Hyperbolic geometry

Duality between Hyperbolic and de Sitter Geometry

In this paper we describe trigonometry on the de Sitter surface. For that a characterization of geodesics is given, leading to various types of triangles. We define lengths and angles of these. Then, transferring the concept of polar triangles from spherical geometry into the Minkowski space, we relate hyperbolic with de Sitter triangles such that the proof of the hyperbolic law of cosines for angles becomes much clearer and easier than it is traditionally. Furthermore, polar triangles turn out to be a powerful tool for describing de Sitter trigonometry.

双曲流形与球面流形的谱几何：类比与开放问题

The spectral geometry of hyperbolic and spherical manifolds: analogies and open problems

本文综述双曲流形的谱几何已知结果，探讨其在球面流形的类比成立程度，并列出开放问题。

Phase Portraits of Hyperbolic Geometry

Phase plotting is a useful way of visualising functions on complex space. We reinvent the method in the context of hyperbolic geometry, and we use it to plot functions on various representative surfaces for hyperbolic space, illustrating with direct motions in particular. The reinvention is nontrivial, and we discuss the essential features for ensuring that such visualisations convey useful information. Our approach is to exploit conformal maps between representative surfaces, in order to uniquely represent the preimages of geodesics. Our considerations and methods are prototypical of what one…

On the asymptotic geometry of the hyperbolic plane

Asymptotic subcone of an unbounded metric space is another metric space, capturing the structure of the original space at infinity. In this paper we define a functional metric space S which is an asymptotic subcone of the hyperbolic plane. This space is a real tree branching at every its point. Moreover, it is a homogeneous metric space such that any real tree with countably many vertices can be isometrically embedded into it. This implies that every such tree is also an asymptotic subcone of the hyperbolic plane.

Differential geometry of monopole moduli spaces

This thesis was motivated by a desire to understand the natural geometry of hyperbolic monopole moduli spaces. We take two approaches. Firstly we develop the twistor theory of singular hyperbolic monopoles and use it to study the geometry of their charge 1 moduli spaces. After this we introduce a new way to study the moduli spaces of both Euclidean and hyperbolic monopoles by applying Kodaira's deformation theory to the spectral curve. We obtain new results in both the Euclidean and hyperbolic cases. In particular we prove new cohomology vanishing theorems and find that the hyperbolic monopole…

The Hyperbolic Geometry of the Sinh-Gordon Equation

This preliminary report studies immersed surfaces of constant mean curvature in $H^3$ through their {\it adjusted Gauss maps} (as harmonic maps in $S^2$) and their {\it adjusted frames} in SU(2). Lawson's correspondence between Euclidean CMC surfaces and their hyperbolic cousins is interpreted here under a different perspective: the equivalence of their Weierstrass representations (normalized potentials). This work also presents a construction algorithm for the moving frame, the adjusted frame, their Maurer-Cartan forms, and ultimately the CMC immersion.

Moebius geometry of surfaces of constant mean curvature 1 in hyperbolic space

Various transformations of isothermic surfaces are discussed and their interrelations are analyzed. Applications to cmc-1 surfaces in hyperbolic space and their minimal cousins in Euclidean space are presented: the Umehara-Yamada perturbation, the classical and Bryant's Weierstrass type representations, and the duality for cmc-1 surfaces are interpreted in terms of transformations of isothermic surfaces. A new Weierstrass type representation is introduced and a Moebius geometric characterization of cmc-1 surfaces in hyperbolic space and minimal surfaces in Euclidean space is given.

Hyperbolic Unfoldings of Minimal Hypersurfaces

We study the intrinsic geometry of area minimizing (and also of almost minimizing) hypersurfaces from a new point of view by relating this subject to quasiconformal geometry. For any such hypersurface we define and construct a so-called S-structure which reveals some unexpected geometric and analytic properties of the hypersurface and its singularity set. In this paper, this is used to prove the existence of hyperbolic unfoldings: canonical conformal deformations of the regular part of these hypersurfaces into complete Gromov hyperbolic spaces of bounded geometry with Gromov boundary homeomorp…

On the geometry of the space of oriented lines of the hyperbolic space

Let H be the n-dimensional hyperbolic space of constant sectional curvature -1 and let G be the identity component of the isometry group of H. We find all the G-invariant pseudo-Riemannian metrics on the space OG_n of oriented geodesics of H (modulo orientation preserving reparametrizations). We characterize the null, time- and space-like curves, providing a relationship between the geometries of OG_n and H. Moreover, we show that OG_3 is Kähler and find an orthogonal almost complex structure on OG_7.

在校准几何中关于双曲性和Schwarz引理

Hyperbolicity and Schwarz Lemmas in Calibrated Geometry

本文在校准几何中定义了双曲性概念并推导了Schwarz引理，扩展了复几何中的相关结果。

Geometry and topology of complex hyperbolic and CR-manifolds

We study geometry, topology and deformation spaces of noncompact complex hyperbolic manifolds (geometrically finite, with variable negative curvature), whose properties make them surprisingly different from real hyperbolic manifolds with constant negative curvature. This study uses an interaction between Kähler geometry of the complex hyperbolic space and the contact structure at its infinity (the one-point compactification of the Heisenberg group), in particular an established structural theorem for discrete group actions on nilpotent Lie groups.

Weil-Petersson geometry for families of hyperbolic conical Riemann Surfaces

We study the Weil-Petersson geometry for holomorphic families of Riemann Surfaces equipped with the unique conical metric of constant curvature -1.

A Geometric transition from hyperbolic to anti de Sitter geometry

We introduce a geometric transition between two homogeneous three-dimensional geometries: hyperbolic geometry and anti de Sitter (AdS) geometry. Given a path of three-dimensional hyperbolic structures that collapse down onto a hyperbolic plane, we describe a method for constructing a natural continuation of this path into AdS structures. In particular, when hyperbolic cone manifolds collapse, the AdS manifolds generated on the "other side" of the transition have tachyon singularities. The method involves the study of a new transitional geometry called half-pipe geometry. We demonstrate these m…

On the Ambartzumian-Pleijel identity in hyperbolic geometry

We describe a hyperbolic version of the Ambartzumian-Pleijel identity. We use this identity to prove the hyperbolic Crofton formula and the hyperbolic isoperimetric inequality. This identity also provides a way to compute the chord length distribution for an ideal polygon in the hyperbolic plane. The analogous results for a maximally symmetric, simply connected, $2$-dimensional Riemannian manifold with constant sectional curvature are given at the end.

Persistence of noncompact normally hyperbolic invariant manifolds in bounded geometry

We prove a persistence result for noncompact normally hyperbolic invariant manifolds in Riemannian manifolds of bounded geometry. The bounded geometry of the ambient manifold is a crucial assumption in order to control the uniformity of all estimates throughout the proof.

Some 4-manifold geometry from hyperbolic knots in S^3

A 4-manifold is constructed with some curious metric properties; or maybe it is many 4-manifolds masquerading as one, which would explain why it looks curious. Anyway, knots in the 3-sphere with complete finite volume hyperbolic metrics on their complements play a role in this story.

$L^2$ 几何中的双曲磁单极子

$L^2$ geometry of hyperbolic monopoles

本文针对双曲磁单极子模空间上 $L^2$ 度量发散的问题，提出受超对称启发的规范固定条件加以消除，得到双曲类比于欧几里得情形的超凯勒几何。

Higgs bundles, pseudo-hyperbolic geometry and maximal representations

These notes are an extended version of a talk given by the author in the seminar "Theorie Spectrale et Geometrie" at the Institut Fourier in No- vember 2016. We present here some aspects of a work in collaboration with B. Collier and N. Tholozan (arXiv:1702.08799). We describe how Higgs bundle theory and pseudo-hyperbolic geometry interfere in the study of maximal representations into Hermitian Lie groups of rank 2.

The Geometry of Hyperbolic and Elliptic CR-manifolds of codimension two

The general theory of parabolic geometries is applied to the study of the normal Cartan connections for all hyperbolic and elliptic 6-dimensional CR-manifolds of codimension two. The geometric meaning of the individual components of the torsion is explained and the chains of dimensions one and two are discussed. This seems to be the first application of the general methods to a geometry with a non-connected and non-simple structure group.

Hyperbolic geometry and non-Kahler manifolds with trivial canonical bundle

We use hyperbolic geometry to construct simply-connected symplectic or complex manifolds with trivial canonical bundle and with no compatible Kahler structure. We start with the desingularisations of the quadric cone in C^4: the smoothing is a natural S^3-bundle over H^3, its holomorphic geometry is determined by the hyperbolic metric; the small-resolution is a natural S^2-bundle over H^4 with symplectic geometry determined by the metric. Using hyperbolic geometry, we find orbifold quotients with trivial canonical bundle; smooth examples are produced via crepant resolutions. In particular, we …

Pluricomplex geometry and hyperbolic monopoles

Motivated by strong desire to understand the natural geometry of moduli spaces of hyperbolic monopoles, we introduce and study a new type of geometry: pluricomplex geometry. It is a generalisation of hypercomplex geometry: we still have a 2-sphere of complex structures, but they no longer behave like unit imaginary quaternions. We still require, however, that the 2-sphere of complex structures determines a decomposition of the complexified tangent space as tensor product of C^{2n} and C^2. Among interesting properties of pluricomplex manifold is the existence of a canonical torsion free connec…

Selberg's zeta function and the spectral geometry of geometrically finite hyperbolic surfaces

For hyperbolic Riemann surfaces of finite geometry, we study Selberg's zeta function and its relation to the relative scattering phase and the resonances of the Laplacian. As an application we show that the conjugacy class of a finitely generated, torsion-free, discrete subgroup of SL(2,R) is determined by its trace spectrum up to finitely many possibilities, thus generalizing results of McKean and Mueller to groups which are not necessarily cofinite.

双曲流形的分支覆盖几何

The geometry of branched coverings of hyperbolic manifolds

研究双曲流形的分支覆盖几何性质，重点探讨高维闭合双曲流形沿完全测地子流形的覆盖结构及其几何特性。

Geometry of the smallest 1-form Laplacian eigenvalue on hyperbolic manifolds

We relate small 1-form Laplacian eigenvalues to relative cycle complexity on closed hyperbolic manifolds: small eigenvalues correspond to closed geodesics no multiple of which bounds a surface of small genus. We describe potential applications of this equivalence principle toward proving optimal torsion homology growth in families of hyperbolic 3-manifolds Benjamini-Schramm converging to $\mathbb{H}^3.$

双曲背景几何中组合Ricci流收敛于退化圆填充的研究

Convergences of Combinatorial Ricci Flows to Degenerated Circle Packings in Hyperbolic Background Geometry

该研究针对双曲背景几何中退化圆填充能否实现给定全测地曲率数据的问题，刻画了充要条件并证明唯一性，还借鉴已有方法引入组合Ricci流求解目标退化圆填充曲面。

The symplectic geometry of polygons in hyperbolic 3-space

We study the symplectic geometry of the moduli space of closed n-gons with fixed side-lengths in hyperbolic 3-space. We prove that these moduli spaces have a symplectic structure coming from Poisson Lie theory. We construct completely integrable systems on these moduli spaces by bending n-gons along their diagonals. The results of this paper are the analogues of the results of the first two authors for n-gon linkages in Euclidean 3-space in JDG 44. We conclude by proving by a deformation argument that the moduli spaces of n-gon linkages in hyperbolic 3-space and Euclidean 3-space with the same…

复双曲几何中的Orbifold与orbibundle

Orbifolds and orbibundles in complex hyperbolic geometry

本文利用diffeology从几何角度发展orbibundle理论，提供新工具以计算4-流形几何中底空间为2-orbifold的复双曲圆盘orbibundle的欧拉数与PU(2,1)-表示的Toledo不变量。

Explicit constructions of universal R-trees and asymptotic geometry of hyperbolic spaces

We present some explicit constructions of universal R-trees with applications to the asymptotic geometry of hyperbolic spaces. In particular, we show that any asymptotic cone of a complete simply connected manifold of negative curvature is a complete homogeneous R-tree with the valency $2^{\aleph_0}$ at every point. It implies that all these asymptotic cones are isometric depending neither on a manifold nor on an ultrafilter. It is also proved that the same R-tree can be isometrically embedded at infinity into such a manifold or into a non-abelian free group.

Persistence of noncompact normally hyperbolic invariant manifolds in bounded geometry

We prove a persistence result for noncompact normally hyperbolic invariant manifolds in the setting of Riemannian manifolds of bounded geometry. Bounded geometry of the ambient manifold is a crucial assumption required to control the uniformity of all estimates throughout the proof. The $C^{k,α}$-smoothness result is optimal with respect to the spectral gap condition involved. The core of the persistence proof is based on the Perron method. In the process we derive new results on noncompact submanifolds in bounded geometry: a uniform tubular neighborhood theorem and uniform smooth approximatio…

2-Dimensional Combinatorial Calabi Flow in Hyperbolic Background Geometry

For triangulated surfaces locally embedded in the standard hyperbolic space, we introduce combinatorial Calabi flow as the negative gradient flow of combinatorial Calabi energy. We prove that the flow produces solutions which converge to ZCCP-metric (zero curvature circle packing metric) if the initial energy is small enough. Assuming the curvature has a uniform upper bound less than $2π$, we prove that combinatorial Calabi flow exists for all time. Moreover, it converges to ZCCP-metric if and only if ZCCP-metric exists.

A Discrete Ricci Flow on Surfaces in Hyperbolic Background Geometry

In this paper, we generalize our results in \cite{GX3} to triangulated surfaces in hyperbolic background geometry, which means that all triangles can be embedded in the standard hyperbolic space. We introduce a new discrete Gaussian curvature by dividing the classical discrete Gauss curvature by an area element, which could be taken as the area of the hyperbolic disk packed at each vertex. We prove that the corresponding discrete Ricci flow converges if and only if there exists a circle packing metric with zero curvature. We also prove that the flow converges if the initial curvatures are all …

Synthetic Geometry in Hyperbolic Simplices

Let $τ$ be an $n$-simplex and let $g$ be a metric on $τ$ with constant curvature $κ$. The lengths that $g$ assigns to the edges of $τ$, along with the value of $κ$, uniquely determine all of the geometry of $(τ, g)$. In this paper we focus on hyperbolic simplices ($κ= -1$) and develop geometric formulas which rely only on the edge lengths of $τ$. Our main results are distance and projection formulas in hyperbolic simplices, as well as a projection formula in Euclidean simplices. We also provide analogous formulas in simplices with arbitrary constant curvature $κ$.

Lorentzian Spectral Geometry for Globally Hyperbolic Surfaces

The fermionic signature operator is analyzed on globally hyperbolic Lorentzian surfaces. The connection between the spectrum of the fermionic signature operator and geometric properties of the surface is studied. The findings are illustrated by simple examples and counterexamples.

Holography and the geometry of certain convex cocompact hyperbolic 3-manifolds

Krasnov (arXiv: hep-th/0005106) identified the renormalized volume of a Schottky 3-manifold with the action of the Liouville theory on the conformal infiinity. We try to compute the renormalized volume in terms of more transparent geometric quantities.

Critical exponent and Hausdorff dimension in pseudo-Riemannian hyperbolic geometry

The aim of this article is to understand the geometry of limit sets in pseudo-Riemannian hyperbolic geometry. We focus on a class of subgroups of $\mathrm{PO}(p,q+1)$ introduced by Danciger, Guéritaud and Kassel, called $\mathbb{H}^{p,q}$-convex cocompact. We define a pseudo-Riemannian analogue of critical exponent and Hausdorff dimension of the limit set. We show that they are equal and bounded from above by the usual Hausdorff dimension of the limit set. We also prove a rigidity result in $\mathbb{H}^{2,1}=\mathrm{ADS}^3$ which can be understood as a Lorentzian version of a famous Theorem of…

Hypercomplex limits of pluricomplex structures and the Euclidean limit of hyperbolic monopoles

We discuss the Euclidean limit of hyperbolic SU(2)-monopoles, framed at infinity, from the point of view of pluricomplex geometry. More generally, we discuss the geometry of hypercomplex manifolds arising as limits of pluricomplex manifolds.

Variations of the boundary geometry of 3--dimensional hyperbolic convex cores

A fundamental object in a hyperbolic 3-manifold M is its convex core C(M), defined as the smallest closed non-empty convex subset of M. We investigate the way the geometry of the boundary S of C(M) varies as we vary the hyperbolic metric of M. Thurston observed that the intrinsic metric of S is hyperbolic, and that its bending is described by a measured geodesic lamination. We show that, as the hyperbolic metric of the 3--manifold M varies differentiably, the hyperbolic metric of the surface S varies in a C^1, but usually not C^2, manner. Differentiability properties for the bending measured l…

凸核倍流形从双曲到反德西特几何的过渡

Transition of convex core doubles from hyperbolic to Anti-de sitter geometry

本文提出一种构造方法，研究曲面 $Σ\times\mathbb{S}^1$ 上凸核倍流形在弯曲叶状结构均匀坍缩至零时，从双曲几何经Half-pipe几何过渡到Anti-de Sitter几何，并沿链产生锥奇点。

通过双曲动力学研究非Weinstein Liouville几何的正则性与持久性

Regularity and persistence in non-Weinstein Liouville geometry via hyperbolic dynamics

本文在Liouville插值系统与部分双曲流框架下，基于Anosov 3-流探索非Weinstein Liouville几何的构造，证明了其动力学与几何刚性，并应用于弱控制丛的正则性理论。

Contact Geometry of Hyperbolic Equations of Generic Type

We study the contact geometry of scalar second order hyperbolic equations in the plane of generic type. Following a derivation of parametrized contact-invariants to distinguish Monge-Ampere (class 6-6), Goursat (class 6-7) and generic (class 7-7) hyperbolic equations, we use Cartan's equivalence method to study the generic case. An intriguing feature of this class of equations is that every generic hyperbolic equation admits at most a nine-dimensional contact symmetry algebra. The nine-dimensional bound is sharp: normal forms for the contact-equivalence classes of these maximally symmetric gen…

Intrinsic geometry of convex ideal polyhedra in hyperbolic 3-space

The main result is that every complete finite area hyperbolic metric on a sphere with punctures can be uniquely realized as the induced metric on the surface of a convex ideal polyhedron in hyperbolic 3-space. A number of other observations are included.

Fourier and Gegenbauer expansions for a fundamental solution of the Laplacian in the hyperboloid model of hyperbolic geometry

Due to the isotropy $d$-dimensional hyperbolic space, there exist a spherically symmetric fundamental solution for its corresponding Laplace-Beltrami operator. On the $R$-radius hyperboloid model of $d$-dimensional hyperbolic geometry with $R>0$ and $d\ge 2$, we compute azimuthal Fourier expansions for a fundamental solution of Laplace's equation. For $d\ge 2$, we compute a Gegenbauer polynomial expansion in geodesic polar coordinates for a fundamental solution of Laplace's equation on this negative-constant sectional curvature Riemannian manifold. In three-dimensions, an addition theorem for …

3-dimensional Combinatorial Yamabe Flow in Hyperbolic Background Geometry

We study the 3-dimensional combinatorial Yamabe flow in hyperbolic background geometry. For a triangulation of a 3-manifold, we prove that if the number of tetrahedra incident to each vertex is at least 23, then there exist real or virtual ball packings with vanishing (extended) combinatorial scalar curvature, i.e. the (extended) solid angle at each vertex is equal to 4π. In this case, if such a ball packing is real, then the (extended) combinatorial Yamabe flow converges exponentially fast to that ball packing. Moreover, we prove that there is no real or virtual ball packing with vanishing (e…