Boundary behavior of the Kobayashi-Royden metric in smooth pseudoconvex domains

Upper semi-continuity of the Royden-Kobayashi pseudo-norm, a counterexample for Hölderian almost complex structures

If $X$ is an almost complex manifold, with an almost complex structure $J$ of class $\CC^α$, for some $α>0$, for every point $p\in X$ and every tangent vector $V$ at $p$, there exists a germ of $J$-holomorphic disc through $p$ with this prescribed tangent vector. This existence result goes back to Nijenhuis-Woolf. All the $J$ holomorphic curves are of class $\CC^{1,α}$ in this case. Then, exactly as for complex manifolds one can define the Royden-Kobayashi pseudo-norm of tangent vectors. The question arises whether this pseudo-norm is an upper semi-continuous function on the tangent bundle. Fo…

The Kobayashi-Royden metric on punctured spheres

This paper gives an explicit formula of the asymptotic expansion of the Kobayashi-Royden metric on the punctured sphere $\mathbb{CP}^1\backslash\{0,1,\infty\}$ in terms of the exponential Bell polynomials. We prove a local quantitative version of the Little Picard's theorem as an application of the asymptotic expansion. Meanwhile, the approach in the paper leads to the conclusion that the coefficients in the asymptotic expansion are rational numbers. Furthermore, the explicit metric formula and the conclusion regarding the coefficients apply to a more general case as well, the metric on $\math…

Existence of close pseudoholomorphic disks for almost complex manifolds and an applications to Kobayashi-Royden pseudonorm

In this paper we extend the notion of the Kobayashi-Royden pseudonorm for almost complex manifolds. Its basic properties known from the complex analysis are preserved in the nonintegrable case as well. The main theorem on coincidence of the pseudodistance induced by this pseudonorm with the Kobayashi pseudodistance for the almost complex manifold is equivalent to the possibility of deforming slightly a pseudoholomorphic disk in an almost complex manifold. We also describe the result in terms of h-principle and consider a geometric application for moduli spaces of pseudoholomorphic curves.

强伪凸域中的Gehring-Hayman不等式

A Gehring-Hayman inequality for strongly pseudoconvex domains

本文在强伪凸域中建立了Kobayashi-Royden度量下测地线长度与欧几里得距离的关系不等式。

论 Riemannian 流形上的 Kobayashi 度量

On the Kobayashi metrics on Riemannian manifolds

本文在 Riemannian 流形上构造 Kobayashi-Royden 度量的类比，并考察其基本性质。

Some aspects of the Kobayashi and Carathéodory metrics on pseudoconvex domains

The purpose of this article is to consider two themes both of which emanate from and involve the Kobayashi and the Carathéodory metric. First we study the biholomorphic invariant introduced by B. Fridman on strongly pseudoconvex domains, on weakly pseudoconvex domains of finite type in $\mathbf C^2$ and on convex finite type domains in $\mathbf C^n$ using the scaling method. Applications include an alternate proof of the Wong-Rosay theorem, a characterization of analytic polyhedra with noncompact automorphism group when the orbit accumulates at a singular boundary point and a description of th…

C^2 中有限型拟凸域上的 Kobayashi 度量与 Gromov 双曲性

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。

关于严格伪凸域中Kobayashi--Fuks度量的不变量与曲率的边界行为

On the Boundary Behaviour of Invariants and Curvatures of the Kobayashi--Fuks Metric in Strictly Pseudoconvex Domains

本文利用缩放方法研究严格伪凸域上Kobayashi--Fuks度量的边界行为，考察其典范不变量、全纯截面曲率及Ricci曲率，扩展并改进了现有分析。

Boundary behavior of the Kobayashi distance in pseudoconvex Reinhardt domains

We prove that the Kobayashi distance near boundary of a pseudoconvex Reinhardt domain $D$ increases asymptotically at most like $-\log d_D+C$. Moreover, for boundary points from $\text{int}\bar{D}$ the growth does not exceed $1/2\log(-\log d_D)+C$. The lower estimate by $-1/2\log d_D+C$ is obtained under additional assumptions of $\mathcal C^1$-smoothness of a domain and a non-tangential convergence.

Some remarks on the Kobayashi--Fuks metric on strongly pseudoconvex domains

The Ricci curvature of the Bergman metric on a bounded domain $D\subset \mathbb{C}^n$ is strictly bounded above by $n+1$ and consequently $\log (K_D^{n+1}g_{B,D})$, where $K_D$ is the Bergman kernel for $D$ on the diagonal and $g_{B, D}$ is the Riemannian volume element of the Bergman metric on $D$, is the potential for a Kähler metric on $D$ known as the Kobayashi--Fuks metric. In this note we study the localization of this metric near holomorphic peak points and also show that this metric shares several properties with the Bergman metric on strongly pseudoconvex domains.

A Kobayashi metric version of Bun Wong's theorem

We prove that a strongly pseudoconvex domain with noncompact group of Kobayashi/Royden metric isometries must be biholomorphic to the unit ball.

Pseudoconvex regions of finite D'Angelo type in four dimensional almost complex manifolds

Let D be a J-pseudoconvex region in a smooth almost complex manifold (M,J) of real dimension four. We construct a local peak J-plurisubharmonic function at every boundary point p of finite D'Angelo type. As applications we give local estimates of the Kobayashi pseudometric, implying the local Kobayashi hyperbolicity of D at p. In case the point p is of D'Angelo type less than or equal to four, or the approach is nontangential, we provide sharp estimates of the Kobayashi pseudometric.

Localization of the Kobayashi distance for any visibility domain

In this article, we prove localization results for the Kobayashi distance of Kobayashi hyperbolic domains with local visibility property in $\mathbb{C}^d$, $d \geq 1$. This is done by proving a localization result for the Kobayashi-Royden pseudometric, along with some other results for domains satisfying local weak visibility.

On the Gromov hyperbolicity of the Kobayashi metric on strictly pseudoconvex regions in the almost complex case

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.

On the derivatives of the Lempert functions

We show that if the Kobayashi--Royden metric of a complex manifold is continuous and positive at a given point and any non-zero tangent vector, then the "derivatives" of the higher order Lempert functions exist and equal the respective Kobayashi metrics at the point. It is a generalization of a result by M. Kobayashi for taut manifolds.

Bounds for Invariant Distances on Pseudoconvex Levi Corank One Domains and Applications

Let $D \subset \mathbb{C}^n$ be a smoothly bounded pseudoconvex Levi corank one domain with defining function $r$, i.e., the Levi form $\partial \bar {\partial} r$ of the boundary $\partial D$ has at least $(n - 2)$ positive eigenvalues everywhere on $\partial D$. The main goal of this article is to obtain a lower bound for the Carathéodory, Kobayashi and the Bergman distance between a given pair of points $p, q \in D$ in terms of parameters that reflect the Levi geometry of $\partial D$ and the distance of these points to the boundary. Applications include an understanding of Fridman's invari…

Complex geodesics in tube domains and their role in the study of harmonic mappings in the disc

We continue the research on the structure of complex geodesics in tube domains over (bounded) convex bases. In some special cases a more explicit form of the geodesics than the existing ones are provided. As one of the consequences of our study an effective formula for the Kobayashi-Royden metric in the tube domain $T_{\mathbb B_n}$ at the origin is given. The results on the Kobayashi-Royden metric in a natural way provide versions of the Schwarz Lemma for harmonic mappings. We also present a result on harmonic mappings defined on the disc that may be seen as a generalisation of the Radó-Knese…

二维拟凸域上不变度量的Kähler性

K$\ddot{\operatorname{a}}$hlerity of invariant metrics on pseudoconvex domain of dimension two

本文针对二维有界有限型拟凸域，利用三阶拟有限几何的Kähler-Kobayashi度量与Kähler-Carathéodory度量证明单值化定理，并给出Reinhardt域为单位球的Bergman度量标量倍数判据及全纯截面曲率与Lu常数的刚性定理。

Deformations of Strongly Pseudoconvex Domains

We show that two smoothly bounded, strongly pseudoconvex domains which are diffeomorphic may be smoothly deformed into each other, with all intermediate domains being strongly pseudoconvex. This result relates to Lempert's ideas about Kobayashi extremal discs, and also has intrinsic interest.

A Kobayashi and Bergman complete domain without bounded representations

We construct an unbounded strictly pseudoconvex Kobayashi hyperbolic and complete domain in $\mathbb{C}^2$, which also possesses complete Bergman metric, but has no nonconstant bounded holomorphic functions.

Peak functions and boundary behaviour of holomorphically invariant distances on strictly pseudoconvex domains

We give a parameter version of Graham-Kerzman approximation theorem for bounded holomorphic functions on strictly pseudoconvex domains. As an application, we present some uniform estimates for the boundary behaviour of the Kobayashi and Carathéodory pseudodistences on such domains.

C²-光滑边界点附近的Kobayashi距离的最优界

Optimal bounds for the Kobayashi distance near $\mathcal C^2$-smooth boundary points

本文证明了C²强拟凸边界附近Kobayashi距离的最优上下界，推广了已有结果，还给出了非半正边界附近该距离的上下界。

Regularity of Kobayashi metric

We review some recent results on existence and regularity of Monge-Ampère exhaustions on the smoothly bounded strongly pseudoconvex domains, which admit at least one such exhaustion of sufficiently high regularity. A main consequence of our results is the fact that the Kobayashi pseudo-metric k on an appropriare open subset of each of the above domains is actually a smooth Finsler metric. The class of domains to which our result apply is very large. It includes for instance all smoothly bounded strongly pseudoconvex complete circular domains and all their sufficiently small deformations.

Asymptotic behavior of the Kobayashi metric in the normal direction

In this paper, we construct a pseudoconvex domain in $\mathbb C^3$ where the Kobayashi metric does not blow up at a rate of one over distance to the boundary in the normal direction.

The Kobayashi metric, extremal discs, and biholomorphic mappings

We study extremal discs for the Kobayashi metric. Inspired by work of Lempert on strongly convex domains, we present results on strongly pseudoconvex domains. We also consider a useful biholomorphic invariant, inspired by the Kobayashi (and Carathéodory) metric, and prove several new results about biholomorphic equivalence of domains. Some useful results about automorphism groups of complex domains are also established.

On isometries of the Kobayashi and Carathéodory metrics

This article considers isometries of the Kobayashi and Carathéod-ory metrics on domains in $ \mathbf{C}^n $ and the extent to which they behave like holomorphic mappings. First we prove a metric version of Poincaré's theorem about biholomorphic inequivalence of $ \mathbf{B}^n $, the unit ball in $ \mathbf{C}^n $ and $ Δ^n $, the unit polydisc in $ \mathbf{C}^n $ and then provide few examples which \textit{suggest} that $ \mathbf{B}^n $ cannot be mapped isometrically onto a product domain. In addition, we prove several results on continuous extension of isometries $ f : D_1 \rightarrow D_2 $ to…

Estimates of invariant metrics on pseudoconvex domains near boundaries with constant Levi ranks

Estimates of the Bergman kernel and the Bergman and Kobayashi metrics on pseudoconvex domains near boundaries with constant Levi ranks are given.

Landau's theorem for holomorphic curves in projective space and the Kobayashi metric on hyperplane complements

We prove an effective version of a theorem of Dufresnoy: For any set of 2n+1 hyperplanes in general position in n-dimensional complex projective space, we find an explicit constant K such that for every holomorphic map f from the unit disc to the complement of these hyperplanes, the derivative of f at the origin measured with respect to the Fubuni-Study metric is bouned above by K. This result gives an explicit lower bound on the Royden function, i.e., the ratio of the Kobayashi metric on the hyperplane complement to the Fubini-Study metric. Our estimate is based on the potential-theoretic met…

Estimates of the Kobayashi metric on almost complex manifolds

We establish a lower estimate for the Kobayashi-Royden infinitesimal pseudometric on an almost complex manifold $(M,J)$ admitting a bounded strictly plurisubharmonic function. We apply this result to study the boundary behaviour of the metric on a strictly pseudoconvex domain in $M$ and to give a sufficient condition for the complete hyperbolicity of a domain in $(M,J)$. Finally we obtain the regularity up to the bounday of $J$-holomorphic discs attached to a totally real submanifold in $M$.

Unbounded Kobayashi hyperbolic domains in $\mathbb C^n$

We first give a sufficient condition, issued from pluripotential theory, for an unbounded domain in the complex Euclidean space $\mathbb C^n$ to be Kobayashi hyperbolic. Then, we construct an example of a rigid pseudoconvex domain in $\mathbb C^3$ that is Kobayashi hyperbolic and has a nonempty core. In particular, this domain is not biholomorphic to a bounded domain in $\mathbb C^3$ and the mentioned above sufficient condition for Kobayashi hyperbolicity is not necessary.

强伪凸域上不变距离的精确估计

Precise estimates of invariant distances on strongly pseudoconvex domains

本文研究强伪凸域上的不变距离，利用实与复测地线给出 Kobayashi、Lempert、Carathéodory 及 Bergman 距离的尖锐估计。

不是伪凸的可见性域

Visibility domains that are not pseudoconvex

针对已知可见性域均为伪凸域的现状，本文证明了非伪凸可见性域的存在性，并提出一种构造大量非伪凸（因此非Kobayashi完备）可见性域的通用方法。

在$\mathbb{C}^2$中有限型伪凸域上的Gehring-Hayman型定理

The Gehring-Hayman type theorem on pseudoconvex domains of finite type in $\mathbb{C}^2$

本文在$\mathbb{C}^2$中有限型光滑有界伪凸域上证明了Gehring-Hayman型定理，并据此给出了边界点附近全局与局部Kobayashi距离的定量比较。

Sharp estimates of the Kobayashi metric and Gromov hyperbolicity

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.

稳定的丰沛向量丛不具有半正Kobayashi曲率

Stable ample vector bundles without semipositive Kobayashi curvature

本文在Hirzebruch曲面上构造了无半正Kobayashi曲率的稳定丰沛向量丛，否定了Kobayashi问题，并为Fulton问题和Hartshorne--Schneider猜想提供了反例。

Boundary behavior of the Kobayashi metric near a point of infinite type

Under a potential-theoretical hypothesis named $f$-Property with $f$ satisfying $\displaystyle\int_t^\infty \dfrac{da}{a f(a)}<\infty$, we show that the Kobayashi metric $K(z,X)$ on a weakly pseudoconvex domain $\Om$, satisfies the estimate $K(z,X)\ge Cg(δ_\Om(x)^{-1})|X|$ for any $X\in T^{1,0}\Om$ where $(g(t))^{-1}$ denotes the above integral and $δ_\Om(z)$ is the distance from $z$ to $b\Om$.

Localization of the Kobayashi metric and applications

In this paper we introduce a new class of domains -- log-type convex domains, which have no boundary regularity assumptions. Then we will localize the Kobayashi metric in log-type convex subdomains. As an application, we prove a local version of continuous extension of rough isometric maps between two bounded domains with log-type convex Dini-smooth boundary points. Moreover we prove that the Teichmüller space $\mathcal T_{g,n}$ is not biholomorphic to any bounded pseudoconvex domain in $\mathbb C^{3g-3+n}$ which is locally log-type convex near some boundary point.

平面区域上Kobayashi–Fuks度量的测地螺线的存在性

Existence of geodesic spirals for the Kobayashi--Fuks metric on planar domains

本文研究强拟凸域上Kobayashi–Fuks度量是否存在内部螺旋转动的测地线，并证明在平面区域非单连通情形下该问题有肯定答案。

不变度量的标形、极大圆度与域的重标度

Indicatrix of Invariant metrics, Maximal circularity and Scaling of domains

本文利用不变度量的标形重标度双全纯映射序列以确保收敛，并借助Kobayashi-Royden度量定义极大圆度函数，衡量域接近圆域的程度。

$\u2102^2$中拟凸有限型域的Gromov双曲性

Gromov hyperbolicity of pseudoconvex finite type domains in $\mathbb{C}^2$

该研究证明$\u2102^2$中有限d'Angelo型有界光滑域带Kobayashi距离时为Gromov双曲且边界典范同胚，还给出自同构列收敛于有限型边界点时域的Gromov双曲性判定。

Comparison of invariant metrics and distances on strongly pseudoconvex domains and worm domains

We prove that for a strongly pseudoconvex domain $D\subset\mathbb C^n$, the infinitesimal Carathéodory metric $g_C(z,v)$ and the infinitesimal Kobayashi metric $g_K(z,v)$ coincide if $z$ is sufficiently close to $bD$ and if $v$ is sufficiently close to being tangential to $bD$. Also, we show that every two close points of $D$ sufficiently close to the boundary and whose difference is almost tangential to $bD$ can be joined by a (unique up to reparameterization) complex geodesic of $D$ which is also a holomorphic retract of $D$. The same continues to hold if $D$ is a worm domain, as long as the…

Characterizing strong pseudoconvexity, obstructions to biholomorphisms, and Lyapunov exponents

In this paper we consider the following question: For bounded domains with smooth boundary, can strong pseudoconvexity be characterized in terms of the intrinsic complex geometry of the domain? Our approach to answering this question is based on understanding the dynamical behavior of real geodesics in the Kobayashi metric and allows us to prove a number of results for domains with low regularity. For instance, we show that for convex domains with $C^{2,ε}$ boundary strong pseudoconvexity can be characterized in terms of the behavior of the squeezing function near the boundary, the behavior of…

Kobayashi距离的下估计与复测地线的极限

Lower estimates of the Kobayashi distance and limits of complex geodesics

本文证明了强伪凸域中复测地线与紧子集相交的性质，b并以此获得了Kobayashi距离的下界估计。

Positively curved Finsler metrics on vector bundles

We construct a convex and strongly pseudoconvex Kobayashi positive Finsler metric on a vector bundle $E$ under the assumption that the symmetric power of the dual $S^kE^*$ has a Griffiths negative $L^2$-metric for some $k$. The proof relies on the negativity of direct image bundles and the Minkowski inequality for norms. As a corollary, we show that given a strongly pseudoconvex Kobayashi positive Finsler metric, one can upgrade to a \textit{convex} Finsler metric with the same property. We also give an extremal characterization of Kobayashi curvature for Finsler metrics.

Discrete sequences in unbounded domains

Discrete sequences with respect to the Kobayashi distance in a strongly pseudoconvex bounded domain $D$ are related to Carleson measures by a formula that uses the Euclidean distance from the boundary of $D$. Thus the speed of escape at the boundary of such sequence has been studied in details for strongly pseudoconvex bounded domain $D$. In this note we show that such estimations completely fail if the domain is not bounded.

指数平坦无限型域上Bergman不变量与Kobayashi度量的渐近行为

Asymptotic behaviour of the Bergman invariant and Kobayashi metric on exponentially flat infinite type domains

本文研究了指数平坦无限型边界点处Bergman不变量、Ricci曲率、标量曲率及Kobayashi度量的非切向渐近极限，通过局部化与伸缩方法给出证明。

Carleson measures and uniformly discrete sequences in strongly pseudoconvex domains

We characterize using the Bergman kernel Carleson measures of Bergman spaces in strongly pseudoconvex bounded domains in several complex variables, generalizing to this setting theorems proved by Duren and Weir for the unit ball. We also show that uniformly discrete (with respect to the Kobayashi distance) sequences give examples of Carleson measures, and we compute the speed of escape to the boundary of uniformly discrete sequences in strongly pseudoconvex domains, generalizing results obtained in the unit ball by Jevtić, Massaneda and Thomas, by Duren and Weir, and by MacCluer.

On the existence of Kobayashi and Bergman metrics for Model domains

We prove that for a pseudoconvex domain of the form $\mathfrak{A} = \{(z, w) \in \mathbb C^2 : v > F(z, u)\}$, where $w = u + iv$ and F is a continuous function on ${\mathbb C}_z \times {\mathbb R}_u$, the following conditions are equivalent: (1) The domain $\mathfrak{A}$ is Kobayashi hyperbolic. (2) The domain $\mathfrak{A}$ is Brody hyperbolic. (3) The domain $\mathfrak{A}$ possesses a Bergman metric. (4) The domain $\mathfrak{A}$ possesses a bounded smooth strictly plurisubharmonic function, i.e. the core $\mathfrak{c}(\mathfrak{A})$ of $\mathfrak{A}$ is empty. (5) The graph $Γ(F)$ of $F$ c…

Gromov hyperbolicity and the Kobayashi metric on "convex" sets

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…

Conformal equivalence of visual metrics in pseudoconvex domains

We refine estimates introduced by Balogh and Bonk, to show that the boundary extensions of isometries between smooth strongly pseudoconvex domains in $\C^n$ are conformal with respect to the sub-Riemannian metric induced by the Levi form. As a corollary we obtain an alternative proof of a result of Fefferman on smooth extensions of biholomorphic mappings between pseudoconvex domains. The proofs are inspired by Mostow's proof of his rigidity theorem and are based on the asymptotic hyperbolic character of the Kobayashi or Bergman metrics and on the Bonk-Schramm hyperbolic fillings.

Toeplitz operators and Carleson measures in strongly pseudoconvex domains

We study mapping properties of Toeplitz operators associated to a finite positive Borel measure on a bounded strongly pseudoconvex domain D in n complex variables. In particular, we give sharp conditions on the measure ensuring that the associated Toeplitz operator maps the Bergman space A^p(D) into A^r(D) with r>p, generalizing and making more precise results by Cuckovic and McNeal. To do so, we give a geometric characterization of Carleson measures and of vanishing Carleson measures of weighted Bergman spaces in terms of the intrinsic Kobayashi geometry of the domain, generalizing to this se…

作为一族测地线上Kobayashi等距映射的全纯映射

Holomorphic maps acting as Kobayashi isometries on a family of geodesics

本文研究全纯映射在Kobayashi距离测地线族上作为等距映射时何时成为双全纯映射，在完全双曲区域和光滑有界拟凸区域两种情形下给出刻画，并通过例子证明假设的最优性。

The Gehring-Hayman type theorems on complex domains

In this paper we establish Gehring-Hayman type theorems for some complex domains. Suppose that $Ω\subset \mathbb{C}^n$ is a bounded $m$-convex domain with Dini-smooth boundary, or a bounded strongly pseudoconvex domain with $C^2$-smooth boundary. Then we prove that the Euclidean length of Kobayashi geodesic $[x,y]$ in $Ω$ is less than $c_1|x-y|^{c_2}$. Furthermore, if $Ω$ endowed with the Kobayashi metric is Gromov hyperbolic, then we can generalize this result to quasi-geodesics with respect to Bergman metric, Carathéodory metric or Kähler-Einstein metric. As applications, we prove the bi-Höl…

Goldilocks domains, a weak notion of visibility, and applications

In this paper we introduce a new class of domains in complex Euclidean space, called Goldilocks domains, and study their complex geometry. These domains are defined in terms of a lower bound on how fast the Kobayashi metric grows and an upper bound on how fast the Kobayashi distance grows as one approaches the boundary. Strongly pseudoconvex domains and weakly pseudoconvex domains of finite type always satisfy this Goldilocks condition, but we also present families of Goldilocks domains that have low boundary regularity or have boundary points of infinite type. We will show that the Kobayashi …