Nikolay Shcherbina, Liyou Zhang
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.
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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。
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.
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…
Precise estimates for the Bergman distances of Dini-smooth bounded planar domains are given. These estimates imply that on such domains the Bergman distance almost coincides with the Carathéodory and Kobayashi distances.
We study the completeness of a metric which is related to the Bergman metric of a bounded domain. We provide a criterion for its completeness in the spirit of the Kobayashi criterion for the completeness of the Bergman metric. In particular we prove that in hyperconvex domains our metric is complete.
Increasing sequences of complex manifolds with uniform squeezing constants and their Bergman spaces
本文研究具有公共一致挤压常数的$\mathbb{C}^d$中有界开集递增并构成的复流形，通过Kobayashi度量余秩分类流形结构，证明其Bergman空间维数为零或无穷，解决了对应拟凸域的Wiegerinck猜想。
We estimate the boundary behavior of the Kobayashi metric on $\C\sm\set{0,1}$. We also compare the Bergman metric on the ring domain in $\C^{2}$ to the Bergman metric on the ball.
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。
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.
In this paper, we study the holomorphicity of totally geodesic Kobayashi isometric embeddings between bounded symmetric domains. First we show that for a $C^1$-smooth totally geodesic Kobayashi isometric embedding $f\colon Ω\toΩ'$ where $Ω$, $Ω'$ are bounded symmetric domains, if $Ω$ is irreducible and $\text{rank}(Ω) \geq \text{rank}(Ω')$ or more generally, $\text{rank}(Ω) \geq \text{rank}(f_*v)$ for any tangent vector $v$ of $Ω$, then $f$ is either holomorphic or anti-holomorphic. Secondly we characterize $C^1$ Kobayashi isometries from a reducible bounded symmetric domain to itself.
Estimates of the Bergman kernel and the Bergman and Kobayashi metrics on pseudoconvex domains near boundaries with constant Levi ranks are given.
We study the compactness of composition operators on the Bergman spaces of certain bounded pseudoconvex domains in $\mathbb{C}^n$ with non-trivial analytic disks contained in the boundary. As a consequence we characterize that compactness of the composition operator with a holomorphic, continuous symbol (up to the closure) on the Bergman space of the polydisk.
On Bergman type projection in some new analytic function spaces in bounded strongly pseudoconvex domains
本文将单位圆盘上Bergman型投影的有界性结果推广到具有光滑边界的有界强拟凸域中的两类解析函数空间。
We study the problem of the boundary behaviour of the Bergman kernel and the Bergman completeness in some classes of bounded pseudoconvex domains, which contain also non-hyperconvex domains. Among the classes for which we prove the Bergman completeness and the convergence of the Bergman kernel to infinity while tending to the boundary are all bounded pseudoconvex balanced domains, all bounded Hartogs domains with balanced fibers over regular domains and some bounded Laurent-Hartogs domains.
In this paper we investigate a two classes of domains in $\mathbb{C}^n$ generalizing the Hartogs triangle. We prove optimal estimates for the mapping properties of the Bergman projection on these domains.
A domain in $\C^n$ with Levi-flat boundary near a given point is characterized in terms of the boundary behavior of the Kobayashi or Bergman metrics, or of the Bergman kernel. Some results are given in the case of intermediate values of the rank of the Levi form.
In this paper we study the boundedness of Bergman projectors on weighted Bergman spaces on homogeneous Siegel domains of Type II. As it appeared to be a natural approach in the special case of tube domains over irreducible symmetric cones, we study such boundedness on the scale of mixed-norm weighted Lebesgue spaces. The sharp range for the boundedness of such operators is essentially known only in the case of tube domains over Lorentz cones. In this paper we prove that the boundedness of such Bergman projectors is equivalent to variuos notions of atomic decomposition, duality, and characteriz…
A class of pseudoconvex domains in $\mathbb{C}^{n}$ generalizing the Hartogs triangle is considered. The $L^p$ boundedness of the Bergman projection associated to these domains is established, for a restricted range of $p$ depending on the "fatness" of domains. This range of $p$ is shown to be sharp.
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.
Precise behavior of the Caratheodory, Kobayashi and Bergman metrics and distances near smooth boundary points of domains in C is found under different assumptions of regularity.
Using the logarithmic capacity, we give quantitative estimates of the Green function, as well as lower bounds of the Bergman kernel for bounded pseudoconvex domains in $\mathbb C^n$ and the Bergman distance for bounded planar domains. In particular, it is shown that the Bergman kernel satisfies $K_Ω(z)\gtrsim δ_Ω(z)^{-2}$ for any bounded pseudoconvex domain with $C^0-$boundary. An application to holomorphic motions is given.
Holomorphic Bergman isometries between bounded domains and the extension of Lu's theorem
该研究探讨复流形间保持Bergman度量的全纯映射，得出等维情形下全纯Bergman等距映射的性质，推广Lu定理并给出相关应用。
We obtain $L^p$ regularity for the Bergman projection on some Reinhardt domains. We start with a bounded initial domain $Ω$ with some symmetry properties and generate successor domains in higher {dimensions}. We prove: If the Bergman kernel on $Ω$ satisfies appropriate estimates, then the Bergman projection on the successor is $L^p$ bounded. For example, the Bergman projection on successors of strictly pseudoconvex initial domains is bounded on $L^p$ for $1<p<\infty$. The successor domains need not have smooth boundary nor be strictly pseudoconvex.
Bounded domains on Kobayashi hyperbolic manifolds covering compact complex manifolds
本文通过局部化技术证明，具有C^{1,1}边界且覆盖紧复流形的Kobayashi超复流形的universal cover必为欧几里得球。
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.
The quotient of the Szegö and Bergman kernels for a smooth bounded pseudoconvex domains in ${\mathbb C}^n$ is bounded from above by $δ|\logδ|^p$ for any $p>n$, where $δ$ is the distance to the boundary. For a class of domains that includes those of D'Angelo finite type and those with plurisubharmonic defining functions, the quotient is also bounded from below by $δ|\logδ|^p$ for any $p<-1$. Moreover, for convex domains, the quotient is bounded from above and below by constant multiples of $δ$.
Geometric estimates and comparability of Eisenman volume elements with the Bergman kernel on (C-)convex domains
本文在非退化凸域和C-凸域上给出Carathéodory与Kobayashi-Eisenman体积元素的几何上下界，得到商不变量的通用下界，并证明Bergman核与这些体积元素在仅依赖维数的常数下可比较。
On the setting of the Siegel upper half-space we study the spaces of bounded and vanishing mean oscillations which are defined in terms of the Berezin transform, and we use them to characterize bounded and compact Hankel operators on Bergman space.
We give a precise description of Bergman complete bounded pseudoconvex Reinhardt domains.
Regularity and irregularity of the Bergman projection on $L^p$ spaces is established on a natural family of bounded, pseudoconvex domains. The family is parameterized by a real variable $γ$. A surprising consequence of the analysis is that, whenever $γ$ is irrational, the Bergman projection is bounded only for $p=2$.
Infinite-Dimensionality of Bergman Spaces on\ Complete $d$-Bounded Kähler Manifolds
本文证明在Gromov意义下$d$-有界的完全Kähler流形上，其Bergman空间（$L^2$全纯$n$-形式的Hilbert空间）是无限维的，方法结合Gromov线丛构造、参数一致$\bar{\partial}$估计与奇异权$L^2$方法。
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…
We show some lower estimates for the Kobayashi-Royden metric on a class of smooth bounded pseudoconvex domains.
If a bounded domain can be covered by the polydisk through a rational proper holomorphic map, then the Bergman projection is $L^p$-bounded for $p$ in a certain range depending on the ramified rational covering. This result can be applied to the symmetrized polydisk and to the Hartogs triangle with exponent $γ$.
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…
We study Bergman kernels $K_Π$ and projections $P_Π$ in unbounded planar domains $Π$, which are periodic in one dimension. In the case $Π$ is simply connected we write the kernel $K_Π$ in terms of a Riemann mapping $φ$ related to the bounded periodic cell $\varpi$ of the domain $Π$. We also introduce and adapt to the Bergman space setting the Floquet transform technique, which is a standard tool for elliptic spectral problems in periodic domains. We investigate the boundedness properties of the Floquet transform operators in Bergman spaces and derive a general formula connecting $P_Π$ to a pro…
We construct a bounded plane domain which is Bergman complete but for which the Bergman kernel does not tend to infinity as the point approaches the boundary.
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…
We characterize bounded and compact positive Toeplitz operators defined on the Bergman spaces over the Siegel upper half-space.
It is shown that even a weak multidimensional Suita conjecture fails for any bounded non-pseudoconvex domain with $\mathcal C^1$ boundary: the product of the Bergman kernel by the volume of the indicatrix of the Azukawa metric is not bounded below. This is obtained by finding a direction along which the Sibony metric tends to infinity as the base point tends to the boundary. The analogous statement fails for a Lipschitz boundary. For a general $\mathcal C^1$ boundary, we give estimates for the Sibony metric in terms of some directional distance functions. For bounded pseudoconvex domains, the …
A new strong rigidity phenomenon for the Bergman metric
该研究建立了Bergman度量的局部到全局刚性新现象，证明两种情形下局部共形映射可延拓为双全纯映射，回答了Loi--Palmieri和Zimmer的问题，关键工具是新的Calabi型延拓定理。
We generalize the Hartogs triangle to a class of bounded Hartogs domains, and we prove that the corresponding Bergman projections are bounded on $L^p$ if and only if $p$ is in the range $(\frac{2n}{n+1},\frac{2n}{n-1})$.
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.
We introduce the notion of "virtual Bergman kernel" and apply it to the computation of the Bergman kernel of "domains inflated by Hermitian balls", in particular when the base domain is a bounded symmetric domain.
We shall give a variational formula of the full Bergman kernels associated to a family of smoothly bounded strongly pseudoconvex domains. An equivalent criterion for the triviality of holomorphic motions of planar domains in terms of the Bergman kernel is given as an application.
We obtain sharp ranges of $L^p$-boundedness for domains in a wide class of Reinhardt domains representable as sub-level sets of monomials, by expressing them as quotients of simpler domains. We prove a general transformation law relating $L^p$-boundedness on a domain and its quotient by a finite group. The range of $p$ for which the Bergman projection is $L^p$-bounded on our class of Reinhardt domains is found to shrink as the complexity of the domain increases.
$L^p$ boundedness of the Bergman Projection on generalizations of the Hartogs triangle in $\mathbb{C}^{n+1}$
本文研究一类推广的Hartogs三角形域$\Omega^{n+1}_\gamma$，得到了Bergman投影在该域上有界的精确$p$范围，将Edholm和McNeal在$n=1$时的结果推广到任意维数$n$。
Let $Ω\subset \mathbb{C}^n$ for $n\geq 2$ be a bounded pseudoconvex domain with a $C^2$-smooth boundary. We study the compactness of composition operators on the Bergman spaces of smoothly bounded convex domains. We give a partial characterization of compactness of the composition operator (with sufficient regularity of the symbol) in terms of the behavior of the Jacobian on the boundary. We then construct a counterexample to show the converse of the theorem is false.
On the Schwarz Lemma for Bergman metrics of bounded domains
本文提出基于概率论Cauchy-Schwarz不等式的新Schwarz引理，用于有界域Bergman度量的研究。
Sobolev Regularity of the Bergman Projection on a Smoothly Bounded Stein Domain that is not Hyperconvex
本文构造非超凸Stein域，证明Bergman投影在特定Sobolev空间中的正则性临界行为，并给出光滑函数投影非光滑的反例。
This is our third work on Bergman-type operator over bounded domains. In the previous two articles, we systematically study the boundedness, compactness and Schatten membership of Bergman-type on the Hilbert unit ball. In the present paper, we investigate singular integral operators induced by the Bergman kernel and Szegö kernel on the irreducible bounded symmetric domain in its standard Harish-Chandra realization. We completely characterize when Bergman-type operators and Szegö-type operators belong to Schatten class operator ideals by several analytic numerical invariants of the bounded symm…
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。
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.
Recent advances in Bergman type projections in bounded pseudoconvex and tubular domains
本文为综述，汇总多复变中有界伪凸域、对称锥上管型域等场景下不同维数函数空间间Bergman型投影的最新成果，推广经典结论，提出新研究问题并展望其应用场景。
We obtain ceratin estimates for the reproducing kernels of large weighted Bergman spaces. Applications of these estimates to boundedness of the Bergman projection on $L^p(\D,ω^{p/2})$, complex interpolation and duality of weighted Bergman spaces are given.
We study the boundary limits of the Carathéodory and Kobayashi-Eisenman volume elements on smoothly bounded convex finite type domains and also on Levi corank one domains.
In this paper, we study weighted composition operators on Bergman spaces of analytic functions which are square integrable on polydisk. We develop the study in full generality, meaning that the corresponding weighted composition operators are not assumed to be bounded. The properties of weighted composition operators such as real symmetry, unitariness, complex symmetry, are characterized fully in simple algebraic terms, involving their symbols. As it turns out, a weighted composition operator having a symmetric structure must be bounded. We also obtain the interesting consequence that real sym…
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.
We provide a large family of atoms for Bergman spaces on irreducible bounded symmetric domains. This vastly generalizes results by Coifman and Rochberg from 1980. The atomic decompositions are derived using the holomorphic discrete series representations for the domain, and the approach is inspired by recent advances in wavelet and coorbit theory. This approach also settles the relation between atomic decompositions for the bounded and unbounded realizations of the domain.
Bergman kernels of Monomial Polyhedra
本文针对单项式多面体这类有界伪凸Reinhardt域，利用Bell变换公式显式计算其Bergman核，并证明所得核为有理函数。
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