arXiv research

Optimally estimate distances on surfaces using reconstructed meshes.

Delaunay has shown that the Delaunay complex of a finite set of points $P$ of Euclidean space $\mathbb{R}^m$ triangulates the convex hull of $P$, provided that $P$ satisfies a mild genericity property. Voronoi diagrams and Delaunay complexes can be defined for arbitrary Riemannian manifolds. However, Delaunay's generic…

Given a finite set of points in $\mathbb R^n$ and a radius parameter, we study the Čech, Delaunay-Čech, Delaunay (or Alpha), and Wrap complexes in the light of generalized discrete Morse theory. Establishing the Čech and Delaunay complexes as sublevel sets of generalized discrete Morse functions, we prove that the four…

We consider circle packings and, more generally, Delaunay circle patterns - arrangements of circles arising from a Delaunay decomposition of a finite set of points - on surfaces equipped with a complex projective structure. Motivated by a conjecture of Kojima, Mizushima and Tan, we prove that the forgetful map sending …

Classical Delaunay surfaces are highly symmetric constant mean curvature (CMC) submanifolds of space forms. We prove the existence of Delaunay-type hypersurfaces in a large class of compact manifolds, using the geometry of cohomogeneity one group actions and variational bifurcation techniques. Our construction speciali…

Paper defines and evaluates DR complex for persistent homology.

New connection found between shape reconstruction methods and persistent homology.

The notion of Lagrangian $H$-umbilical submanifolds was introduced by B. Y. Chen in 1997, and these submanifolds have appeared in several important problems in the study of Lagrangian submanifolds from the Riemannian geometric point of view. Recently, the author introduced the notion of tangentially biharmonic submanif…

The paper classifies adjacencies in $L^\infty$-Delaunay triangulations of abelian differentials.

Proves rigidity of circle packings in the plane, generalizing previous work.

Convex iso-Delaunay regions found in flat surface strata.

New types of Delaunay hypersurfaces found in spheres.

The paper discusses new Lagrangian constructions and examples.

Given a lattice L of R^n, a polytope D is called a Delaunay polytope in L if the set of its vertices is S\cap L where S is a sphere having no lattice points in its interior. D is called perfect if the only ellipsoid in R^n that contains S\cap L is exactly S. For a vector v of the Leech lattice Λ_{24} we define Λ_{24}(v…

We consider constant mean curvature 1 surfaces in $\mathbb{R}^3$ arising via the DPW method from a holomorphic perturbation of the standard Delaunay potential on the punctured disk. Kilian, Rossman and Schmitt have proven that such a surface is asymptotic to a Delaunay surface. We consider families of such potentials p…

The paper studies complex affine structures near irregular singularities.

Improved rigidity of Delaunay triangulated plane.

The study identifies unique Delaunay surfaces with constant mean curvature.

The generalized Weierstrass representation is used to analyze the asymptotic behavior of a constant mean curvature surface that arises locally from an ordinary differential equation with a regular singularity. We prove that a holomorphic perturbation of an ODE that represents a Delaunay surface generates a constant mea…

Tangential families are 1-parameter families of rays emanating tangentially from smooth curves. We classify tangential family germs up to Left-Right equivalence: we prove that there are two infinite series and four sporadic simple singularities of tangential family germs (in addition to two stable singularities). We gi…

In the following article we discuss Delaunay triangulations for a point cloud on an embedded surface in $\mathbb{R}^3$. We give sufficient conditions on the point cloud to show that the diagonal switch algorithm finds an embedded Delaunay triangulation.

We use a variational principle to prove an existence and uniqueness theorem for planar weighted Delaunay triangulations (with non-intersecting site-circles) with prescribed combinatorial type and circle intersection angles. Such weighted Delaunay triangulations may be interpreted as images of hyperbolic polyhedra with …

We propose a differentiable nonparametric algorithm, the Delaunay triangulation learner (DTL), to solve the functional approximation problem on the basis of a $p$-dimensional feature space. By conducting the Delaunay triangulation algorithm on the data points, the DTL partitions the feature space into a series of $p$-d…

Minimal Delaunay triangulations on hyperbolic surfaces have linear number of vertices.

We study tangential families, i.e. systems of rays emanating tangentially from given curves. We classify, up to Left-Right equivalence, stable singularities of tangential family germs (under deformations among tangential families) and we study their envelopes. We discuss applications of our results to the case of tange…

Plane Delaunay triangulations are rigid under Luo's discrete conformal change.

Study on properties of tangential hypersurfaces in product-like manifolds.

The paper constructs solutions to a critical Dirac equation on spheres.

Construct minimal Lagrangian surfaces in complex projective plane via loop group method.

We show that the of 2-lobed Delaunay tori are stable as constrained Willmore surfaces in the 3-sphere.

A finite subset S of a closed hyperbolic surface F canonically determines a "centered dual decomposition" of F: a cell structure with vertex set S, geodesic edges, and 2-cells that are unions of the corresponding Delaunay polygons. Unlike a Delaunay polygon, a centered dual 2-cell Q is not determined by its collection …

We classify up to diffeomorphism all smooth manifolds homeomorphic to the complex projective m-space $\mathbb{C}P^{m}$ for $m = 5, 6, 7$ and $8$. As an application, for $m = 7$ and $8$, we compute the smooth tangential structure set of $\mathbb{C}P^{m}$ and obtain a bound on the number of smooth homotopy complex projec…

Given a Delaunay decomposition of a compact hyperbolic surface, one may record the topological data of the decomposition, together with the intersection angles between the `empty disks' circumscribing the regions of the decomposition. The main result of this paper is a characterization of when a given topological decom…

An n-dimensional submanifold X of a projective space P^N (C) is called tangentially degenerate if the rank of its Gauss mapping γ: X ---> G (n, N) satisfies 0 < rank γ< n. The authors systematically study the geometry of tangentially degenerate submanifolds of a projective space $P^N (\mathbf{C})$. By means of the foca…

In this paper, we construct Delaunay type constant mean curvature surfaces along a nondegenerate closed geodesic in a 3-dimensional Riemannian manifold.

New matrices link point motions to braid groups.

Study classifies metrics on a twice-punctured sphere, proving Delaunay metrics are complete.

A Delaunay decomposition is a cell decomposition in R^d for which each cell is inscribed in a Euclidean ball which is empty of all other vertices. This article introduces a generalization of the Delaunay decomposition in which the Euclidean balls in the empty ball condition are replaced by other families of regions bou…

We construct constant mean curvature surfaces in euclidean space by gluing n half Delaunay surfaces to a non-degenerate minimal n-noid, using the DPW method.

No compact surfaces with specific curvature can exist near singular limits.

In his seminal 1951 paper "Extreme forms" Coxeter \cite{cox51} observed that for $n \ge 9$ one can add vectors to the perfect lattice $\sfA_9$ so that the resulting perfect lattice, called $\sfA_9^2$ by Coxeter, has exactly the same set of minimal vectors. An inhomogeneous analog of the notion of perfect lattice is tha…

We use the DPW method to obtain the associate family of Delaunay surfaces and derive a formula for the neck size of the surface in terms of the entries of the holomorphic potential.

Proof of existence and uniqueness of weighted Voronoi-Delaunay on polyhedral surfaces.

We show the rigidity of the hexagonal Delaunay triangulated plane under Luo's PL conformality. As a consequence, we obtain a rigidity theorem for a particular type of locally finite convex ideal hyperbolic polyhedra.

Investigates polar tangential angles of curves and their monotonicity.

Study examines tangential real hypersurfaces on Hermite-like manifolds.

A submanifold is said to be tangentially biharmonic if the bitension field of the isometric immersion that defines the submanifold has vanishing tangential component. The purpose of this paper is to prove that a surface in Euclidean $3$-space has tangentially biharmonic normal bundle if and only if it is either minimal…

Optimal Euclidean structure minimizes energy in weighted toroidal graphs.