arXiv research

Researchers find highest volumes for isospectral spherical orbifolds and space forms.

Study on hyperbolic knotoids, proving their volumes add and providing tables.

We present in this paper a \boundary version" for theorems about minimality of volume and energy functionals on a spherical domain of threedimensional Euclidean sphere.

Spherical Plateau problem studies minimal surfaces in quotients of spheres.

In this paper, we study the symplectic volume of the moduli space of polygons by using Witten's formula. We propose to use this volume as a measure for the flexibility of a polygon with fixed side-lengths. The main result of our is that among all the Spherical and Euclidean polygons with fixed perimeter the regular one…

We show how to compute the circular area invariant of planar curves, and the spherical volume invariant of surfaces, in terms of line and surface integrals, respectively. We use the Divergence Theorem to express the area and volume integrals as line and surface integrals, respectively, against particular kernels; our r…

Study spherical Fourier transform on hypergeometric type harmonic manifolds.

Study shows negatively curved manifolds' spherical volume equals minimal surface area.

The Willmore flow preserves surface volume, leading to convergence to a sphere.

In this paper, we define a certain "proportional volume property" for an unit vector field on a spherical domain in S3. We prove that the volume of these vector fields has an absolute minimum and this value is equal to the volume of the Hopf vector field. Some examples of such vector fields are given. We also study the…

We present two identities (contiguity relation and variation formula) concerning the volume of a spherically faced simplex in the Euclidean space. These identities are described in terms of Cayley-Menger determinants and their differentials involved with hypersphere arrangements. They are derived as a limit of fundamen…

The present paper considers volume formulae, as well as trigonometric identities, that hold for a tetrahedron in 3-dimensional spherical space of constant sectional curvature +1. The tetrahedron possesses a certain symmetry: namely rotation through angle $π$ in the middle points of a certain pair of its skew edges.

For a convex body on the Euclidean unit sphere the spherical convex floating body is introduced. The asymptotic behavior of the volume difference of a spherical convex body and its spherical floating body is investigated. This gives rise to a new spherical area measure, the floating area. Remarkably, this floating area…

Study shows how curved surfaces evolve smoothly to spherical shapes.

Study a flow in a ball that preserves volume and converges to spherical caps.

The present paper considers two infinite families of cone-manifolds endowed with spherical metric. The singular strata is either the torus knot ${\rm t}(2n+1, 2)$ or the torus link ${\rm t}(2n, 2)$. Domains of existence for a spherical metric are found in terms of cone angles and volume formulæ are presented.

Survey on 4-manifolds with specific curvature properties.

Nearly spherical, positively curved surfaces are mapped from a sphere.

It is proved that the volume of spherical or hyperbolic simplices, when considered as a function of the dihedral angles, can be extended continuously to degenerated simplices.

New theorem shows nearly spherical manifolds can be mapped from spheres.

Linear bound on Betti numbers of negatively curved orbifolds.

We prove two rigidity results for complete Riemannian three-manifolds of higher rank. Complete three-manifolds have higher spherical rank if an only if they are spherical space forms. Complete finite volume three-manifolds have higher hyperbolic rank if and only if they are hyperbolic space forms.

We establish the second part of Milnor's conjecture on the volume of simplexes in hyperbolic and spherical spaces. A characterization of the closure of the space of the angle Gram matrices of simplexes is also obtained.

Proves stability of cone-volume measure with nearly constant density.

The paper proves rigidity theorems for area widths of Riemannian manifolds.

Study finds volume minimization principle for conical Calabi-Yau structures on horospherical cones.

We prove global rigidity for compact hyperbolic and spherical cone-3-manifolds with cone-angles $\leq π$ (which are not Seifert fibered in the spherical case), furthermore for a class of hyperbolic cone-3-manifolds of finite volume with cone-angles $\leq π$, possibly with boundary consisting of totally geodesic hyperbo…

Formula for transgressions on polyhedral manifolds, linking face volumes and outer angles.

In this paper, we consider generic corank 2 sub-Riemannian structures, and we show that the Spherical Hausdorf measure is always a C^1-smooth volume, which is in fact generically C^2- smooth out of a stratified subset of codimension 7. In particular, for rank 4, it is generically C^2 . This is the continuation of a pre…

New flow for capillary surfaces converges to spherical caps.

We give sufficient conditions for a noncompact Riemannian manifold, which has quadratic curvature decay, to have finite topological type with ends that are cones over spherical space forms.

Simplified proof of mass center system's uniqueness and generalized Pappus' theorem across Euclidean, spherical, and hyperbolic geometries.

We investigate weighted floating bodies of polytopes. We show that the weighted volume depends on the complete flags of the polytope. This connection is obtained by introducing flag simplices, which translate between the metric and combinatorial structure. Our results are applied in spherical and hyperbolic space. This…

Proves Li-Yau inequality for Helfrich functional, ensuring embeddedness in spherical cases.

We prove that the maximal development of any spherically symmetric spacetime with collisionless matter (obeying the Vlasov equation) or a massless scalar field (obeying the massless wave equation) and possessing a constant mean curvature $S^1 \times S^2$ Cauchy surface also contains a maximal Cauchy surface. Combining …

The paper proves the existence and uniqueness of Calabi-Yau metrics on affine spherical varieties.

Weyl's tube formula holds for various cross-sections under symmetry conditions.

Let M be a closed 5-manifold of pinched curvature 0<δ\le \text{sec}_M\le 1. We prove that M is homeomorphic to a spherical space form if M satisfies one of the following conditions: (i) δ=1/4 and the fundamental group is a non-cyclic group of order at least C, a constant. (ii) The center of the fundamental group has in…

Exact formulas for volumes of specific knot cone-manifolds.

Study spherical convex bodies using $L_p$-floating areas and curvature entropy.

Proves stability of spacetime Penrose inequality for spherical symmetric initial data.

In his paper "On the Schlafli differential equality", J. Milnor conjectured that the volume of n-dimensional hyperbolic and spherical simplices, as a function of the dihedral angles, extends continuously to the closure of the space of allowable angles. A proof of this has recently been given by F. Luo (see math.GT/0412…

3-manifolds with hyperbolic handlebody complements are studied.

For a regular sub-Riemannian manifold we study the Radon-Nikodym derivative of the spherical Hausdorff measure with respect to a smooth volume. We prove that this is the volume of the unit ball in the nilpotent approximation and it is always a continuous function. We then prove that up to dimension 4 it is smooth, whil…

The paper shows how to use hyperplanes and hyperballs interchangeably using inversive geometry.

The paper studies the past inextendibility of FLRW spacetimes using the VDR asymptote.

Study on curvature measures in non-Euclidean spaces linked to Euclidean geometry.

Paper solves capillary Orlicz-Minkowski problem with new inequalities.