We establish a connection between recent developments in the study of vortices in the abelian Higgs models, and in the theory of structure-preserving discrete conformal maps. We explain how both are related via conformal mapping problems involving prescribed linear combinations of the curvature and volume form, and sho…
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48 results for “discrete conformal theory”
We establish a connection between two previously unrelated topics: a particular discrete version of conformal geometry for triangulated surfaces, and the geometry of ideal polyhedra in hyperbolic three-space. Two triangulated surfaces are considered discretely conformally equivalent if the edge lengths are related by s…
Study on deforming discrete conformal structures on surfaces with boundaries.
problem Deforming discrete conformal structures on surfaces with boundaries.
method Introduce combinatorial Ricci flow and combinatorial Calabi flow, establish longtime existence and global convergence of solutions.
result Effective algorithms for finding discrete hyperbolic metrics on surfaces with totally geodesic boundaries of prescribed lengths.
Finding appropriate notions of discrete holomorphic maps and, more generally, conformal immersions of discrete Riemann surfaces into 3-space is an important problem of discrete differential geometry and computer visualization. We propose an approach to discrete conformality that is based on the concept of holomorphic l…
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.
The paper introduces a new discretization of Gaussian curvature on surfaces.
problem Discretizing Gaussian curvature on surfaces with conic singularities.
method Discrete conformal theory and variational principles with constraints.
result Established a discrete uniformization theorem for surfaces with non-positive Euler number.
We study surfaces with decorations and prove uniformization in non-Euclidean geometries.
problem Discrete conformal equivalence in non-Euclidean geometries.
method Variational principle and continuous deformation.
result One master theory of discrete conformal equivalence across different geometries.
Discrete Riemann Surfacesmath.CV
We detail the theory of Discrete Riemann Surfaces. It takes place on a cellular decomposition of a surface, together with its Poincaré dual, equipped with a discrete conformal structure. A lot of theorems of the continuous theory follow through to the discrete case, we define the discrete analogs of period matrices, Ri…
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.
A Darboux transformation for polarized space curves is introduced and its properties are studied, in particular, Bianchi permutability. Semi-discrete isothermic surfaces are described as sequences of Darboux transforms of polarized curves in the conformal n-sphere and their transformation theory is studied. Semi-discre…
The elastic flow, which is the -gradient flow of the elastic energy, has several applications in geometry and elasticity theory. We present stable discretizations for the elastic flow in two-dimensional Riemannian manifolds that are conformally flat, i.e.\ conformally equivalent to the Euclidean space. Examples in…
Paper proves discrete uniformizations converge to continuous for surfaces of genus ≥1.
problem Computing uniformizations for surfaces of genus >1.
method Discrete conformality and uniformization on triangle meshes.
result Discrete uniformizations approximate continuous uniformization for closed surfaces of genus ≥1.
New theorem proves convergence of various discrete conformal structures to conformal maps.
problem Proving convergence of discrete conformal structures to conformal maps.
method General theorem using piecewise linear discrete conformal mappings and Riemannian barycentric coordinates.
result Discrete conformal mappings converge to conformal maps under certain conditions.
Plane triangulations remain rigid under discrete conformal changes.
problem Rigidity of acute triangulations under discrete conformal changes.
method Maximum principles, discrete Liouville theorem, extremal lengths, Euclidean to hyperbolic discrete conformality.
result Uniformly acute triangulations are rigid under Luo's discrete conformal change.
Plane Delaunay triangulations are rigid under Luo's discrete conformal change.
problem Rigidity of Delaunay triangulations under discrete conformal changes.
method Developed discrete Schwarz lemma and Liouville theorem, used conformal modulus and extremal length.
result Discrete analogue of conformal rigidity of the plane.
Extends rigidity and existence results for discrete conformal structures on surfaces with boundary.
problem Rigidity and existence of discrete conformal structures on surfaces with boundary.
method Axiomatic framework and classification of discrete conformal structures.
result Extends results by Guo-Luo and Guo to a general context.
This paper completes the classification of discrete conformal structures on surfaces.
problem Classifying discrete conformal structures on surfaces.
method Axiomatic approach and study of existing structures.
result Find new classes of discrete conformal structures, including generalized circle packing metrics.
Survey on discrete minimal surfaces and their properties.
problem Discretizing minimal surfaces in Euclidean space.
method Polyhedral surfaces with parallel face offsets and circle patterns.
result All simply connected discrete minimal surfaces can be constructed from circle patterns.
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 …
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.
The paper proves convergence of discrete maps to Riemann mappings for polyhedral surfaces.
problem Discrete conformal geometry of polyhedral surfaces.
method Establishing rigidity for hexagonal triangulations and estimating quasiconformal constants.
result Discrete conformal maps converge to Riemann mappings for Jordan domains.
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.
We consider the interpretation in classical geometry of conformal field theories constructed from orbifolds with discrete torsion. In examples we can analyze, these spacetimes contain ``stringy regions'' that from a classical point of view are singularities that are to be neither resolved nor blown up. Some of these mo…
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.
Paper generalizes discrete uniformization for genus-zero surfaces.
problem Discrete uniformization for surfaces of genus zero.
method Reduction to planar cases via stereographic projections.
result Generalization of discrete uniformization to genus-zero surfaces.
New method approximates Gaussian curvature on discrete surfaces.
problem Approximating solutions to the prescribed Gaussian curvature problem.
method Discrete conformality and convex functional minimization.
result Efficient numerical method to compute solutions.
This paper classifies discrete conformal structures on surfaces with boundary.
problem Classifying discrete conformal structures on surfaces with boundary.
method Axiomatic approach ensuring good geometric structure, classification based on triangulation and axioms.
result Unified and generalized existing discrete conformal structures on surfaces with boundary.
We recall the theory of linear discrete Riemann surfaces and show how to use it in order to interpret a surface embedded in R^3 as a discrete Riemann surface and compute its basis of holomorphic forms on it. We present numerical examples, recovering known results to test the numerics and giving the yet unknown period m…
New discrete conformal structures on surfaces with boundary, proving global rigidity and constructing hyperbolic metrics.
problem Creating new discrete conformal structures on surfaces with boundary.
method Introducing new discrete conformal structures, proving global rigidity using variational principles, and introducing combinatorial curvature flows.
result Global rigidity of new discrete conformal structures and effective algorithms for constructing hyperbolic metrics.
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 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.
Surfaces from Circlesmath.DG
In the search for appropriate discretizations of surface theory it is crucial to preserve such fundamental properties of surfaces as their invariance with respect to transformation groups. We discuss discretizations based on Möbius invariant building blocks such as circles and spheres. Concrete problems considered in t…
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.
The conformal geometry of the Schwarzian Davey-Stewartson II hierarchy and its discrete analogue is investigated. Connections with discrete and continuous isothermic surfaces and generalised Clifford configurations are recorded. An interpretation of the Schwarzian Davey-Stewartson II flows as integrable deformations of…
Unique metric found for discrete curvature on spherical cone-metrics.
problem Finding a unique metric with prescribed curvature on spherical cone-metrics.
method Discrete conformal approach to spherical cone-metrics.
result Existence of a unique metric realizing prescribed curvature in each conformal class.
Duality structures and discrete conformal variations of piecewise constant curvature surfacesmath.GT
A piecewise constant curvature manifold is a triangulated manifold that is assigned a geometry by specifying lengths of edges and stipulating that for a chosen background geometry (Euclidean, hyperbolic, or spherical), each simplex has an isometric embedding into the background geometry with the chosen edge lengths. Ad…
5D SCFTs can have confining vacua with strings and unbroken symmetries.
problem Investigating phases of 5D SCFTs by varying couplings.
method Using geometric realisation of M-theory on metrically conical Calabi-Yau threefolds.
result Many 5D SCFTs have couplings leading to massive, confining vacua with strings and unbroken symmetries.
Conformal Surface Splinesmath.DG
New boundary and point constraints for controlling conformal surfaces.
problem Controlling the geometry of surfaces defined by minimizers of conformal variational problems.
method Introducing new boundary conditions, point constraints, and flux constraints to control the metric and conformal scale factor.
result Introduces intuitive controls for exploring a subspace of conformal immersions.
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…
In differential geometry of surfaces the Dirac operator appears intrinsically as a tool to address the immersion problem as well as in an extrinsic flavour (that comes with spin transformations to comformally transfrom immersions) and the two are naturally related. In this paper we consider a corresponding pair of disc…
New interpretation of discrete conformality using polyhedral convex hulls.
problem Understanding discrete conformality in 3D.
method Epstein-Penner convex hull construction and induced metrics.
result New bijections and interpretations of discrete conformality.
Extremal length is a conformal invariant that transfers naturally to the discrete setting, giving square tilings as a natural combinatorial analog of conformal mappings. Recent work by S. Hersonsky has explored generalizing these ideas to three-dimensional cube tilings. The connections between discrete extremal length …
Liouville's theorem says that in dimension greater than two, all conformal maps are Möbius transformations. We prove an analogous statement about simplicial complexes, where two simplicial complexes are considered discretely conformally equivalent if they are combinatorially equivalent and the lengths of corresponding …
A new chaotic financial system is proposed by considering ethics involvement in a four-dimensional financial system with market confidence. A five-dimensional conformable derivative financial system is presented by introducing conformable fractional calculus to the integer-order system. A discretization scheme is propo…
Discrete approximation solves Björling's minimal surface problem.
problem Constructing minimal surfaces from real-analytic curves with specified normal fields.
method Approximate solution by discrete minimal surfaces and discrete isothermic surfaces.
result Approximation error is proportional to the square of the mesh size.
Develop conformal prediction for dyadic regression under complex missingness.
problem Conformal prediction for dyadic regression under complex missingness mechanisms.
method Developing general technical tools and conformal prediction procedures for dyadic regression under complex missingness.
result Establishing asymptotic validity of weighted conformal prediction under a nonparametric graphon model for missingness mechanism.
Discrete Laplacians defined for spherical and hyperbolic surfaces.
problem Defining discrete Laplacians for non-Euclidean geometries.
method Definitions close to Euclidean, structure-preserving properties proven.
result Connection between discrete and smooth Laplacians in non-Euclidean settings.
Combinatorial curvature flows with surgery for inversive distance circle packings on surfacesmath.DG
New method finds metrics on surfaces with prescribed curvatures using circle packings and surgery.
problem Finding piecewise Euclidean metrics on surfaces with prescribed combinatorial curvatures.
method Combinatorial curvature flows with surgery for inversive distance circle packings.
result Longtime existence and global convergence of combinatorial curvature flows with surgery.