On Bergman completeness of non-hyperconvex domains

Bergman completeness is not a quasi-conformal invariant

We show that Bergman completeness is not a quasi-conformal invariant for general Riemann surfaces.

An example concerning Bergman completeness

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.

Completeness on the worm domain and the Müntz-Szász problem for the Bergman space

In this paper we are concerned with the problem of completeness in the Bergman space of the worm domain $\mathcal{W}_μ$ and its truncated version $\mathcal{W}'_μ$. We determine some orthogonal systems and show that they are not complete, while showing that the union of two particular of such systems is complete. In order to prove our completeness result we introduce the Muentz-Szasz problem for the 1-dimensional Bergman space of the disk $\{ζ: |ζ-1|<1\}$ and find a sufficient condition for its solution.

加权Bergman空间中的密度与Hartogs域的Bergman完备性

Density in weighted Bergman spaces and Bergman completeness of Hartogs domains

本文研究带多重次调和权的加权Bergman空间中边界邻域全纯函数的密度，并证明由平面域边界距离定义的Hartogs域Bergman完备当且仅当底域边界点均非孤立。

Weakly 1-completeness of holomorphic fiber bundles over compact Kähler manifolds

In 1985 Diederich and Ohsawa proved that every disc bundle over a compact Kähler manifold is weakly 1-complete. In this paper, under certain conditions we generalize this result to the case of fiber bundles over compact Kähler manifolds whose fibers are bounded symmetric domains. Moreover if the bundle is obtained by the diagonal action on the product of irreducible bounded symmetric domains, we show that it is hyperconvex.

Some properties of the $p-$Bergman kernel and metric

The $p-$Bergman kernel $K_p(\cdot)$ is shown to be of $C^{1,1/2}$ for $1<p<\infty$. An unexpected relation between the off-diagonal $p-$Bergman kernel $K_p(\cdot,z)$ and certain weighted $L^2$ Bergman kernel is given for $1\le p\le 2$. As applications, we show that for each $1\le p\le 2$, $K_p(\cdot,z)\in L^q(Ω)$ for $q< \frac{2pn}{2n-α(Ω)}$ and $|K_s(z)-K_p(z)| \lesssim |s-p||\log |s-p||$ whenever the hyperconvexity index $α(Ω)$ is positive. Counterexamples for $2<p<\infty$ are given respectively. An optimal upper bound for the holomorphic sectional curvature of the $p-$Bergman metric when $2…

Bergman-Calabi diastasis and Kähler metric of constant holomorphic sectional curvature

We prove that for a bounded domain in $\mathbb C^n$ with the Bergman metric of constant holomorphic sectional curvature being biholomorphic to a ball is equivalent to the hyperconvexity or the exhaustiveness of the Bergman-Calabi diastasis. By finding its connection with the Bergman representative coordinate, we give explicit formulas of the Bergman-Calabi diastasis and show that it has bounded gradient. In particular, we prove that any bounded domain whose Bergman metric has constant holomorphic sectional curvature is Lu Qi-Keng. We also extend a theorem of Lu towards the incomplete situation…

$p-$Bergman核的正则性

Regularity of the $p-$Bergman kernel

本文证明了有界域上p-Bergman核的局部C^{1,1}正则性，并给出了全局非正则性例子及Levi形式上界估计和对数Lipschitz连续性结果。

Positivity and completeness of invariant metrics

We present a method for constructing global holomorphic peak functions from local holomorphic support functions for broad classes of unbounded domains. As an application, we establish a method for showing the positivity and completeness of invariant metrics including the Bergman metric mainly for the unbounded domains.

齐性Siegel域上的Bergman空间理论研究

On the Theory of Bergman Spaces on Homogeneous Siegel Domains

针对齐性Siegel域上混合范数Bergman空间现有两种研究思路下结果难对比的问题，本文梳理已有成果并在一种框架下补充了包含关系、对偶性等多方面结论，完善了相关理论。

单连通Kähler双曲流形的Kähler双曲模量

Kähler Hyperbolicity Modulus for Simply-connected Kähler Hyperbolic manifolds

本文研究完备Kähler流形上的Kähler双曲模量，利用多重次调和函数梯度长度的边界行为建立下界，并应用于有界对称域及带Kähler-Einstein度量或Bergman度量的有界强拟凸域。

与一类广义解析函数相关的 Bergman 和 Hardy 空间的某些方面

Some aspects of the Bergman and Hardy spaces associated with a class of generalized analytic functions

本文定义了基于 Dunkl 算子的 $λ$-解析函数，并研究了相关 Bergman 和 Hardy 空间的投影有界性、函数增长、稠密性、完备性、对偶空间及刻画与插值等问题。

Bergman kernels and local holomorphic Morse inequalities

Let X be a hermitian manifold and let L^k be a high power of a hermitian line bundle over X. Local versions of Demailly's holomorphic Morse inequalities are presented - after integration they yield the usual inequalities. The local weak inequalities hold on any hermitian manifold X, regardless of compactness and completeness. The proofs, which are elementary, are based on a new approach to pointwise Bergman kernel estimates, where the kernels are estimated by a model kernel in the standard complex space C^n.

带有界系数的双复Vekua方程的Bergman空间

Bergman spaces for the bicomplex Vekua equation with bounded coefficients

该研究针对有界域上带界系数的双复Vekua方程，构建其广义$L_p$-解的Bergman空间理论，分析空间完备性、解正则性等性质，还探讨主Vekua方程与电导率方程的关联等内容。

Equality in Suita's conjecture and metrics of constant Gaussian curvature

Without using the $L^2$ extension theorem, we provide a new proof of the equality part in Suita's conjecture, which states that for any open Riemann surface admitting a Green's function, the Bergman kernel and the logarithmic capacity coincide at one point if and only if the surface is biholomorphic to a disc possibly less a relatively closed polar set. In comparison with Guan and Zhou's proof, our proof essentially depends on Maitani and Yamaguchi's variation formula for the Bergman kernel, and we explore the harmonicity in such variations. As applications, we characterize the above surface b…

Linear invariants of complex manifolds and their plurisubharmonic variations

For a bounded domain $D$ and a real number $p>0$, we denote by $A^p(D)$ the space of $L^p$ integrable holomorphic functions on $D$, equipped with the $L^p$- pseudonorm. We prove that two bounded hyperconvex domains $D_1\subset \mc^n$ and $D_2\subset \mc^m$ are biholomorphic (in particular $n=m$) if there is a linear isometry between $A^p(D_1)$ and $A^p(D_2)$ for some $0<p<2$. The same result holds for $p>2, p\neq 2,4,\cdots$, provided that the $p$-Bergman kernels on $D_1$ and $D_2$ are exhaustive. With this as a motivation, we show that, for all $p>0$, the $p$-Bergman kernel on a strongly pseu…