Geometrically finite groups,Patterson-Sullivan measures and Ratner's ridigity theorem

Summary The rigidity properties of the horospherical foliations of geometrically finite hyperbolic manifolds are investigated. Ratner's theorem generalizes to these foliations with respect to the Patterson-Sullivan measure. In the spirit of Mostow, we prove the nonexistence of invariant measurable distributions on the boundary of hyperbolic space for geometrically finite groups. Finally, we show that the frame flow on geometrically finite hyperbolic manifolds is Bernoulli.partially supported by NSF Grant No. DMS-820-04024partially supported by NSF Grant No. DMS-85-02319     

Hyperbolic cone-manifolds, short geodesics, and Schwarzian derivatives

Given a geometrically finite hyperbolic cone-manifold, with the cone-singularity sufficiently short, we construct a one-parameter family of cone-manifolds decreasing the cone-angle to zero. We also control the geometry of this one-parameter family via the Schwarzian derivative of the projective boundary and the length of closed geodesics.

     

Deformations of hyperbolic convex polyhedra and cone-3-manifolds

The Stoker problem, first formulated in Stoker (Commun. Pure Appl. Math. 21:119–168, 1968), consists in understanding to what extent a convex polyhedron is determined by its dihedral angles. By means of the double construction, this problem is intimately related to rigidity issues for 3-dimensional cone-manifolds. In Mazzeo and Montcouquiol (J. Differ. Geom. 87(3):525–576, 2011), two such rigidity results were proven, implying that the infinitesimal version of the Stoker conjecture is true in the hyperbolic and Euclidean cases. In this second article, we show that local rigidity holds and prove that the space of convex hyperbolic polyhedra with given combinatorial type is locally parametrized by the set of dihedral angles, together with a similar statement for hyperbolic cone-3-manifolds.     

Limit sets of relatively hyperbolic groups

In this paper, we prove a limit set intersection theorem in relatively hyperbolic groups. Our approach is based on a study of dynamical quasiconvexity of relatively quasiconvex subgroups. Using dynamical quasiconvexity, many well-known results on limit sets of geometrically finite Kleinian groups are derived in general convergence groups. We also establish dynamical quasiconvexity of undistorted subgroups in finitely generated groups with nontrivial Floyd boundaries.     

Limit sets of geometrically finite groups acting on Busemann spaces

In this paper, we investigate limit sets of geometrically finite groups acting on Busemann spaces. We show a Busemann space analogue of several results proved by A. Ranjbar-Motlagh for geometrically finite groups acting on hyperbolic spaces in the sense of Gromov.     

Rigidity and Positivity of Mass for Asymptotically Hyperbolic Manifolds

The Witten spinorial argument has been adapted in several works over the years to prove positivity of mass in the asymptotically AdS and asymptotically hyperbolic settings in arbitrary dimensions. In this paper we prove a scalar curvature rigidity result and a positive mass theorem for asymptotically hyperbolic manifolds that do not require a spin assumption. The positive mass theorem is reduced to the rigidity case by a deformation construction near the conformal boundary. The proof of the rigidity result is based on a study of minimizers of the BPS brane action. Submitted: March 16, 2007. Accepted: June 14, 2007.     

Minimal surfaces and particles in 3-manifolds

We consider 3-dimensional anti-de Sitter manifolds with conical singularities along time-like lines, which is what in the physics literature is known as manifolds with particles. We show that the space of such cone-manifolds is parametrized by the cotangent bundle of Teichmüller space, and that moreover such cone-manifolds have a canonical foliation by space-like surfaces. We extend these results to de Sitter and Minkowski cone-manifolds, as well as to some related “quasifuchsian” hyperbolic manifolds with conical singularities along infinite lines, in this later case under the condition that they contain a minimal surface with principal curvatures less than 1. In the hyperbolic case the space of such cone-manifolds turns out to be parametrized by an open subset in the cotangent bundle of Teichmüller space. For all settings, the symplectic form on the moduli space of 3-manifolds that comes from parameterization by the cotangent bundle of Teichmüller space is the same as the 3-dimensional gravity one. The proofs use minimal (or maximal, or CMC) surfaces, along with some results of Mess on AdS manifolds, which are recovered here in a different way, using differential-geometric methods and a result of Labourie on some mappings between hyperbolic surfaces, that allows an extension to cone-manifolds.      

Geometrically infinite surfaces with discrete length spectra

It is well-known that the length spectrum of a geometrically finite hyperbolic manifold is discrete. In this paper, we begin a study of the length spectrum for geometrically infinite hyperbolic surfaces. In this generality, it is possible that the spectrum is not discrete and the main focus of this work is to find necessary and sufficient conditions for a geometrically infinite surface to have a discrete spectrum. After deriving a number of properties of the length spectrum, we show that every topological surface of infinite type admits both an infinite dimensional family of quasiconformally distinct hyperbolic structures having a discrete length spectrum, and an infinite dimensional family of quasiconformally distinct structures with a nondiscrete spectrum. Moreover, there exists such an infinite dimensional subspace arbitrarily close to (in the Fenchel-Nielsen topology) any hyperbolic structure.      

A Bowen type rigidity theorem for non-cocompact hyperbolic groups

We establish a Bowen type rigidity theorem for the fundamental group of a noncompact hyperbolic manifold of finite volume (with dimension at least 3).      

Non-realizability and ending laminations: Proof of the density conjecture

We give a complete proof of the Bers?CSullivan?CThurston density conjecture. In the light of the ending lamination theorem, it suffices to prove that any collection of possible ending invariants is realized by some algebraic limit of geometrically finite hyperbolic manifolds. The ending invariants are either Riemann surfaces or filling laminations supporting Masur domain measured laminations and satisfying some mild additional conditions. With any set of ending invariants we associate a sequence of geometrically finite hyperbolic manifolds and prove that this sequence has a convergent subsequence. We derive the necessary compactness theorem combining the Rips machine with non-existence results for certain small actions on real trees of free products of surface groups and free groups. We prove then that the obtained algebraic limit has the desired conformal boundaries and the property that none of the filling laminations is realized by a pleated surface. In order to be able to apply the ending lamination theorem, we have to prove that this algebraic limit has the desired topological type and that these non-realized laminations are ending laminations. That this is the case is the main novel technical result of this paper. Loosely speaking, we show that a filling Masur domain lamination which is not realized is an ending lamination.     

Maximal tubes under the deformations of 3-dimensional hyperbolic cone-manifolds

Hodgson and Kerckhoff found a small bound on Dehn surgered 3-manifolds from hyperbolic knots not admitting hyperbolic structures using deformations of hyperbolic cone-manifolds. They asked whether the area normalized meridian length squared of maximal tubular neighborhoods of the singular locus of the cone-manifold is decreasing and that summed with the cone-angle squared is increasing as we deform the cone-angles. We confirm this near 0 cone-angles for an infinite family of hyperbolic cone-manifolds obtained by Dehn surgeries along the Whitehead link complements. The basic method rests on explicit holonomy computations using the A-polynomials and finding the maximal tubes. One of the key tools is the Taylor expansion of a geometric component of the zero set of the A-polynomial in terms of the cone-angles. We also show that a sequence of Taylor expansions for Dehn surgered manifolds converges to 1 for the limit hyperbolic manifold.     

Symmetric differentials, Kähler groups and ball quotients

While Margulis’ superrigidity theorem completely describes the finite dimensional linear representations of lattices of higher rank simple real Lie groups, almost nothing is known concerning the representation theory of complex hyperbolic lattices. The main result of this paper (Theorem 1.3) is a strong rigidity theorem for a certain class of cocompact arithmetic complex hyperbolic lattices. It relies on the following two ingredients:

• Theorem 1.6 showing that the representations of the topological fundamental group of a compact Kähler manifold X are controlled by the global symmetric differentials on X.
• An arithmetic vanishing theorem for global symmetric differentials on certain compact ball quotients using automorphic forms, in particular deep results of Clozel on base change (Theorem 1.11).

     

Rees–Shishikura’s theorem for geometrically finite rational maps

In this paper, we generalize Rees–Shishikura’s theorem to the class of geometrically finite rational maps.     

On the Gauss map of complete CMC hypersurfaces in the hyperbolic space

In this paper, as suitable applications of the so-called Omori–Yau generalized maximum principle, we obtain rigidity results concerning to complete hypersurfaces with constant mean curvature in the hyperbolic space, under appropriated restrictions on their Gauss image. Furthermore, by supposing a linear dependence between support functions naturally attached to such hypersurfaces, we establish a characterization theorem.     

Projective deformations of hyperbolic Coxeter 3-orbifolds

By using Klein??s model for hyperbolic geometry, hyperbolic structures on orbifolds or manifolds provide examples of real projective structures. By Andreev??s theorem, many 3-dimensional reflection orbifolds admit a finite volume hyperbolic structure, and such a hyperbolic structure is unique. However, the induced real projective structure on some such 3-orbifolds deforms into a family of real projective structures that are not induced from hyperbolic structures. In this paper, we find new classes of compact and complete hyperbolic reflection 3-orbifolds with such deformations. We also explain numerical and exact results on projective deformations of some compact hyperbolic cubes and dodecahedra.     

Spontaneous Surgery on the Borromean Rings

We study the cone-manifolds whose singular sets are obtained by orbifold and spontaneous surgeries on components of the Borromean rings. We establish existence of geometric structures on these manifolds. For manifolds with hyperbolic structure we obtain an integral representation for volumes.     

On the Isometry Groups of Hyperbolic Orbifolds

A generic, geometrically finite, hyperbolic n-orbifold is proved to have a finite group of isometries.     

在有限区间上的简单双曲样条

在插值和曲线拟合中,简单双曲样条和三次样条相比,由前者得到的曲线更好些.因为把三次样条用作“流线型”的内插曲线,在某些应用中,特别是船线的光顺中,会产生多余的拐点.而简单双曲样条则不会产生这种情况.     

局部对称的黎曼流形中的极小子流形

主要研究了局部对称的黎曼流形中的定向紧致无边极小子流形的内蕴刚性问题,利用一个矩阵不等式,得到了这类子流形的一个刚性定理．所得结果部分改进了已有的一个结论．     

Effective Circle Count for Apollonian Packings and Closed Horospheres

The main result of this paper is an effective count for Apollonian circle packings that are either bounded or contain two parallel lines. We obtain this by proving an effective equidistribution of closed horospheres in the unit tangent bundle of a geometrically finite hyperbolic 3-manifold, whose fundamental group has critical exponent bigger than 1. We also discuss applications to affine sieves. Analogous results for surfaces are treated as well.