Related papers: Hyperbolic distance to submanifolds in an almost-c…

We derive an explicit lower bound on the radius of a ball embedded in a quaternionic hyperbolic manifold.

In this paper, we prove the following result : let X be a complex manifold, hyperbolic for the Carath\'eodory distance and let U be an open set relatively compact in X. Then, there exists k<1 such that we get, for the Carath\'eodory…

In this paper we study the global geometry of the Kobayashi metric on domains in complex Euclidean space. We are particularly interested in developing necessary and sufficient conditions for the Kobayashi metric to be Gromov hyperbolic. For…

In this paper, we state two combination theorems for relatively quasiconvex subgroups in a relatively hyperbolic group. Applications are given to the separability of double cosets of certain relatively quasiconvex subgroups and the…

It is conjectured that every cusped hyperbolic 3-manifold has a decomposition into positive volume ideal hyperbolic tetrahedra (a "geometric" triangulation of the manifold). Under a mild homology assumption on the manifold we construct…

We prove $\epsilon$-closeness of hypersurfaces to a sphere in Euclidean space under the assumption that the traceless second fundamental form is $\delta$-small compared to the mean curvature. We give the explicit dependence of $\delta$ on…

An interesting question about quasiconvexity in a hyperbolic group concerns finding classes of quasiconvex subsets that are closed under finite intersections. A known example is the class of all quasiconvex subgroups. However, not much is…

We prove sharp bounds for the product and the sum of two hyperbolic distances between the opposite sides of hyperbolic Lambert quadrilaterals in the unit disk. Furthermore, we study the images of Lambert quadrilaterals under quasiconformal…

Let $1 \to K \longrightarrow G \stackrel{\pi}\longrightarrow Q$ be an exact sequence of hyperbolic groups. Let $Q_1 < Q$ be a quasiconvex subgroup and let $G_1=\pi^{-1}(Q_1)$. Under relatively mild conditions (e.g. if $K$ is a closed…

We show that any compact orientable hyperbolic 3-cone-manifold with cone angle at most \pi can be continuously deformed to a complete hyperbolic manifold homeomorphic to the complement of the singularity. This together with the local…

In this paper we prove that every quasi-projective base space $V$ of smooth family of minimal projective manifolds with maximal variation is pseudo Kobayashi hyperbolic, i.e. $V$ is Kobayashi hyperbolic modulo a proper subvariety…

Among other things, we prove the following two topologcal statements about closed hyperbolic 3-manifolds. First, every rational second homology class of a closed hyperbolic 3-manifold has a positve integral multiple represented by an…

This paper builds one-cusped complex hyperbolic $2$-manifolds by an explicit geometric construction. Specifically, for each odd $d \ge 1$ there is a smooth projective surface $Z_d$ with $c_1^2(Z_d) = c_2(Z_d) = 6d$ and a smooth irreducible…

We study the geometry of hyperconvex representations of hyperbolic groups in ${\rm PSL}(d,\mathbb{C})$ and establish two structural results: a group admitting a hyperconvex representation is virtually isomorphic to a Kleinian group, and its…

In this paper, we continue our study of quasihyperbolic metric in Banach spaces. The main results of the paper present a criterion for smoothness of geodesics of quasihyperbolic type metrics in Banach spaces, under a Dini type condition on…

Recent research in representation learning has shown that hierarchical data lends itself to low-dimensional and highly informative representations in hyperbolic space. However, even if hyperbolic embeddings have gathered attention in image…

We prove that if a proper metric space is quasi-isometric to a finitely generated group and to a space with a horoball over a finitely generated group, then that space is quasi-isometric to a rank-one symmetric space or the real line.

We prove that all hierarchically hyperbolic spaces have finite asymptotic dimension and obtain strong bounds on these dimensions. One application of this result is to obtain the sharpest known bound on the asymptotic dimension of the…

We give an explicit construction of special strongly pseudoconvex domains in C^n of handlebody type, i.e., domains which are small tubes surrounding the union of a quadratic strongly pseudoconvex domain with an attached totally real handle.…

We describe a new and robust method to prove rigidity results in complex dynamics. The new ingredient is the geometry of the critical puzzle pieces: under control of geometry and ``complex bounds'', two generalized polynomial-like maps…