Ninh Van Thu, Nguyen Thi Kim Son, Chu Van Tiep
In this paper, we prove that the general ellipsoid $D_P$ is holomorphically homogeneous regular provided that it is a $WB$-domain. Then, the uniform lower bound for the squeezing function near a $(P,r)$-extreme point is also given.
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It is shown that any non-degenerate $\mathbb C$-convex domain in $\mathbb C^n$ is uniformly squeezing. It is also found the precise behavior of the squeezing function near a Dini-smooth boundary point of a plane domain.
It is shown that if the squeezing function tends to one at an h-extendible boundary point of a $\mathcal C^\infty$-smooth, bounded pseudoconvex domain, then the point is strictly pseudoconvex.
We describe the boundary behaviors of the squeezing functions for all bounded convex domains in $\mathbb{C}^n$ and bounded domains with a $C^2$ strongly convex boundary point.
J. E. Fornaess has posed the question whether the boundary point of smoothly bounded pseudoconvex domain is strictly pseudoconvex, if the asymptotic limit of the squeezing function is 1. The purpose of this paper is to give an affirmative answer when the domain is in C^2 with smooth boundary of finite type in the sense of D'Angelo.
An extension of the estimates for the squeezing function of strictly pseudoconvex domains obtained recently by J. E. Fornæss and E. Wold in \cite{FW1} is applied to derive a sharp boundary behaviour of invariant metrics and Bergman curvatures.
In recent work by Zimmer it was proved that if $Ω\subset\mathbb C^n$ is a bounded convex domain with $C^\infty$-smooth boundary, then $Ω$ is strictly pseudoconvex provided that the squeezing function approaches one as one approaches the boundary. We show that this result fails if $Ω$ is only assumed to be $C^2$-smooth.
The purpose of this article is twofold. The first aim is to prove that if there exist a sequence $\{φ_j\}\subset \mathrm{Aut}(Ω)$ and $a\in Ω$ such that $\lim_{j\to\infty}φ_j(a)=ξ_0$ and $\lim_{j\to\infty}σ_Ω(φ_j(a))=1$, where $ξ_0$ is a linearly convex boundary point of finite type, then $ξ_0$ must be strongly pseudoconvex. Then, the second aim is to investigate the boundary behaviour of the squeezing function of a general ellipsoid.
Let $Ω$ be a domain in $\mathbb C^n$. Suppose that $\partialΩ$ is smooth pseudoconvex of D'Angelo finite type near a boundary point $ξ_0\in \partialΩ$ and the Levi form has corank at most $1$ at $ξ_0$. Our goal is to show that if the squeezing function $s_Ω(η_j)$ tends to $1$ or the Fridman invariant $h_Ω(η_j)$ tends to $0$ for some sequence $\{η_j\}\subset Ω$ converging to $ξ_0$, then this point must be strongly pseudoconvex.
The central purpose of the present paper is to study boundary behavior of squeezing functions on bounded domains. We prove that the squeezing function of a strongly pseudoconvex domain tends to 1 near the boundary. In fact, such an estimate is proved for the squeezing function on any domain near its globally strongly convex boundary points. We also study the stability of squeezing functions on a sequence of bounded domains, and give comparisons of intrinsic measures and metrics on bounded domains in terms of squeezing functions. As applications, we give new and simple proofs of several well kn…
We construct a strictly pseudoconvex domain with smooth boundary whose squeezing function is not plurisubharmonic.
Properties of squeezing functions on $h$-extendible domains
本文研究了$h$-可扩展域上挤压函数的行为，证明了其在强伪凸边界点和有限类型伪凸边界点附近趋近于1的性质。
In this article we continue the study of properties of squeezing functions and geometry of bounded domains. The limit of squeezing functions of a sequence of bounded domains is studied. We give comparisons of intrinsic positive forms and metrics on bounded domains in terms of squeezing functions. To study the boundary behavior of squeezing functions, we introduce the notions of (intrinsic) ball pinching radius, and give boundary estimate of squeezing functions in terms of these datum. Finally, we use these results to study geometric and analytic properties of some interesting domains, includin…
We show that if a bounded domain $Ω$ is exhausted by a bounded strictly pseudoconvex domain $D$ with $C^2$ boundary, then $Ω$ is holomorphically equivalent to $D$ or the unit ball, and show that a bounded domain has to be holomorphically equivalent to the unit ball if its Fridman's invariant has certain growth condition near the boundary.
We construct a class of bounded domains, on which the squeezing function is not uniformly bounded from below near a smooth and pseudoconvex boundary point.
For a domain $D \subset \mathbb C^n$, the relationship between the squeezing function and the Fridman invariants is clarified. Furthermore, localization properties of these functions are obtained. As applications, some known results concerning their boundary behavior are extended.
We provide explicit expression of squeezing function for infinitely connected planar domain obtained by removing a convergent sequence of points from the unit disk converging to the boundary of unit disk. We also discuss Fridman invariant associated with this domain as well as some examples of squezzing functions corresponding to polydisk.
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.
Exhaustion of hyperbolic complex manifolds and relations to the squeezing function
本文通过穷竭序列刻画了被域序列穷竭的n维双曲复流形，并证明了任何球面极端边界点必为强伪凸的。
More precise estimates for the Bergman metric on strongly pseudoconvex domains are given, based on the use of the squeezing function.
The main purpose of the present paper is to introduce the notion of squeezing functions of bounded domains and study some properties of them. The relation to geometric and analytic structures of bounded domains will be investigated. Existence of related extremal maps and continuity of squeezing functions are proved. Holomorphic homogeneous regular domains are exactly domains whose squeezing functions have positive lower bounds. Completeness of certain intrinsic metrics and pseudoconvexity of holomorphic homogeneous regular domains are proved by alternative method. In dimension one case, we get…
We prove that if a holomorphic self-map $f\colon Ω\to Ω$ of a bounded strongly convex domain $Ω\subset \mathbb C^q$ with smooth boundary is hyperbolic then it admits a natural semi-conjugacy with a hyperbolic automorphism of a possibly lower dimensional ball $\mathbb B^k$. We also obtain the dual result for a holomorphic self-map $f\colon Ω\to Ω$ with a boundary repelling fixed point. Both results are obtained by rescaling the dynamics of $f$ via the squeezing function.
The squeezing problem on $\mathbb{C}$ can be stated as follows. Suppose that $Ω$ is a multiply connected domain in the unit disk $\mathbb{D}$ containing the origin $z=0$. How far can the boundary of $Ω$ be pushed from the origin by an injective holomorphic function $f:Ω\to \mathbb{D}$ keeping the origin fixed? In this note, we discuss recent results on this problem obtained by Ng, Tang and Tsai (Math. Anal. 2020) and by Gumenyuk and Roth (arXiv:2011.13734, 2020) and also prove few new results using a method suggested in one of our previous papers (Zapiski Nauchn. Sem. POMI 1993).
Strong localization of invariant metrics
本文研究了复空间域边界点附近不变度量与挤压函数的局部化问题，给出了峰值点处的定量强局部化及反峰值点处的弱局部化，并分析了严格伪凸边界点附近度量的行为。
Very recently, the Fridman function of a complex manifold $X$ has been identified as a dual of the squeezing function of $X$. In this paper, we prove that the Fridman function for certain hyperbolic complex manifold $X$ is bounded above by the injectivity radius function of $X$. This result also suggests us to use the Fridman function to extend the definition of uniform thickness to higher-dimensional hyperbolic complex manifolds. We also establish an expression for the Fridman function (with respect to the Kobayashi metric) when $X = \mathbb{D} \diagup Γ$ and $Γ$ is a torsion-free discrete su…
Geometry of bounded generic domains with piecewise smooth boundary
研究有界分段光滑边界域的几何特性，探讨极化函数与Levi平坦性关系，并证明二维有界通用凸域在有限体积商下与双盘双射，同时证明Teichmuller空间不能与有界通用域双射。
In this paper we establish a gap theorem for the complex geometry of smoothly bounded convex domains which informally says that if the complex geometry near the boundary is close to the complex geometry of the unit ball, then the domain must be strongly pseudoconvex. One consequence of our general result is the following: for any dimension there exists some $ε> 0$ so that if the squeezing function on a smoothly bounded convex domain is greater than $1-ε$ outside a compact set, then the domain is strongly pseudoconvex (and hence the squeezing function limits to one on the boundary). Another con…
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…
The purpose of this article is twofold. The first aim is to characterize $h$-extendibility of smoothly bounded pseudoconvex domains in $\mathbb C^{n+1}$ by their noncompact automorphism groups. Our second goal is to show that if the squeezing function tends to $1$ at an $h$-extendible boundary point of a smooth pseudoconvex domain in $\mathbb C^{n+1}$, then this point must be strongly pseudoconvex.
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…
Boundary Behavior of Bisectional Curvatures for Weighted Bergman Metrics
本文研究了加权伯格曼度量的全纯双 sectional 曲率在边界处的渐进行为，通过L²正交投影刻画极值函数，推导出加权双 sectional 曲率的显式公式，并利用挤压函数得到有界伪凸域上曲率的定量上界和下界，进一步证明在强伪凸边界点处，双 sectional 曲率渐近趋于单位球的曲率，作为应用，这些结果为已知的Kähler-Einstein度量双 sectional 曲率渐进行为提供了简洁统一的证明。
Given an iterated function system of affine dilations with fixed points the vertices of a regular polygon, we characterize which points in the limit set lie on the boundary of its convex hull.
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