigidity of embedded ICPs and infinite ideal polyhedra (IIP).
Properties of a parametric curve in R^3 are often determined by analysis of its piecewise linear (PL) approximation. For Bezier curves, there are standard algorithms, known as subdivision, that recursively create PL curves that converge to the curve in distance . The exterior angles of PL curves under subdivision are s…
Study exterior maximal surfaces with precise asymptotic behavior.
problem Maximal surfaces in exterior domains with prescribed asymptotic behavior.
method Analyze the maximal surface equation and prove the exterior Dirichlet problem's uniqueness.
result Uniquely solvable exterior Dirichlet problem with given asymptotic behavior.
Weakly Inscribed Polyhedramath.MG
Study of polyhedra on a sphere in projective 3-space.
problem Characterizing polyhedra with vertices on a sphere.
method Purely combinatorial and linear programming approaches.
result Characterization of dihedral angles and hyperbolic-de Sitter structure.
Characterizes alternating link exteriors using cubed complexes.
problem Understanding the structure of alternating link exteriors.
method Introducing signed BW cubed-complexes and characterizing their homeomorphism to link exteriors.
result Characterization of alternating link exteriors in terms of cubed complexes.
Classifies essential annuli in a genus two handlebody exterior.
problem Classifying essential annuli in a genus two handlebody exterior.
method Building on JSJ-graph classification and essential annuli classification.
result Characterizes the numbers of different types of essential annuli in an infinite family.
The paper classifies symmetries and determines exterior types of knotted handlebodies.
problem Classifying symmetries and determining exterior types of knotted handlebodies.
method Analysis of hyperbolic knots with non-integral toroidal Dehn surgeries.
result Determines symmetries and exterior types of knotted genus two handlebodies.
We give a topological characterisation of alternating knot exteriors based on the presence of special spanning surfaces. This shows that alternating is a topological property of the knot exterior and not just a property of diagrams, answering an old question of Fox. We also give a characterisation of alternating link e…
The study classifies area-maximizing hypersurfaces with singularities and exterior domains.
problem Classifying area-maximizing hypersurfaces with singularities and exterior domains.
method Complete classification for entire area maximizing hypersurfaces with isolated singularities. Construction of an example. Partial result on asymptotic behavior for exterior domains. Solvability of exterior Dirichlet problems.
result Complete classification and partial results on asymptotic behavior for area maximizing hypersurfaces.
Discrete exterior calculus shows natural properties of wedge product and averaging.
problem Naturalness of discrete exterior calculus operations.
method Showed naturalness of discrete wedge product and averaging interpretation.
result Discrete wedge product is natural and equals Wilson's cochain product.
The study finds infinite knot exteriors with meridional surfaces of any genus and boundary components.
problem Existence of meridional essential surfaces in knot exteriors.
method Analyzing knot exteriors and their surfaces.
result Existence of infinitely many knot exteriors with meridional essential surfaces of any genus and boundary components.
A brief introduction to exterior differential systems for graduate students familiar with manifolds and differential forms. For complete files, see https://github.com/Ben-McKay/introduction-to-exterior-differential-systems
Extends exterior diff. sys. to Lie algebroids with examples.
problem Invariant inverse problem of the calculus of variations
method Extends exterior differential systems to Lie algebroids, defines integral manifolds.
result Defines integral manifolds for exterior diff. systems on Lie algebroids.
The study finds infinite meridional essential surfaces in hyperbolic knot exteriors of various genera.
problem Existence of meridional essential surfaces in hyperbolic knot exteriors.
method Essential tangle decompositions and meridional essential embeddings.
result Infinite collection of hyperbolic knots with meridional essential surfaces of arbitrary genus and boundary components.
Algorithm finds knot diagrams from exterior triangulations.
problem Finding knot diagrams from their exteriors.
method First practical algorithm for finding diagrams from triangulations of exteriors.
result First diagrams for 23 link exteriors with over 2,500 crossings.
It is shown that two definitions for an exterior differential in superspace, giving the same exterior calculus, yet lead to different results when applied to the Poisson bracket. A prescription for the transition with the help of these exterior differentials from the given Poisson bracket of definite Grassmann parity t…
Evolutionary forms, as well as exterior forms, are skew-symmetric differential forms. But in contrast to the exterior forms, the basis of evolutionary forms is deforming manifolds (with unclosed metric forms). Such forms possess a peculiarity, namely, the closed inexact exterior forms are obtained from that. The closur…
The ropelength of a space curve is usually defined as the quotient of its length by its thickness: the radius of the largest embedded tube around the knot. This idea was extended to space polygons by Eric Rawdon, who gave a definition of ropelength in terms of doubly-critical self-distances (local minima of the distanc…
Paper introduces an invariant to distinguish handlebody-knot exteriors.
problem Challenges in distinguishing handlebody-knots with homeomorphic exteriors.
method Defined an invariant (annulus diagram) using characteristic submanifold theory and Koda-Ozawa classification for essential annuli.
result The annulus diagram can differentiate handlebody-knot families.
A new discrete calculus for bundle-valued forms is proposed and validated.
problem Discretization of exterior calculus for bundle-valued forms.
method Discretization of Cartan's exterior calculus for differential forms with values in vector bundles.
result The proposed discrete operator mimics the continuous exterior covariant derivative and ensures numerical convergence.
New framework classifies high-dimensional shapes using ray intersections, establishing data requirements.
problem Classifying high-dimensional shapes in real-world data.
method Ray-based classification (RBC) framework using intersections of one-dimensional representations (rays) with shape boundaries.
result Established bounds on the number of rays necessary for shape classification, defined by key angular metrics.
Paper shows non-convexity in solutions to Hessian equations.
problem Non-convexity of solutions to -Hessian equations in exterior domains.
method Examples and new proof for quasiconvexity of harmonic functions.
result Solutions to -Hessian equations are not quasiconvex in exterior domains.
Extends Minkowski stability proof to minimal decay assumptions.
problem Global stability of Minkowski spacetime with minimal decay.
method Extends Christodoulou-Klainerman's proof to minimal decay assumptions.
result Exterior stability of Minkowski holds with borderline decay.
A generalization of exterior calculus is considered by allowing the partial derivatives in the exterior derivative to assume fractional orders. That is, a fractional exterior derivative is defined. This is found to generate new vector spaces of finite and infinite dimension, fractional differential form spaces. The def…
Study of group orderability using tensor and exterior squares.
problem Circular orderability of groups and torsion elements.
method Analysis of tensor and exterior squares of groups.
result Circularly orderable groups have left-orderable tensor and exterior squares under certain conditions.
We give an elegant formulation of the structure equations (of Cartan) and the Bianchi identities in terms of exterior calculus without reference to a particular basis and without the exterior covariant derivative. This approach allows both structure equations and the Bianchi identities to be expressed in terms of forms…
Paper proves a Penrose inequality in extrinsic geometry.
problem Proving a Penrose inequality in extrinsic geometry.
method Analyzing minimal capillary surfaces and their free energy.
result Established an extrinsic Penrose inequality.
New disks found with similar outer shapes.
problem Finding similar outer shapes for Lagrangian disks.
method Modified construction of Abe and Tange.
result Arbitrarily large families of disks found.
Paper solves overdetermined -Hessian equation in exterior domains.
problem Overdetermined problem for -Hessian equation in exterior domains.
method Combining integral identities and geometric inequalities, derived general monotone formulas.
result Established general monotone formulas for -admissible solutions.
Maxwell meets Korn: A New Coercive Inequality for Tensor Fields with Square-Integrable Exterior Derivative
Survey revisits vector calculus results using exterior derivative and provides a new formulation of Stokes' theorem.
problem Classical results in vector calculus and analysis.
method Generalised perspective on the exterior derivative and a higher-dimensional Mean Value Theorem.
result Provides a natural formulation of Stokes' theorem and a practical algorithm for exterior differentiation.
Geometric DEC solves Poisson on general triangulations.
problem Solving Poisson equation on arbitrary triangulations.
method Revisited Discrete Exterior Calculus (DEC) for general triangulations using Vector Calculus and Matrix Algebra.
result DEC solutions match FEML for Poisson equation.
Extends Young integral to Hölder differential forms in arbitrary dimensions.
problem Extending the Young integral to Hölder differential forms in arbitrary dimensions.
method Introducing a complex of cochains, α-fractional charges, and defining the exterior product between them.
result The exterior product between α-fractional and β-fractional charges is defined when α + β > 1.
Exterior differential systems are given, and their Cartan characters calculated, for Maxwell and SU(2)-Yang-Mills equations in dimensions from three to six.
The Whitehead link exterior lacks most Euler class taut foliations.
problem Existence of co-orientable taut foliations with specific Euler classes.
method Combining foliation theory techniques and topological obstructions.
result Most lattice points in the dual Thurston ball cannot be Euler classes of co-orientable taut foliations.
Study of Lee form exterior derivative in almost Hermitian manifolds.
problem Understanding the exterior derivative of Lee form in almost Hermitian manifolds.
method Analyzes the Lee form of almost Hermitian manifolds and its exterior derivative in terms of intrinsic torsion and Riemannian curvature tensor components.
result Proves that the -component of is always zero and provides expressions for other components.
We study in detail Hodge-Helmholtz decompositions in non-smooth exterior domains filled with inhomogeneous and anisotropic media. We show decompositions of alternating differential forms belonging to weighted Sobolev spaces into irrotational and solenoidal forms. These decompositions are essential tools, for example, i…
We construct a two-parameter covariant differential calculus on the quantum -exterior plane. We also give a deformation of the two-dimensional fermionic phase space.
Brunnian theta curves in 3D spheres have hyperbolic exteriors.
problem Characterizing Brunnian theta curves in 3D spheres.
method Sutured manifold theory, classification of annuli, and Thurston's hyperbolic geometry.
result Brunnian theta curves of low bridge number have hyperbolic exteriors with totally geodesic boundary.
Researchers prove constant mean curvature graphs in hyperbolic 3-space for specific domains.
problem Existence of hyperbolic Killing graphs with constant mean curvature in exterior domains.
method Existence proof using CMC graphs and Killing vector fields.
result Existence of hyperbolic Killing graphs of constant mean curvature H in exterior domains.
We provide a classification of the essential surfaces of non-negative Euler characteristic in the exteriors of genus two handlebodies embedded in the 3-sphere.