Discrete version of Liouville's theorem for simplicial complexes.
problem Finding equivalent simplicial complexes under discrete conformal equivalence.
method Proving an analogous statement for simplicial complexes, considering combinatorial equivalence and scale factors associated with vertices.
result All discretely conformally equivalent simplicial complexes are combinatorially equivalent.
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.
Paper proves continuous flow converges to constant curvature metrics.
problem Discrete conformal factors on surfaces and their deformation.
method Discrete Yamabe flow extension without surgeries.
result Flow converges exponentially to constant curvature PL-metric.
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…
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.
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.
Paper addresses travel time tomography stability and statistical inversion.
problem Determining conformal factors of metrics from geodesic lengths.
method Established forward and inverse stability estimates; applied to Bayesian statistical inversion.
result Consistency of statistical inversion technique for travel time tomography.
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.
Study connects vortices in abelian Higgs models to discrete conformal maps.
problem Understanding vortices in abelian Higgs models.
method Relating vortices to discrete conformal maps via curvature and volume form.
result Constructing discrete vortex solutions using discrete conformal theory.
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 …
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.
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.
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.
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.
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.
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.
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.
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 introduces new curvature flows and uniformization theorems for polyhedral surfaces.
problem Discrete uniformization and rigidity of polyhedral surfaces.
method Parameterized discrete curvature, uniformization theorem, Yamabe flow, Calabi flow.
result The flows converge to metrics with constant discrete curvature, confirming conjectures.
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.
New geometric variant of factorization homology for conformally flat manifolds.
problem Defining invariants of conformally flat manifolds.
method Introducing a metric-dependent geometric variant of factorization homology.
result Left Kan extensions of conformally flat d-disk algebras define invariants of conformally flat manifolds. 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 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…
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.
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…
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.
Stable discretizations for elastic flow on Riemannian manifolds.
problem Discretizing elastic flow on curved spaces.
method Conformally flat Riemannian manifolds discretization.
result Robust and quadratic convergence of the method.
Nonexistence theorem for product type manifolds, proving no locally conformally flat metrics.
problem Proving nonexistence of locally conformally flat metrics on product manifolds.
method Nonexistence theorem for product type manifolds.
result 4-manifold Σ_g×Σ_h does not admit locally conformally flat metrics for g≥2 and h≥1.
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 …
A conformal map from a Riemann surface to the Euclidean four-space is explained in terms of its twistor lift. A local factorization of a differential of a conformal map is obtained. As an application, the factorization of a differential provides an upper bound of the area of a super-conformal map around a branch point.
A method for disentangling discrete and continuous factors of data without using a discriminator network.
problem Unsupervised disentanglement of discrete and continuous factors of data.
method A procedure that minimizes total correlation of continuous latent variables and a separate discrete inference procedure.
result The method significantly outperforms current disentanglement methods based on disentanglement score and inference network classification score.
For a complete Riemannian metric, a pointwise conformal transformation may lead to a complete or incomplete transformed Riemannian metric, depending on the behavior of the conformal factor. We establish conditions on the growth of the conformal factor towards the infinity of the Riemannian metric, such that the conform…
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.
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.
A new financial system with ethics risk modeled using fractional calculus.
problem Modeling financial systems with ethical considerations and market confidence.
method Introduced a five-dimensional conformable derivative financial system and a discretization scheme.
result Numerical solutions of the conformable derivative system were tested for hyperchaos.
Discond-VAE separates continuous and discrete factors in data.
problem Separating shared and class-specific variations in real-world data.
method Introduces private and public latent variables to represent continuous and discrete factors, respectively.
result Discond-VAE successfully disentangles class-dependent continuous factors from discrete factors.
Skew parallelogram nets factorize, encompassing discrete differential geometry.
problem Factorization of polynomials in discrete differential geometry.
method Lax representation, Bäcklund transformations, factorization of polynomials.
result Skew parallelogram nets encompass all systems with polynomial representations.
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.
A new method uncovers discrete and continuous factors in gene expression data.
problem Jointly identifying discrete and continuous factors of variability without supervision.
method cpl-mixVAE framework using multiple interacting networks.
result The method successfully uncovers discrete and continuous factors in gene expression data.