Lipschitz property of harmonic mappings with respect to the pseudo-hyperbolic metric

Gromov hyperbolicity of pseudo-convex Levi corank one domains

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.

Bi-Lipschitz extension from boundaries of certain hyperbolic spaces

Tukia and Vaisala showed that every quasi-conformal map of $\R^n$ extends to a quasi-conformal self-map of $\R^{n+1}$. The restriction of the extended map to the upper half-space $\R^n \times \R^+$ is, in fact, bi-Lipschitz with respect to the hyperbolic metric. More generally, every homogeneous negatively curved manifold decomposes as $M = N \rtimes \R^+$ where $N$ is a nilpotent group with a metric on which $\R^+$ acts by dilations. We show that under some assumptions on $N$, every quasi-symmetry of $N$ extends to a bi-Lipschitz map of $M$. The result applies to a wide class of manifolds $M$…

Lipschitz Estimates for Conformal Maps from the Unit Disk to Convex Domains

We obtain an explicit uniform upper bound for the derivative of a conformal mapping of the unit disk onto a convex domain. This estimate depends only on the outer and inner radii of the domain, and on a curvature radius of its boundary. Its proof is based on a Möbius invariant metric of hyperbolic type, introduced by Kulkarni and Pinkall in 1994.

$L^p$ Mapping Properties for the Cauchy-Riemann Equations on Lipschitz Domains Admitting Subelliptic Estimates

We show that on bounded Lipschitz pseudoconvex domains that admit good weight functions the $\overline{\partial}$-Neumann operators $N_q, \overline{\partial}^* N_{q}$, and $\overline{\partial} N_{q}$ are bounded on $L^p$ spaces for some values of $p$ greater than 2.

Picard hyperbolicity of manifolds admitting nilpotent harmonic bundles

For a quasi-compact Kähler manifold $U$ endowed with a nilpotent harmonic bundle whose Higgs field is injective at one point, we prove that $U$ is pseudo-algebraically hyperbolic, pseudo-Picard hyperbolic, and is of log general type. Moreover, we prove that there is a finite unramified cover $\tilde{U}$ of $U$ from a quasi-projective manifold $\tilde{U}$ so that any projective compactification of $\tilde{U}$ is pseudo-algebraically hyperbolic, pseudo-Picard hyperbolic and is of general type. As a byproduct, we establish some criterion of pseudo-Picard hyperbolicity and pseudo-algebraic hyperbo…

Hyperbolic Distance versus Quasihyperbolic Distance in Plane Domains

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.

Pseudo Kobayashi hyperbolicity of base spaces of families of minimal projective manifolds with maximal variation

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 $Z\subsetneq V$. In particular, $V$ is algebraically degenerate, that is, every nonconstant entire curve $f:\mathbb{C}\to V$ has image $f(\mathbb{C})$ which lies in that proper subvariety $Z\subsetneq V$. As a direct consequence, we prove the Brody hyperbolicity of moduli spaces of minimal projective manifolds, which answers a question by Viehweg-Zuo in 2003.

几乎保持双曲面积的单位圆盘解析映射

Analytic mappings of the unit disk which almost preserve hyperbolic area

本文研究对大双曲圆盘面积扭曲有界的单位圆盘解析自映射，通过角导数、Lipschitz扩张、Möbius扭曲、临界点分布及Aleksandrov-Clark测度等对其进行刻画，并探讨了相关测度的Lyapunov指数。

The Thurston metric on hyperbolic domains and boundaries of convex hulls

We show that the nearest point retraction is a uniform quasi-isometry from the Thurston metric on a hyperbolic domain in the Riemann sphere to the boundary of the convex hull of its complement. As a corollary, one obtains explicit bounds on the quasi-isometry constant of the nearest point retraction with respect to the Poincare metric when the domain is uniformly perfect. We also establish Marden and Markovic's conjecture that a hyperbolic domain is uniformly perfect if and only if the nearest point retraction is Lipschitz with respect to the Poincare metric.

实欧几里得空间中的双曲区域

Hyperbolic domains in real Euclidean spaces

本文研究了实欧几里得空间中由共形调和圆盘定义的极小伪度量的性质，给出了区域为完备双曲的充分条件，并证明了极小伪度量的局部化定理。