Peter Pflug, Wlodzimierz Zwonek
In the paper we give some necessary conditions for a mapping to be a $κ$-geodesic in non-convex complex ellipsoids. Using these results we calculate explicitly the Kobayashi metric in the ellipsoids $\{|z_1|^2+|z_2|^{2m}<1\}\subset\bold C^2$, where $m<\frac12$.
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We construct families of convex domains that are biholomorphic to bounded domains, but not bounded convex domains. This is accomplished by finding an obstruction related to the Gromov hyperbolicity of the Kobayashi metric.
A pure geometric description of the Kobayashi balls of C-convex domains is given in terms of the so-called minimal basis.
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.
Kobayashi visibility of strictly convex domains
该研究证明了复空间ℂ^N中带Kobayashi距离的任意有界严格凸域均具有可见性。
Let $(Ω,K_Ω)$ be a convex domain in $\mathbb C^d$ with the Kobayashi metric $K_Ω$. In this paper we prove that $m$-convexity is a necessary condition for $(Ω, K_Ω)$ to be CAT(0) if $d=2$. Moreover, when $Ω\subset \mathbb{C}^d, \: d\geq3$, we obtain a similar result with the further smoothness assumption on its boundary.
In this paper, we calculate estimates for invariant metrics on a finite type convex domain in $\mathbb C^n$ using the Sibony metric. We also discuss a possible modification of the Sibony metric.
It has been recently shown that for a convex domain $Ω$ in $\mathbb C^n$ and $w\inΩ$ the function $F_Ω(w):=\big(K_Ω(w)λ(I_Ω(w))\big)^{1/n}$, where $K_Ω$ is the Bergman kernel on the diagonal and $I_Ω(w)$ the Kobayashi indicatrix, satisfies $1\leq F_Ω\leq 4$. While the lower bound is optimal, not much more is known about the upper bound. In general it is quite difficult to compute $F_Ω$ even numerically and the highest value of it obtained so far is $1.010182\dots$ In this paper we present precise, although rather complicated formulas for the ellipsoids $Ω=\{|z_1|^{2m}+|z_2|^2<1\}$ (with $m\geq…
It is shown that a lower bound of the Kobayashi metric of convex domains in C^n does not hold for non-convex domains.
We present a form of Schwarz's lemma for holomorphic maps between convex domains $D_1$ and $D_2$. This result provides a lower bound on the distance between the images of relatively compact subsets of $D_1$ and the boundary of $D_2$. This is a natural improvement of an old estimate by Bernal-González that takes into account the geometry of $\partial{D_1}$. Using similar techniques, we also provide a new estimate for the Kobayashi metric on bounded convex domains.
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…
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.
Let $D\subset \mathbb C^n$ be a bounded domain. A pair of distinct boundary points $\{p,q\}$ of $D$ has the visibility property provided there exist a compact subset $K_{p,q}\subset D$ and open neighborhoods $U_p$ of $p$ and $U_q$ of $q$, such that the real geodesics for the Kobayashi metric of $D$ which join points in $U_p$ and $U_q$ intersect $K_{p,q}$. Every Gromov hyperbolic convex domain enjoys the visibility property for any couple of boundary points. The Goldilocks domains introduced by Bharali and Zimmer and the log-type domains of Liu and Wang also enjoy the visibility property. In th…
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 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.
We study the Carathéodory and Kobayashi metrics by way of the method of dual extremal problems in functional analysis. Particularly incisive results are obtained for convex domains.
We use convex geometry tools, in particular John ellipsoids, to obtain a size estimate for the Szegő kernel on the boundary of a class of unbounded convex domains in $\mathbb{C}^n.$ Given a polynomial $b:\mathbb{R}^n \rightarrow \mathbb{R}$ satisfying a certain growth condition, we consider domains of the type $Ω_b = \{ z\in\mathbb{C}^{n+1}\,:\, {\rm Im}[z_{n+1}] > b({\rm Re}[z_1],\ldots,{\rm Re}[z_n]) \}.$
On non-visibility of Kobayashi geodesics and the geometry of the boundary
本文研究了凸域边界几何与不可见Kobayashi测地线的关系，证明了其极限集包含于复面中，并在二维下与边界非Goldilocks点的复切线段存在充要构造关系。
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…
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.
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…
Positively curved Finsler metrics on vector bundles III
本文证明了小林正向量丛的商丛和张量积仍为小林正，从而加强了丰富性、格里菲斯正性与小林正性之间的猜想等价关系。
We provide several equivalent characterizations of Kobayashi hyperbolicity in unbounded convex domains in terms of peak and anti-peak functions at infinity, affine lines, Bergman metric and iteration theory.
We extend the upper estimates obtained by M. Carlehed and B.-Y. Chen about the ratio of the classical and pluricomplex Green functions to the case of $\mathcal C^2$-smooth locally $\mathbb C$-convexifiable domains of finite type. We also give some lower estimates. In order to obtain those results, and because it is of independent interest, we refine and unify some classical estimates about the Kobayashi distance and the Lempert function in such domains.
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.
An example of a space $L^{p(\cdot)}$ on which the Cauchy-Leray-Fantappiè operator for complex ellipsoid is not bounded
本文构造了一个变指数Lebesgue空间，证明复椭球上的Cauchy-Leray-Fantappiè算子在该空间上无界，从而将对数连续性条件的尖锐性推广到非严格凸区域。
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.
In this paper we study the following "slice rigidity property": given two Kobayashi complete hyperbolic manifolds $M, N$ and a collection of complex geodesics $\mathcal F$ of $M$, when is it true that every holomorphic map $F:M\to N$ which maps isometrically every complex geodesic of $\mathcal F$ onto a complex geodesic of $N$ is a biholomorphism? Among other things, we prove that this is the case if $M, N$ are smooth bounded strictly (linearly) convex domains, every element of $\mathcal F$ contains a given point of $\overline{M}$ and $\mathcal F$ spans all of $M$. More general results are pro…
In the paper we generalize the notion of problem (P) introduced by Poletsky. We introduce the notion of (P_m) extremals. For example, geodesics are (P_1) extremals. Using obtained results we present a description of (P_m) extremals in arbitrary complex ellipsoids. It is a generalization of the result obtained by Jarnicki-Pflug-Zeinstra. We also have a proof of conjecture put forward by Pflug-Zwonek concerning the formulas for geodesics in non-convex complex ellipsoids.
We provide a sufficient condition for the continuous extension of isometries for the Kobayashi distance between bounded convex domains in complex Euclidean spaces having boundaries that are only slightly more regular than $\mathcal{C}^1$. This is a generalization of a recent result by A. Zimmer.
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.
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…
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。
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 give a description (direct formulas) of all complex geodesics in a convex tube domain in $\CC^n$ containing no complex affine lines, expressed in terms of geometric properties of the domain. We next apply that result to give formulas (a necessary condition) for extremal mappings with respect to the Lempert function and the Kobayashi-Royden metric in a big class of bounded, pseudoconvex, complete Reinhardt domains: for all of them in $\CC^2$ and for those of them in $\CC^n$ which logarithmic image is strictly convex in geometric sense.
We provide a class of geometric convex domains on which the Carathéodory-Reiffen metric, the Bergman metric, the complete Kähler-Einstein metric of negative scalar curvature are uniformly equivalent, but not proportional to each other. In a two-dimensional case, we provide a full description of curvature tensors of the Bergman metric on the weakly pseudoconvex boundary point and show that invariant metrics are proportional to each other if and only if the geometric convex domain is the Poincaré-disk.
Let $M$ be a Kobayashi hyperbolic homogenous manifold. Let $\mathcal F$ be a holomorphic foliation on $M$ invariant under a transitive group $G$ of biholomorphisms. We prove that the leaves of $\mathcal F$ are the fibers of a holomorphic $G$-equivariant submersion $π\colon M \to N$ onto a $G$-homogeneous complex manifold $N$. We also show that if $\mathcal Q$ is an automorphism family of a hyperbolic convex (possibly unbounded) domain $D$ in $\mathbb C^n$, then the fixed point set of $\mathcal Q$ is either empty or a connected complex submanifold of $D$.
On the exponential convergence of Kobayashi geodesics in strongly convex domains
研究强凸域中 Kobayashi 测地线的指数收敛性，通过证明特定条件下两条测地线的指数收敛不等式，利用此性质借助广义挤压函数刻画强拟凸域，给出了强拟凸域的一种新判定方法。
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。
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…
In the spirit of Kobayashi's applications of methods of invariant metrics to questions of projective geometry, we introduce a projective analogue of the complex squeezing function. Using Frankel's work, we prove that for convex domains it stays uniformly bounded from below. In the case of strongly convex domains, we show that it tends to 1 at the boundary. This is applied to get a new proof of a projective analogue of the Wong-Rosay theorem.
Let $M$ be a complex manifold which admits an exhaustion by open subsets $M_j$ each of which is biholomorphic to a fixed domain $Ω\subset \mathbb C^n$. The main question addressed here is to describe $M$ in terms of $Ω$. Building on work of Fornaess--Sibony, we study two cases namely, $M$ is Kobayashi hyperbolic and the other being the corank one case in which the Kobayashi metric degenerates along one direction. When $M$ is Kobayashi hyperbolic, its complete description is obtained when $Ω$ is one of the following domains -- (i) a smoothly bounded Levi corank one domain, (ii) a smoothly bound…
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…
We prove that Kobayashi isometries between strongly convex domains are holomorphic or anti-holomorphic. More precisely, let $n_1, n_2$ be positive integers and let $Ω_i \subset \C^{n_i}, \ i=1,2$, be bounded $C^3$ strongly convex domains. If $ϕ: (Ω_1, d^K_{Ω_1}) \rightarrow (Ω_2, d^K_{Ω_2})$ is an isometry, i.e. $ d^K_Ω_{n_2}(f(ζ),f(η)) = d^K_{n_1} (ζ,η)$ for all $ζ,η\in Ω_1,$ then $ϕ$ is either holomorphic or anti-holomorphic.
Precise estimates for certain distances in $\mathbb{R}^d$
本文研究了$\mathbb{R}^d$中Finsler度量的精确距离估计，包括Kobayashi-Hilbert度量、最小度量和k-准双曲度量，并在凸几何中给出了k-准双曲度量的特征化结果。
Small exponent Schatten class Toeplitz operators on convex domains of finite type
本文通过离散 Kobayashi 格和概率选择方法，在有限型凸域上刻画了 Schatten p-类 Toeplitz 算子，并给出了加权复合算子的几何与解析刻画，回答了 Xiao-Yang-Yuan 提出的问题。
We prove that a backward orbit with bounded Kobayashi step for a hyperbolic or strongly elliptic holomorphic self-map of a bounded strongly convex domain in the d-dimensional complex Euclidean space necessarily converges to a boundary fixed point, generalizing previous results obtained by Poggi-Corradini in the unit disk and by Ostapyuk in the unit ball.
In this paper we study the automorphism group of smoothly bounded convex domains. We show that such a domain is biholomorphic to a "polynomial ellipsoid" (that is, a domain defined by a weighted homogeneous balanced polynomial) if and only if the limit set of the automorphism group intersects at least two closed complex faces of the set. The proof relies on a detailed study of the geometry of the Kobayashi metric and ideas from the theory of non-positively curved metric spaces. We also obtain a number of other results including the Greene-Krantz conjecture in the case of uniform non-tangential…
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