Related papers: Balls in complex hyperbolic manifolds

In this paper we study regularity and topological properties of volume constrained minimizers of quasi-perimeters in $\sf RCD$ spaces where the reference measure is the Hausdorff measure. A quasi-perimeter is a functional given by the sum…

We provide a detailed proof of the following folklore theorem: Let mu > 0 be a Margulis constant for 3-dimensional hyperbolic space. Then for any d>0 there exists a constant K>0, depending on mu and d, so that for any complete finite volume…

Let $(M, \partial M)$ be a compact 3-manifold with boundary which admits a complete, convex co-compact hyperbolic metric. For each hyperbolic metric $g$ on $M$ such that $\dr M$ is smooth and strictly convex, the induced metric on $\dr M$…

As was pointed out by Nikulin [8] and Vinberg [10], a right-angled polyhedron of finite volume in hyperbolic n-space $\mathbb{H}^n$ has at least one cusp for $n\geq 5$. We obtain non-trivial lower bounds on the number of cusps of such…

We survey some properties of Gromov--Hausdorff--Prokhorov convergent sequences $(\mathsf{X}_n, d_{\mathsf{X}_n}, \nu_{\mathsf{X}_n})_{n \ge 1}$ of random compact metric spaces equipped with Borel probability measures. We formalize that if…

We establish a min-max estimate on the volume width of a closed Riemannian manifold with nonnegative Ricci curvature. More precisely, we show that every closed Riemannian manifold with nonnegative Ricci curvature admits a PL Morse function…

A compact hyperbolic "cobweb" manifold (hyperbolic space form) of symbol $Cw(6,6,6)$ will be constructed in Fig.1,4,5 as a representant of a presumably infinite series $Cw(2p,2p,2p)$ $(3 \le p \in \bN$ natural numbers). This is a by-product…

This is the second paper of two in a series under the same title ([CRX]); both study the quantitative volume space form rigidity conjecture: a closed $n$-manifold of Ricci curvature at least $(n-1)H$, $H=\pm 1$ or $0$ is diffeomorphic to a…

We prove that there are only finitely many closed hyperbolic 3-manifolds with injectivity radius and first eigenvalue of the Laplacian bounded below whose fundamental groups can be generated by a given number of elements. An application to…

We show that, for horoconvex domains in the hyperbolic space, the product of their fundamental gap with the square of their diameter has no positive lower bound. The result follows from the study of the fundamental gap of geodesic balls as…

Let $P$ be a submanifold properly immersed in a rotationally symmetric manifold having a pole and endowed with a weight $e^h$. The aim of this paper is twofold. First, by assuming certain control on the $h$-mean curvature of $P$, we…

In hyperbolic space density cannot be defined by a limit as we define it in Euclidean space. We describe the local density bounds for sphere packings and we discuss the different attempts to define optimal arrangements in hyperbolic space.

A convex body $R$ in the hyperbolic plane is reduced if any convex body $K\subset R$ has a smaller minimal width than $R$. We examine the area of a family of hyperbolic reduced $n$-gons, and prove that, within this family, regular $n$-gons…

If M is a closed simple 3-manifold whose fundamental group contains a genus-g surface group for some g>1, and if the dimension of H_1(M;Z_2) is at least max(3g-1,6), we show that M contains a closed, incompressible surface of genus at most…

We obtain an upper bound to the packing density of regular tetrahedra. The bound is obtained by showing the existence, in any packing of regular tetrahedra, of a set of disjoint spheres centered on tetrahedron edges, so that each sphere is…

Let $A$ be a compact $d$-dimensional $C^2$ Riemannian manifold with boundary, embedded in ${\bf R}^m$ where $m \geq d \geq 2$, and let $B$ be a nice subset of $A$ (possibly $B=A$). Let $X_1,X_2, \ldots $ be independent random uniform points…

We obtain upper bounds for the isoperimetric quotients of extrinsic balls of submanifolds in ambient spaces which have a lower bound on their radial sectional curvatures. The submanifolds are themselves only assumed to have lower bounds on…

We prove that the covolume of any quasi-arithmetic hyperbolic lattice (a notion that generalizes the definition of arithmetic subgroups) is a rational multiple of the covolume of an arithmetic subgroup. As a corollary, we obtain a good…

Quasifuchsian hyperbolic manifolds, or more generally convex co-compact hyperbolic manifolds, have infinite volume, but they have a well-defined ``renormalized'' volume. We outline some relations between this renormalized volume and the…

We introduce the concept of Bergman bundle attached to a hermitian manifold X, assuming the manifold X to be compact - although the results are local for a large part. The Bergman bundle is some sort of infinite dimensional very ample…