Related papers: Diameters of 3-Sphere Quotients

We show that there exist hyperbolic knots in the 3-sphere such that the set of points of large injectivity radius in the complement take up the bulk of the volume. More precisely, given a finite volume hyperbolic manifold, for any bound R>0…

Let $G$ be a complex connected reductive algebraic group and $G/B$ denote the flag variety of $G$. A $G$-homogeneous space $G/H$ is said to be {\it spherical} if $H$ acts on $G/B$ with finitely many orbits. A class of spherical homogeneous…

In this paper we consider min-max minimal surfaces in three-manifolds and prove some rigidity results. For instance, we prove that any metric on a 3-sphere which has scalar curvature greater than or equal to 6 and is not round must have an…

Motivated by problems on apparent horizons in general relativity, we prove the following theorem on minimal surfaces: Let $g$ be a metric on the three-sphere $S^3$ satisfying $Ric(g) \geq 2 g$. If the volume of $(S^3, g)$ is no less than…

In this paper, the following two theorems are proved: $(1)$ every spherical convex body $W$ of constant width $\Delta (W) \geq \frac{\pi}{2}$ may be covered by a disk of radius $\Delta(W) + \arcsin \left( \frac{2\sqrt{3}}{3} \cdot \cos…

P\'al's isominwidth theorem states that for a fixed minimal width, the regular triangle has minimal area. A spherical version of this theorem was proven by Bezdek and Blekherman, if the minimal width is at most $\tfrac \pi 2$. If the width…

We give a lower bound for the degree of a finite cover of a hyperbolic 3-manifold which fibers over the circle, in terms of volume, the diameter of the manifold and other new invariants.

For three dimensional complete Riemannian manifolds with scalar curvature no less than one, we obtain the sharp upper bound of complete stable minimal surfaces' diameter.

Employing ideas of noncommutative geometry, certain dimensional invariant for quantum homogeneous spaces has been proposed and here we take up its computation for quaternion spheres.

We prove the following three statements: 1) Let $(A, \bar A)$ be a partition of the spherical surface $S^n$ into two measurable sets. Let $st_A$ and $st_{\bar A}$ be their measure density functions of distance. Then $|st_A - st_{\bar A}|$…

In this paper, we prove that the systolic volume of a closed aspherical 3-manifold is bounded below in terms of complexity. Systolic volume is defined as the optimal constant in a systolic inequality. Babenko showed that the systolic volume…

We show that the topological complexity of an aspherical space $X$ is bounded below by the cohomological dimension of the direct product $A\times B$, whenever $A$ and $B$ are subgroups of $\pi_1(X)$ whose conjugates intersect trivially. For…

A small polygon is a polygon of unit diameter. The maximal width of an equilateral small polygon with $n=2^s$ vertices is not known when $s \ge 3$. This paper solves the first open case and finds the optimal equilateral small octagon. Its…

We prove that the cardinality of the torsion subgroups in homology of a closed hyperbolic manifold of any dimension can be bounded by a doubly exponential function of its diameter. It would follow from a conjecture by Bergeron and Venkatesh…

In this note we provide several lower bounds for the volume of a geodesic ball within the injectivity radius in a $3$-dimensional Riemannian manifold assuming only upper bounds for the Ricci curvature.

We obtain exact lower bounds for Kolmogorov $n$-widths in spaces $C$ and $L$ of classes of convolutions with Neumann kernel $N_{q,\beta}(t)=\sum\limits_{k=1}^{\infty}\dfrac{q^k}{k}\cos\left(kt-\dfrac{\beta\pi}{2}\right)$, ${q\in(0,1)}$,…

Let $E(x;\omega)$ be the error term for the number of integer lattice points lying inside a $3$-dimensional Cygan-Kor\'anyi spherical shell of inner radius $x$ and gap width $\omega(x)>0$. Assuming that $\omega(x)\to0$ as $x\to\infty$, and…

We study isoperimetric inequalities on "slabs", namely weighted Riemannian manifolds obtained as the product of the uniform measure on a finite length interval with a codimension-one base. As our two main applications, we consider the case…

This is the sixth in a series of papers giving a proof of the Kepler conjecture, which asserts that the density of a packing of congruent spheres in three dimensions is never greater than $\pi/\sqrt{18}\approx 0.74048...$. This is the…

Let $(S, m)$ be an $n$-dimensional regular local ring essentially of finite type over a field and let $I$ be an ideal of $S$. We prove that if $\text{depth} S/I \ge 3$, then the cohomological dimension $\mathrm{cd}(S, I)$ of $I$ is less…