Higher dimensional Kleinian groups of small Hausdorff dimension are classical Schottky groups II

Yong Hou

Home/Archive/arXiv

arXiv · 2611.03335

Higher dimensional Kleinian groups of small Hausdorff dimension are classical Schottky groups II

Yong Hou

View original source ↗Read source document ↗Export metadata ↓

Abstract

This is the higher dimensional continuation of our works on Hausdorff dimension of Kleinian groups \cite{Hou2010}, \cite{Hou2020}. In this paper we prove that for all $r\ge 2$ there exists $λ_r>0$, which is independent of the dimension $N$ of the hyperbolic space, such that any finitely generated non-elementary torsion-free Kleinian group of $\Isom(\mathbb{H}^N)$ generated by $r$ elements with limit set of Hausdorff dimension $<λ_r$ is a classical Schottky group. For two generators the bound can be taken to be $1/3$. The proof relies on our previous works in \cite{Hou2010}, \cite{Hou2020} and on a metric carrier graph reduction. Our result can also be considered as the complete converse to the well-known theorem of Phillips-Sarnak \cite{PS} for spaces of any dimension $N$.

Explore related subjects

math.DG

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Yong Hou. 2026-11-02. Higher dimensional Kleinian groups of small Hausdorff dimension are classical Schottky groups II. https://arxiv.org/abs/2611.03335

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Rotational symmetry of complete gradient Einstein-type manifolds with $α=0$ under vanishing conditions on several tensor fields

To provide a unified framework for various geometric solitons, Catino, Mastrolia, Monticelli, and Rigoli introduced gradient Einstein-type manifolds, which include gradient Yamabe solitons, gradient almost Yamabe solitons, gradient $k$-Yamabe solitons and conformal gradient solitons. In this paper, we show that any nontrivial complete four-dimensional gradient Einstein-type manifold with $α=0$, nonnegative scalar curvature, and vanishing Bach tensor is rotationally symmetric. The Bach-flat condition arises naturally in four-dimensional geometry and physics. We prove the main theorem by exploiting the conformal invariance of Bach-flatness and establishing a relation between two functionals. We also provide classification results under vanishing conditions on the Weyl tensor, the Cotton tensor, and the Cao-Chen-type tensor.

math.DG↗

arXiv2026

The min-max width of spheres associated to the distance function

What one obtains when the min-max methods for the distance function are applied on the space of pairs of points of a Riemannian two-sphere? This question is studied in details in the present article. We show that the associated min-max width do not always coincide with half of the length of a simple closed geodesic which is the union of two minimizing geodesics with the same endpoints. Therefore, it is a new geometric invariant. We study the structure of the set of minimizing geodesics joining a pair of points realizing the width, and relationships between this invariant and the diameter. The extrinsic case of an embedded Riemannian sphere is also considered.

math.DG↗

arXiv2026

On the $\mathcal{D}^+_J$ operator on higher-dimensional almost Kähler manifolds

In this paper, we introduce $\mathcal{D}^+_J$, a generalization of $\partial\bar{\partial}$ operator on higher dimensional almost Kähler manifolds. Using the $\mathcal{D}^+_J$ operator, we investigate the $\bar{\partial}$-problem in almost Kähler geometry and explore the generalized Monge-Ampère equation on almost Kähler manifolds. We establish uniqueness up to an additive constant. We also prove a local existence theorem for this equation on almost Kähler four-manifold. At last, we find an elliptical system for $\mathcal{D}^+_J$ operator. As an application, we reorganize the result of Tosatti-Weinkove-Yau in \cite{TWY}.

math.DG↗