arXiv research

We prove that spherical spectral analysis and synthesis hold in Damek-Ricci spaces and derive two-radius theorems.

Study compares eigenvalues on spherically symmetric manifolds to Euclidean balls.

Study on spherical bodies of constant width on the unit sphere, proving bounds on their relative effective radius.

Let B be a thick spherical building equipped with its natural CAT(1) metric and let M be a proper, convex subset of B. If M is open or if M is a closed ball of radius pi/2, then the maximal subcomplex supported by the complement of M is spherical and non contractible.

New minimal surfaces found with spherical curvature lines.

The study proves a rigidity theorem for compact manifolds with boundary.

We define a capacity which measures the size of Weinstein tubular neighbourhoods of Lagrangian submanifolds. In symplectic vector spaces this leads to bounds on the codisc radius for any closed Lagrangian submanifold in terms of Viterbo's isoperimetric inequality. Moreover, we prove a generalization of Gromov's packing…

For spherically symmetric distributions, efficient quantisation can be achieved with moderate sample sizes.

Abstract. In this paper we prove several rigidity theorems related to and including Lytchak's problem. The focus is on Alexandrov spaces with \curv\geq1, nonempty boundary, and maximal radius \fracπ{2}. We exhibit many such spaces that indicate that this class is remarkably flexible. Nevertheless, we also show that whe…

Uniqueness theorem for extremal charged black holes in de Sitter space.

In this paper, we study Mannheim surface offsets in dual space. By the aid of the E. Study Mapping, we consider ruled surfaces as dual unit spherical curves and define the Mannheim offsets of the ruled surfaces by means of dual geodesic trihedron (dual Darboux frame). We obtain the relationships between the invariants …

Let M be a complete Riemannian manifold whose sectional curvature is bounded above by 1. We say that M has positive spherical rank if along every geodesic one hits a conjugate point at t=π. The following theorem is then proved: If M is a complete, simply connected Riemannian manifold with upper curvature bound 1 and po…

Critical spherical catenoids have Robin nullity and asymptotic radius determined.

It is known that the so-called rotation minimizing (RM) frames allow for a simple and elegant characterization of geodesic spherical curves in Euclidean, hyperbolic, and spherical spaces through a certain linear equation involving the coefficients that dictate the RM frame motion (da Silva, da Silva in Mediterr J Math …

This paper connects Laguerre minimal surfaces to Weierstrass representations.

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

Sharp focal radius estimate for hypersurfaces in manifolds with positive curvature.

Paper offers a method for finding the smallest sphere enclosing a set in d-dimensional space.

Small bubbles sliding on a boundary maintain half-spherical shape.

The paper proves reverse inequalities in various geometric settings using curvature radius data.

Artin-Tits groups act on a certain delta-hyperbolic complex, called the "additional length complex". For an element of the group, acting loxodromically on this complex is a property analogous to the property of being pseudo-Anosov for elements of mapping class groups. By analogy with a well-known conjecture about mappi…

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…

H. Hotelling proved that in the n-dimensional Euclidean or spherical space, the volume of a tube of small radius about a curve depends only on the length of the curve and the radius. A. Gray and L. Vanhecke extended Hotelling's theorem to rank one symmetric spaces computing the volumes of the tubes explicitly in these …

Closed and broken electromagnetic orbits in Kerr-Newman spacetime

The paper proves properties of open manifolds with positive isotropic curvature.

The study proves manifolds with positive scalar curvature can be decomposed into spherical and toroidal pieces.

An almost-Fuchsian group is a quasi-Fuchsian group such that the quotient hyperbolic manifold contains a closed incompressible minimal surface with principal curvatures contained in (-1,1). We show that the domain of discontinuity of an almost-Fuchsian group contains many balls of a fixed spherical radius in the visual…

Researchers found cylindrical steady gradient solitons in 3D.

Information mapping is a popular application of Multivoxel Pattern Analysis (MVPA) to fMRI. Information maps are constructed using the so called searchlight method, where the spherical multivoxel neighborhood of every voxel (i.e., a searchlight) in the brain is evaluated for the presence of task-relevant response patte…

Due to the isotropy $d$-dimensional hyperbolic space, there exist a spherically symmetric fundamental solution for its corresponding Laplace-Beltrami operator. On the $R$-radius hyperboloid model of $d$-dimensional hyperbolic geometry with $R>0$ and $d\ge 2$, we compute azimuthal Fourier expansions for a fundamental so…

We initiate the mathematical study of spherical collapse of self-gravitating charged scalar fields. The main result gives a complete characterization of the future boundary of spacetime, providing a starting point for studying the cosmic censorship conjectures. In general, the boundary includes two null components, one…

During an operation of surgery on a Riemannian manifold and along a given embedded submanifold, one needs to replace the (old) metric induced by the exponential map on a tubular neighborhood of the submanifold by the Sasakian metric. So a good understanding of the behavior of these two metrics is important, this is our…

Under mean radius of curvature flow, a closed convex surface in Euclidean space is known to expand exponentially to infinity. In the 3-dimensional case we prove that the oriented normals to the flowing surface converge to the oriented normals of a round sphere whose centre is determined by the initial surface. To prove…

The large-N limit of Segal-Bargmann transform on spheres is studied.

The study proves properties of optimizers for sets maximizing perimeter under fixed volume constraints.

Sharp bounds and rigidity theorems for eigenvalues on manifolds.

In this paper we prove the following pointwise and curvature-free estimates on convexity radius, injectivity radius and local behavior of geodesics in a complete Riemannian manifold $M$: 1) the convexity radius of $p$, $\operatorname{conv}(p)\ge \min\{\frac{1}{2}\operatorname{inj}(p),\operatorname{foc}(B_{\operatorname…

New estimates quantify blow-up rates of spacelike singularities in gravitational collapse.

Positive injectivity radius for manifolds with Lie structure at infinity.

The ratio of convexity radius over injectivity radius may be made arbitrarily small within the class of compact Riemannian manifolds of any fixed dimension at least two. This is proved using Gulliver's method of constructing manifolds with focal points but no conjugate points. The approach is suggested by a characteriz…

Upper bound for conjugate radius in open manifolds with scalar curvature and spectrum constraints.

Develops efficient algorithms for learning latent-variable models using implicit moment tensor computation.

Proves upper bound on filling radius for manifolds with positive scalar curvature.

Compact theorem for minimal surfaces with lower injectivity radius.

Injectivity radius on Stiefel manifold is π.

Mathematical framework for minimum enclosing ball problem.

Lower bound on boundary injectivity radius for specific tubes.

Study gives bounds on filling radius for Riemannian manifolds.