Xiaoxue Xu, Gendi Wang, Xiaohui Zhang
We compare a Gromov hyperbolic metric with the hyperbolic metric in the unit ball or in the upper half space, and prove sharp comparison inequalities between the Gromov hyperbolic metric and some hyperbolic type metrics. We also obtain several sharp distortion inequalities for the Gromov hyperbolic metric under some families of Möbius transformations.
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Worm domains are not Gromov hyperbolic
该研究在Kobayashi距离框架下，证明了Worm域不具有Gromov双曲性，为复分析中域的几何性质研究提供了新结论。
We establish a version of a classical theorem of Pommerenke, which is a diameter version of the Gehring-Hayman inequality on Gromov hyperbolic domains of $\mathbb{R}^n$. Two applications are given. Firstly, we generalize Ostrowski's Faltensatz to quasihyperbolic geodesics of Gromov hyperbolic domains. Secondly, we prove that unbounded uniform domains can be characterized in the terms of Gromov hyperbolicity and a naturally quasisymmetric correspondence on the boundary, where the Gromov boundary is equipped with a Hamenstädt metric (defined by using a Busemann function).
We prove the non-hyperbolicity of the Kobayashi distance for $\mathcal{C}^{1,1}$-smooth convex domains in $\mathbb{C}^{2}$ which contain an analytic disc in the boundary or have a point of infinite type with rotation symmetry. Moreover, examples of smooth, non pseudoconvex, Gromov hyperbolic domains are given; we prove that the symmetrized polydisc and the tetrablock are not Gromov hyperbolic and write down some results about Gromov hyperbolicity of product spaces.
We show that every quasihyperbolic geodesic in a John space admitting a roughly starlike Gromov hyperbolic quasihyperbolization is a cone arc. This result provides a new approach to the elementary metric geometry question, formulated in \cite[Question 2]{Hei89}, which has been studied by Gehring, Hag, Martio and Heinonen. As an application, we obtain a simple geometric condition connecting uniformity of the space with the existence of Gromov hyperbolic quasihyperbolization.
Uniformization of intrinsic Gromov hyperbolic spaces with Busemann functions
该研究针对内蕴Gromov双曲空间，建立共形形变空间的Gehring-Hayman型定理，证明Gromov边界含至少两点的完备内蕴双曲空间可通过Busemann函数诱导密度单值化，且其Gromov边界与形变空间度量边界自然同构。
After a study of the Kobayashi metrics on certain scaled domains, we show the stabilities of the infinitesimal Kobayashi metrics and the integrated distances in different scaling processes. As an application, we prove that bounded pseudo-convex domains of finite type where the Levi form of every boundary point has corank one are Gromov hyperbolic with respect to the Kobayashi metric. The results in this note generalize Zimmer's and Fiacchi's related works on Gromov hyperbolicity of weakly pseudo-convex domains.
In this paper we give an local estimate for the Kobayashi distance on a bounded convex domain of finite type, which relates to a local pseudodistance near the boundary. The estimate is precise up to a bounded additive term. Also we conclude that the domain equipped with the Kobayashi distance is Gromov hyperbolic which gives another proof of the result of Zimmer.
Let D be a smooth relatively compact and strictly J-pseudoconvex domain in a four dimensional almost complex manifold (M,J). We give sharp estimates of the Kobayashi metric. Our approach is based on an asymptotic quantitative description of both the domain D and the almost complex structure J near a boundary point. Following Z.M.Balogh and M.Bonk, these sharp estimates provide the Gromov hyperbolicity of the domain D.
In this paper, we introduce a concept of quasihyperbolic John spaces (with center) and provide a criteria to determine spaces to be quasihyperbolic John. As an application, we provide a simple proof to show that a John space with a Gromov hyperbolic quasihyperbolization is quasihyperbolic John, quantitatively. This gives an affirmative answer to an open question posed by Heinonen (Rev.~Math.~Iber, 1989), which has been studied by Gehring et al. (Math.~Scand, 1989).
Neargeodesics in Gromov hyperbolic John domains in Banach spaces
本文研究 Banach 空间中 Gromov 双曲 John 区域的近测地线，证明其为锥弧，从而改进了 Li 于 2014 年给出的相关定理。
In this paper, we introduce a concept of quasihyperbolic John spaces and provide a necessary and sufficient condition for a space to be quasihyperbolic John. Using this criteria, we exhibit a simple proof to show that a John space with a Gromov hyperbolic quasihyperbolization is quasihyperbolic John, quantitatively. This answers affirmatively to an open question proposed by Heinonen (Rev.~Math.~Iber, 1989), which was studied by Gehring et al. (Math.~Scand, 1989). As a tool, we study its connection between several geometric conditions.
The Kobayashi metric and Gromov hyperbolicity on pseudoconvex domains of finite type in $\mathbb{C}^2$
本文改进了 C^2 中光滑有界有限型拟凸域的 Catlin 型距离估计，并据此给出该域在 Kobayashi 距离下具有 Gromov 双曲性的新证明。
Busemann functions and uniformization of Gromov hyperbolic spaces
本文通过Busemann函数研究无界域上的Gromov双曲空间均匀化，证明变形空间为无界均匀空间，并建立了粗略星形双曲空间与无界均匀空间之间的双Lipschitz类与拟相似类的一一对应。
With the aid of a Gromov hyperbolic characterization of uniform domains, we first give an affirmative answer to an open question arisen by Väisälä under weaker assumption. Next, we show that the three-point condition introduced by Väisälä is necessary to obtain quasisymmetry for quasimöbius maps between bounded connected spaces in a quantitative way. Based on these two results, we investigate the boundary behavior of freely quasiconformal and quasihyperbolic mappings on uniform domains of Banach spaces and partially answer another question raised by Väisälä in different ways.
Geometric characterizations of inner uniformity through Gromov hyperbolicity
本文证明 $\IR^n$ 有界域的内一致性等价于两种结合 Gromov 双曲性的几何条件，从而解决 Bonk、Heinonen 和 Koskela 于 2001 年提出的三个问题。
The main purpose of this paper is to investigate the properties of a mapping which is required to be roughly bilipschitz with respect to the Apollonian metric (roughly Apollonian bilipschitz) of its domain. We prove that under these mappings the uniformity, $φ$-uniformity and $δ$-hyperbolicity (in the sense of Gromov with respect to quasihyperbolic metric) of proper domains of $\mathbb{R}^n$ are invariant. As applications, we give four equivalent conditions for a quasiconformal mapping which is defined on a uniform domain to be roughly Apollonian bilipschitz, and we conclude that $φ$-uniformit…
In this paper, we study certain applications of sphericalization in Gromov hyperbolic metric spaces. We first show that the doubling property regarding two classes of metrics on the Gromov boundary of hyperbolic spaces are coincided. Next, we obtain a characterization of unbounded Gromov hyperbolic domains via metric spaces sphericalization. Finally, we investigate the topological equivalence of Gromov hyperbolic $φ$-uniform domains between the Gromov boundary and the inner metric boundary.
In this paper we study the global geometry of the Kobayashi metric on "convex" sets. We provide new examples of non-Gromov hyperbolic domains in $\mathbb{C}^n$ of many kinds: pseudoconvex and non-pseudocon \newline -vex, bounded and unbounded. Our first aim is to prove that if $Ω$ is a bounded weakly linearly convex domain in $\mathbb{C}^n,\,n\geq 2,$ and $S$ is an affine complex hyperplane intersecting $Ω,$ then the domain $Ω\setminus S$ endowed with the Kobayashi metric is not Gromov hyperbolic (Theorem 1.3). Next we localize this result on Kobayashi hyperbolic convex domains. Namely, we sho…
Let $\mathcal{T}_K(D)$ be the class of $K$-quasiconformal automorphisms of a domain $D\subsetneq \mathbb{R}^n$ with identity boundary values. Teichmüller's problem is to determine how far a given point $x\in D$ can be mapped under a mapping $f\in \mathcal{T}_K(D)$. We estimate this distance between $x$ and $f(x)$ from the above by using two different metrics, the distance ratio metric and the quasihyperbolic metric. We study Teichmüller's problem for Gromov hyperbolic domains in $\mathbb{R}^n$ with identity values at the boundary of infinity. As applications, we obtain results on Teichmüller's…
Generalizations of four hyperbolic-type metrics and Gromov hyperbolicity
本文将欧氏空间开集上的四种双曲型度量推广到一般度量空间，用闭子集和Lipschitz函数替代边界与边界距离，证明推广后的空间具有Gromov双曲性且恒等映射拟共形，并改进了部分Gromov常数。
We give a necessary complex geometric condition for a bounded smooth convex domain in Cn, endowed with the Kobayashi distance, to be Gromov hyperbolic. More precisely, we prove that if a smooth bounded convex domain contains an analytic disk in its boundary, then the domain is not Gromov hyperbolic for the Kobayashi distance. We also provide examples of bounded smooth convex domains that are not strongly pseudoconvex but are Gromov hyperbolic.
We characterize Gromov hyperbolicity of the quasihyperbolic metric space (Ω,k) by geometric properties of the Ahlfors regular length metric measure space (Ω,d,μ). The characterizing properties are called the Gehring--Hayman condition and the ball--separation condition.
We obtain explicit and simple conditions which in many cases allow one decide, whether or not a Denjoy domain endowed with the Poincare or quasihyperbolic metric is Gromov hyperbolic. The criteria are based on the Euclidean size of the complement. As a corollary, the main theorem allows to deduce the non-hyperbolicity of any periodic Denjoy domain.
Optimal Linear Dependence on Boundary Type for Local Gromov Hyperbolicity of the Kobayashi Metric
本文证明 Kobayashi 度量局部 Gromov 双曲常数对边界 D'Angelo 型的最优普适依赖是线性的，并指出全局情形无类似界。
We deliver examples of non-Gromov hyperbolic tube domains with convex bases (equipped with the Kobayashi distance). This is shown by providing a criterion on non-Gromov hyperbolicity of (non-smooth) domains.The results show the similarity of geometry of the bases of non-Gromov hyperbolic tube domains with the geometry of non-Gromov hyperbolic convex domains. A connection between the Hilbert metric of a convex domain $Ω$ in $\mathbb R^n$ with the Kobayashi distance of the tube domain over the domain $Ω$ is also shown. Moreover, continuity properties up to the boundary of complex geodesics in tu…
Local and global visibility and Gromov hyperbolicity of domains with respect to the Kobayashi distance
本文在$\mathbb C^d$中引入局部可见与局部Gromov双曲域概念，证明有界域的局部与全局性质等价，从而可通过边界局部信息检测Gromov双曲性及双全纯映射的连续延拓性。
The horofunction boundary of a Gromov hyperbolic space
本研究提出Gromov双曲空间的逼近测地线性质以保证horofunction紧化与Gromov紧化拓扑等价，证明两类复有界域满足相关性质、其上Abate定义的大小球面相重合，并推广Julia引理应用于非扩张映射动力学。
In this paper we prove necessary and sufficient conditions for the Kobayashi metric on a convex domain to be Gromov hyperbolic. In particular we show that for convex domains with $C^\infty$ boundary being of finite type in the sense of D'Angelo is equivalent to the Gromov hyperbolicity of the Kobayashi metric. We also show that bounded domains which are locally convexifiable and have finite type in the sense of D'Angelo have Gromov hyperbolic Kobayashi metric. The proofs use ideas from the theory of the Hilbert metric.
Gromov hyperbolicity of intrinsic metrics from isoperimetric inequalities
本文利用线性等周不等式刻画Gromov双曲性，证明具有边界扩张性质的Kobayashi、Hilbert及极小度量均为Gromov双曲，并给出新证明及极小度量双曲凸域的刻画。
We prove that every bounded strictly $J$-convex region equipped with the Kobayashi metric is hyperbolic in the sense of Gromov. We apply this result to the study of the dynamics of pseudo-holomorphic maps.
Uniformization of intrinsic Gromov hyperbolic spaces
本文针对非测地、非紧合的Gromov双曲空间提出一致化方法，证明其共形形变为有界一致空间，且Gromov边界与形变空间度量边界存在自然拟等距，推广了已有经典结果。
Horofunction compactifications and local Gromov model domains
本文研究了复欧几里得空间中双曲域的horofunction紧化，并给出了身份映射扩展的充分条件及一类满足条件的平面双曲域。
Uniformization of Gromov hyperbolic domains by circle domains
本文证明了黎曼球面上的域为Gromov双曲当且仅当共形等价于均匀圆域，解决了Bonk-Heinonen-Koskela猜想并验证了Koebe猜想，且均匀化映射在Möbius变换下唯一。
On the Gromov hyperbolicity of the minimal metric
本文研究$\u211d^d$中有界强极小凸域关于最小度量的Gromov双曲性，证明其Gromov紧化等价于欧几里得闭包，并探讨了与Hilbert度量的关系。
It is known that the $\tilde j$-metric, half-apollonian metric and scale-invariant Cassinian metric are not Gromov hyperbolic. These metrics are defined as a supremum of one-point metrics (i.e., metrics constructed using one boundary point) and the supremum is taken over all boundary points. The aim of this paper is to show that taking the average instead of the supremum yields a metric that preserves the Gromov hyperbolicity. Moreover, we show that the Gromov hyperbolicity constant of the resulting metric does not depend on the number of metrics used in taking the average. We also provide an …
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 general domains, it has been suggested that a non-trivial complex affine disk in the boundary is an obstruction to Gromov hyperbolicity. This is known to be the case when the set in question is convex. In this paper we first extend this result to $\mathbb{C}$-convex sets with $C^1$-smooth boundary. We will then show that some boundary regularity is necessar…
In this paper we prove: if the complete Kähler-Einstein metric on a bounded convex domain (with no boundary regularity assumptions) is Gromov hyperbolic, then the $\bar{\partial}$-Neumann problem satisfies a subelliptic estimate. This is accomplished by constructing bounded plurisubharmonic function whose Hessian grows at a certain rate (which implies a subelliptic estimate by work of Catlin and Straube). We also provide a characterization of Gromov hyperbolicity in terms of orbit of the domain under the group of affine transformations. This characterization allows us to construct many example…
Rough isometry between Gromov hyperbolic spaces and unbounded uniformization
本文证明了两个proper、roughly starlike的Gromov双曲空间在粗等距下，其共形变形的一致性是同时性质，为Shanmugalingam和Lindquist的工作提供了对应。
Geometry of horospheres in Kobayashi hyperbolic domains
本文研究了Kobayashi超几何域中horospheres的几何性质，证明了模型-格罗莫夫-超几何域具有特定交点性质，并探讨了其在双全像映射扩展中的应用。
Uniformizing non-proper Gromov Hyperbolic Spaces
本研究将经典单值化理论推广至非紧非测地长度空间，证明完备粗星型Gromov双曲空间拟等距类与有界一致空间拟相似类一一对应，解决了公开问题，核心创新是用$(c,μ)$-拟测地线替代拟双曲测地线。
Gromov hyperbolization of unbounded noncomplete spaces and Hamenstädt metric
本文针对无界、局部完备但不完备的度量空间，考察其在三类双曲型度量下的 Gromov 双曲化，并据此给出 Banach 空间中无界一致域的 Gromov 双曲刻画。
We examine Euclidean plane domains with their hyperbolic or quasihyperbolic distance. We prove that the associated metric spaces are quasisymmetrically equivalent if and only if they are bi-Lipschitz equivalent. On the other hand, for Gromov hyperbolic domains, the two corresponding Gromov boundaries are always quasisymmetrically equivalent. Surprisingly, for any finitely connected hyperbolic domain, these two metric spaces are always quasiisometrically equivalent. We construct an example where the spaces are not quasiisometrically equivalent.
Backward dynamics of non-expanding maps in Gromov hyperbolic metric spaces
本文通过引入稳定膨胀概念，研究Gromov双曲空间中非扩张映射的向后动力学，解决了全纯抛物型自映射逃逸向后轨道收敛性的开放问题。
This paper is devoted to study of transformations on metric spaces. It is done in an effort to produce qualitative version of quasi-isometries which takes into account the asymptotic behavior of the Gromov product in hyperbolic spaces. We characterize a quotient semigroup of such transformations on Teichmüller space by use of simplicial automorphisms of the complex of curves, and we will see that such transformation is recognized as a "coarsification" of isometries on Teichmüller space which is rigid at infinity. We also show a hyperbolic characteristic that any finite dimensional Teichmüller …
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