arXiv research

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…

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…

Optimal Euclidean structure minimizes energy in weighted toroidal graphs.

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 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 …

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

Optimal discrete harmonic maps between hyperbolic surfaces are found via minimizing energy.

The paper studies complex affine structures near irregular singularities.

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

The Delaunay tessellation of a locally finite subset of hyperbolic space is constructed using convex hulls in Euclidean space of one higher dimension. For finite and lattice-invariant sets it is proven to be a polyhedral decomposition, and versions (necessarily modified from the Euclidean setting) of the empty circumsp…

The article describes how decorations on hyperbolic surfaces lead to unique tessellations and decompositions.

We study the moduli space of euclidean structures with cone points on a surface, and describe a decomposition into cells each of which corresponds to a given combinatorial type of Delaunay tessellation. We use some of the ideas to study hyperbolic structures on three-dimensional manifolds

Convex iso-Delaunay regions found in flat surface strata.

New types of Delaunay hypersurfaces found in spheres.

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…

A Delaunay cell decomposition of a surface with constant curvature gives rise to a circle pattern, consisting of the circles which are circumscribed to the facets. We treat the problem whether there exists a Delaunay cell decomposition for a given (topological) cell decomposition and given intersection angles of the ci…

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…

Improved rigidity of Delaunay triangulated plane.

We consider ``hyperideal'' circle patterns, i.e. patterns of disks appearing in the definition of the Delaunay decomposition associated to a set of disjoint disks, possibly with cone singularities at the center of those disks. Hyperideal circle patterns are associated to hyperideal hyperbolic polyhedra. We describe the…

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…

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…

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…

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.

A famous construction of Gelfand, Kapranov and Zelevinsky associates to each finite point configuration $A \subset \mathbb{R}^d$ a polyhedral fan, which stratifies the space of weight vectors by the combinatorial types of regular subdivisions of $A$. That fan arises as the normal fan of a convex polytope. In a complete…

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

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

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

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.

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.

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.

New connection found between shape reconstruction methods and persistent homology.

Paper defines and evaluates DR complex for persistent homology.

Suppose $C$ is a compact, $n$-edged two-cell of the centered dual decomposition of a locally finite set in the hyperbolic plane, a coarsening of the Delaunay tessellation which was introduced in the author's prior work. We describe an effectively computable lower bound on the area of $C$, given an $n$-tuple of positive…

Four constructions of constant mean curvature (CMC) hypersurfaces in the (n+1)-sphere are given, which should be considered analogues of `classical' constructions that are possible for CMC hypersurfaces in Euclidean space. First, Delaunay-like hypersurfaces, consisting roughly of a chain of hyperspheres winding multipl…

We prove that Delaunay surfaces, except the plane and the catenoid, are the only surfaces in Euclidean space with nonzero constant mean curvature that can be expressed as an implicit equation of type $f(x)+g(y)+h(z)=0$, where $f$, $g$ and $h$ are smooth real functions of one variable.

New method constructs surfaces with constant mean curvature.

The paper confirms conjectures about the topology of triangulated polyhedra and geodesic triangulations on spheres.

In this paper we produce families of complete non compact Riemannian metrics with positive constant $σ_k$-curvature by performing the connected sum of a finite number of given $n$-dimensional Delaunay type solutions, provided $2 \leq 2k < n$. The problem is equivalent to solve a second order fully nonlinear elliptic eq…

We give two numerical methods for computing the first bifurcation point for Delaunay nodoids. With regard to methods for constructing constant mean curvature surfaces, we conclude that the bifurcation point in the analytic method of Mazzeo-Pacard is the same as a limiting point encountered in the integrable systems met…