arXiv research

Researchers compute the full spectrum of Laplace operator on distance spheres in symmetric spaces.

The area distance to a convex plane curve is an important concept in computer vision. In this paper we describe a strong link between area distances and improper affine spheres. This link makes possible a better understanding of both theories. The concepts of the theory of affine spheres lead to a new definition of an …

In this paper, we characterize all links in the 3-sphere with bridge number at least three that have a bridge sphere of distance two. We show that a link L has a bridge sphere of distance at most two then it falls into at least one of three categories: (1) The exterior of L contains an essential meridional sphere. (2) …

Spheres can be stretched to have larger diameter than antipodal distance.

Sphere theorems for specific manifolds with curvature constraints.

Given a pair of planar curves, one can define its generalized area distance, a concept that generalizes the area distance of a single curve. In this paper, we show that the generalized area distance of a pair of planar curves is an improper indefinite affine spheres with singularities, and, reciprocally, every indefini…

Topology helps estimate chromatic numbers of random graphs on spheres.

We study left-invariant distances on Lie groups for which there exists a one-parameter family of homothetic automorphisms. The main examples are Carnot groups, in particular the Heisenberg group with the standard dilations. We are interested in criteria implying that, locally and away from the diagonal, the distance is…

We show that the Hausdorff distance between any forward and any backward surgery paths in the sphere graph is at most 2. From this it follows that the Hausdorff distance between any two surgery paths with the same initial sphere system and same target sphere system is at most 4. Our proof relies on understanding how su…

New geometric invariant from min-max width of spheres on Riemannian 2-spheres.

We calculate the bridge distance for $m$-bridge knots/links in the $3$-sphere with sufficiently complicated $2m$-plat projections. In particular we show that if the underlying braid of the plat has $n - 1$ rows of twists and all its exponents have absolute value greater than or equal to three then the distance of the b…

Study distance maps on spaces with curvature bound, proving regularity and sphere theorem.

Bounds and constructions for Gromov-Hausdorff distance between spheres.

A new spherical Sliced-Wasserstein distance for data on spheres.

Defines contact surgery distance and shows it's bounded by topological surgery distance by 5.

The paper defines catenary curves in spheres and hyperbolic planes.

The paper shows how to make certain sets on a sphere smooth and flat.

This paper classifies embedded, codimension-one spheres which are null homotopic. This information is used to show that all null homotopic, immersed codimension-one spheres which are taut in the sense of Terng and Thorbergsson are actually distance spheres.

Here we study geodesics connecting two given points on odd-dimensional spheres respecting the Hopf fibration. This geodesic boundary value problem is completely solved in the case of 3-dimensional sphere and some partial results are obtained in the general case. The Carnot-Carathéodory distance is calculated. We also p…

The setting for this brief paper is R^3. Distance between two spheres is understood as distance delta between spherical centers. For instance, a Reuleaux tetrahedron T is the intersection of four unit balls satisfying delta=1 pairwise. Volume and surface area of T are already well-known; our humble contribution is to c…

A new method using spherical harmonics approximates the Sliced-Wasserstein distance.

In this paper we study the geometry of metric spheres in the curve complex of a surface, with the goal of determining the "average" distance between points on a given sphere. Averaging is not technically possible because metric spheres in the curve complex are countably infinite and do not support any invariant probabi…

Continuous curve evolution depends on initial shape on sphere.

The paper connects geometric and topological concepts to bound distances between metric spaces.

The paper extends Gluck and Warner's result on fibrations of spheres by great subspheres.

3D spheres with certain properties approach the round sphere.

For a hyperbolic knot in the 3-sphere, the distance between toroidal surgeries is at most 5, except the figure eight knot. In this paper, we determine all hyperbolic knots that admit two toroidal surgeries with distance 5.

Research explores Lorentzian distances on a specific geometric plane.

We show that if a closed surface in $\mathbb{R}^3$ has entropy near to that of the unit two-sphere, then the surface is close to a round two-sphere in the Hausdorff distance.

A half-geodesic is a closed geodesic realizing the distance between any pair of its points. All geodesics in a round sphere are half-geodesics. Conversely, this note establishes that Riemannian spheres with all geodesics closed and sufficiently many half-geodesics are round.

We show that if the entropy of any closed hypersurface is close to that of a round hyper-sphere, then it is close to a round sphere in Hausdorff distance. Generalizing the result of \cite{BW1} to higher dimensions.

Define the 1-handle stabilization distance between two surfaces properly embedded in a fixed 4-dimensional manifold to be the minimal number of 1-handle stabilizations necessary for the surfaces to become ambiently isotopic. For every nonnegative integer $m$ we find a pair of 2-knots in the 4-sphere whose stabilization…

Model tracks structural changes in Brownian particle configurations on a sphere.

Study uses equivariant topology to measure distances between G metric spaces.

Many statistical and machine learning approaches rely on pairwise distances between data points. The choice of distance metric has a fundamental impact on performance of these procedures, raising questions about how to appropriately calculate distances. When data points are real-valued vectors, by far the most common c…

This article explains how to construct immersed Lagrangian submanifolds in C^2 that are asymptotic at large distance from the origin to a given braid in the 3-sphere. The self-intersections of the Lagrangians are related to the crossings of the braid. These Lagrangians are then used to construct immersed Lagrangians in…

In this paper the regularity of optimal transportation potentials defined on round spheres is investigated. Specifically, this research generalises the calculations done by Loeper, where he showed that the strong (A3) condition of Trudinger and Wang is satisfied on the round sphere, when the cost-function is the geodes…

We construct knots in S^3 with Heegaard splittings of arbitrarily high distance, in any genus. As an application, for any positive integers t and b we find a tunnel number t knot in the three-sphere which has no (t,b)-decomposition.

Spheres in curve complexes are almost simply connected.

Study investigates induced geometry on surfaces in 3D contact manifolds.

We define "fat" train tracks and use them to give a combinatorial criterion for the Hempel distance of Heegaard splittings for closed orientable 3-manifolds. We apply this criterion to 3-manifolds obtained from surgery on knots in the three sphere.

We give sufficient conditions on initial and target measures supported on the sphere $§^n$ to ensure the solution to the optimal transport problem with the cost $|x-y|^2/2$ is a diffeomorphism.

In this paper, we present an InSphereNet method for the problem of 3D object classification. Unlike previous methods that use points, voxels, or multi-view images as inputs of deep neural network (DNN), the proposed method constructs a class of more representative features named infilling spheres from signed distance f…

Tautness of submanifolds in spheres is preserved under Lie sphere transformations.

This paper introduces repulsive Monte Carlo methods for computing the sliced Wasserstein distance.

In this paper we establish a relationship between geodesic nets and critical points of the distance function. We bound the number of balanced points for certain minimizing geodesic nets on manifolds homeomorphic to the $n$-sphere. We also bound the length of certain minimizing geodesic nets.

Singular Riemannian Foliations are particular types of foliations on Riemannian manifolds, in which leaves locally stay at a constant distance from each other. Singular Riemannian Foliations in round spheres play a special role, since they provide "infinitesimal information" about general Singular Riemannian Foliations…

For a hyperbolic knot in the 3-sphere, at most finitely many Dehn surgeries yield non-hyperbolic 3-manifolds. As a typical case of such an exceptional surgery, a toroidal surgery is one that yields a closed 3-manifold containing an incompressible torus. The slope corresponding to a toroidal surgery, called a toroidal s…