Related papers: Balls in complex hyperbolic manifolds

Let $O$ be a closed $n$-dimensional arithmetic (real or complex) hyperbolic orbifold. We show that the diameter of $O$ is bounded above by $$\frac{c_1\log vol(O) + c_2}{h(O)},$$ where $h(O)$ is the Cheeger constant of $O$, $vol(O)$ is its…

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.

We show that the infimum of the dual volume of the convex core of a convex co-compact hyperbolic $3$-manifold with incompressible boundary coincides with the infimum of the Riemannian volume of its convex core, as we vary the geometry by…

In this note we prove that for any compact subset $S$ of a Busemann surface $({\mathcal S},d)$ (in particular, for any simple polygon with geodesic metric) and any positive number $\delta$, the minimum number of closed balls of radius…

Since the set of volumes of hyperbolic 3-manifolds is well ordered, for each fixed g there is a genus-g surface bundle over the circle of minimal volume. Here, we introduce an explicit family of genus-g bundles which we conjecture are the…

We give an upper bound for the number of compact essential orientable non-isotopic surfaces, with Euler characteristic at least some constant $\chi$, properly embedded in a finite-volume hyperbolic 3-manifold $M$, closed or cusped. This…

The smallest $r$ so that a metric $r$-ball covers a metric space $M$ is called the radius of $M$. The volume of a metric $r$-ball in the space form of constant curvature $k$ is an upper bound for the volume of any Riemannian manifold with…

Let $\Sigma$ be a $k$-dimensional complete proper minimal submanifold in the Poincar\'{e} ball model $B^n$ of hyperbolic geometry. If we consider $\Sigma$ as a subset of the unit ball $B^n$ in Euclidean space, we can measure the Euclidean…

It is known that the volume function for hyperbolic manifolds of dimension $\geq 3$ is finite-to-one. We show that the number of nonhomeomorphic hyperbolic 4-manifolds with the same volume can be made arbitrarily large. This is done by…

We compare the volume of a hyperbolic 3-manifold $M$ of finite volume and the complexity of its fundamental group.

Let $\mathcal{M}_{g,\epsilon}$ be the $\epsilon$-thick part of the moduli space $\mathcal{M}_g$ of closed genus $g$ surfaces. In this article, we show that the number of balls of radius $r$ needed to cover $\mathcal{M}_{g,\epsilon}$ is…

We establish Euclidean-type lower bounds for the codimension-1 Hausdorff measure of sets that separate points in doubling and linearly locally contractible metric manifolds. This gives a quantitative topological isoperimetric inequality in…

We prove a volume-rigidity theorem for fuchsian representations of fundamental groups of hyperbolic k-manifolds into Isom(H^n). Namely, we show that if M is a complete hyperbolic k-manifold with finite volume, then the volume of any…

A theorem of W. Derrick ensures that the volume of any Riemannian cube $([0,1]^n,g)$ is bounded below by the product of the distances between opposite codimension-1 faces. In this paper, we establish a discrete analog of Derrick's…

In this paper, we present upper bounds for the depth of some classes of polyhedra, including: polyhedra with finite fundamental group, polyhedra $P$ with abelian or free $\pi_1(P)$ and finitely generated $H_i(tilde{P};\mathbb{Z}$,…

We prove that for $n\geq2$ there exists a compact subset $X$ of the closed ball in $R^{2n}$ of radius $\sqrt{2}$, such that $X$ has Hausdorff dimension $n$ and does not symplectically embed into the standard open symplectic cylinder. The…

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…

In this paper, we study the problem of hyperball (hypersphere) packings in $n$-dimensional hyperbolic space ($n \ge 4$). We prove that to each $n$-dimensional congruent saturated hyperball packing, there is an algorithm to obtain a…

We show that, the solutions of the isoperimetric problem for small volumes are $C^{2,\alpha}$-close to small spheres. On the way, we define a class of submanifolds called pseudo balls, defined by an equation weaker than constancy of mean…

This paper concerns with deformations of noncompact complex hyperbolic manifolds (with locally Bergman metric), varieties of discrete representations of their fundamental groups into $PU(n,1)$ and the problem of (quasiconformal) stability…