Bottom of spectrum of Kahler manifolds with strongly pseudoconvex boundary

Holomorphic family of strongly pseudoconvex domains in a Kähler manifold

Let $p:X\rightarrow Y$ be a surjective holomorphic mapping between Kähler manifolds. Let $D$ be a bounded smooth domain in $X$ such that every generic fiber $D_y:=D\cap p^{-1}(y)$ for $y\in Y$ is a strongly pseudoconvex domain in $X_y:=p^{-1}(y)$, which admits the complete Kähler-Einstein metric. This family of Kähler-Einstein metrics induces a smooth $(1,1)$-form $ρ$ on $D$. In this paper, we prove that $ρ$ is positive-definite on $D$ if $D$ is strongly pseudoconvex. We also discuss the extensioin of $ρ$ as a positive current across singular fibers.

在有界伪凸域上的Kähler-Ricci孤立子

The Kähler-Ricci soliton on bounded pseudoconvex domains

本文研究有界伪凸域中C²边界上的Kähler-Ricci孤立子，证明其必为Kähler-爱因斯坦度量，并基于Huang和Xiao对Cheng猜想的解决，进一步建立Bergman Kähler-Ricci孤立子的类似结果。

Note on the paper of Fu and Wong on strictly pseudoconvex domains with Kähler--Einstein Bergman metrics

It is shown that the Ramadanov conjecture implies the Cheng conjecture. In particular it follows that the Cheng conjecture holds in dimension two.

A Schwarz lemma for weakly Kähler-Finsler manifolds

In this paper, we first establish several theorems about the estimation of distance function on real and strongly convex complex Finsler manifolds and then obtain a Schwarz lemma from a strongly convex weakly Kähler-Finsler manifold into a strongly pseudoconvex complex Finsler manifold. As applications, we prove that a holomorphic mapping from a strongly convex weakly Kähler-Finsler manifold into a strongly pseudoconvex complex Finsler manifold is necessary constant under an extra condition. In particular, we prove that a holomorphic mapping from a complex Minkowski space into a strongly pseud…

Kähler-Einstein metrics on strictly pseudoconvex domains

The metrics of S. Y. Cheng and S.-T. Yau are considered on a strictly pseudoconvex domains in a complex manifold. Such a manifold carries a complete Kähler-Einstein metric if and only if its canonical bundle is positive. We consider the restricted case in which the CR structure on $\partial M$ is normal. In this case M must be a domain in a resolution of the Sasaki cone over $\partial M$. We give a condition on a normal CR manifold which it cannot satisfy if it is a CR infinity of a Kähler-Einstein manifold. We are able to mostly determine those normal CR 3-manifolds which can be CR infinities…

Locally conformally Kahler metrics obtained from pseudoconvex shells

A locally conformally Kahler (LCK) manifold is a complex manifold admitting a Kahler covering M, such that its monodromy acts on this covering by homotheties. A compact LCK manifold is called LCK with potential if M admits an authomorphic Kahler potential. It is known that in this case it is an algebraic cone, that is, the set of all non-zero vectors in the total space of an anti-ample line bundle over a projective orbifold. We start with an algebraic cone C, and show that the set of Kahler metrics with potential which could arise from an LCK structure is in bijective correspondence with the s…

Fiberwise Kähler-Ricci flows on families of bounded strongly pseudoconvex domains

Let $π:\mathbb{C}^n\times\mathbb{C}\rightarrow\mathbb{C}$ be the projection map onto the second factor and let $D$ be a domain in $\mathbb{C}^{n+1}$ such that for $y\inπ(D)$, every fiber $D_y:=D\capπ^{-1}(y)$ is a smoothly bounded strongly pseudoconvex domain in $\mathbb{C}^n$ and is diffeomorphic to each other. By Chau's theorem, the Kähler-Ricci flow has a long time solution $ω_y(t)$ on each fiber $D_y$. This family of flows induces a smooth real (1,1)-form $ω(t)$ on the total space $D$ whose restriction to the fiber $D_y$ satisfies $ω(t)\vert_{D_y}=ω_y(t)$. In this paper, we prove that $ω(t…

二维伪凸域上的Kähler-Einstein Bergman度量

Kähler-Einstein Bergman metrics on pseudoconvex domains of dimension two

本文证明具有Kähler-Einstein Bergman度量的二维有限型伪凸域必双全纯等价于单位球，回答了Yau的老问题，证明基于Bergman核导数沿临界相切路径的渐近分析。

渐近局部复双曲流形的渐近严格伪凸 CR 结构

Asymptotic strictly pseudoconvex CR structure for asymptotically locally complex hyperbolic manifolds

该研究为曲率渐近于复双曲空间的完备非紧 Kähler 流形，提出利用严格伪凸 CR 结构实现紧化的构造方法。

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。

On Complete Noncompact Kähler Manifolds with Positive Bisectional Curvature

We prove that a complete noncompact Kähler manifold $M^{n}$of positive bisectional curvature satisfying suitable growth conditions is biholomorphic to a pseudoconvex domain of {\bf C}$^{n}$ and we show that the manifold is topologically {\bf R}$^{2n}$. In particular, when $M^{n}$ is a Kähler surface of positive bisectional curvature satisfying certain natural geometric growth conditions, it is biholomorphic to {\bf C}$^{2}$.

完备复Finsler度量与Kobayashi度量的一致等价性

Complete complex Finsler metrics and uniform equivalence of the Kobayashi metric

本文通过研究Bergman曲率有界性和Kähler流形曲率对称性，结合Schwarz引理，证明了有界强拟凸域或有界凸域上存在全纯截面曲率被负常数上界控制的完备强拟凸复Finsler度量，并进一步建立了有界强凸域上Kobayashi度量与Carathéodory度量的一致等价性。

Asymptotic behavior of the complete Kahler-Einstein metric in tube domains

We study the complete Kahler-Einstein metric in tube domains. We obtain estimates of this metric and its holomorphic bisectional curvatures near the weakly pseudoconvex boundary points.

Autodual Einstein versus Kahler-Einstein

Any strictly pseudoconvex domain in C2 carries a complete Kahler-Einstein metric, the Cheng-Yau metric, with ``conformal infinity'' the CR structure of the boundary. It is well known that not all CR structures on the 3-sphere arise in this way. In this paper, we study CR structures on the 3-sphere satisfying a different filling condition: boundaries at infinity of (complete) selfdual Einstein metrics. We prove that (modulo contactomorphisms) they form an infinite dimensional manifold, transverse to the space of CR structures which are boundaries of complex domains (and therefore of Kahler-Eins…

Complex Plateau problem in non Kähler manifolds

We consider a complex Plateau problem for strongly pseudoconvex contours in non Kähler manifolds. A positive solution in the case of manifolds carrying a pluriclosed Hermitian metric forms is given. For the general case we propose a conjecture.

A sharp estimate for the bottom of the spectrum of the Laplacian on Kähler manifolds

On a complete noncompact Kähler manifold we prove that the bottom of the spectrum for the Laplacian is bounded from above by $m^2$ if the Ricci curvature is bounded from below by $-2(m+1)$. Then we show that if this upper bound is achieved then the manifold has at most two ends. These results improve previous results on this subject proved by P. Li and J. Wang in \cite {L-W3} and \cite{L-W} under assumptions on the bisectional curvature.

伪凸域中自由边界圆盘的 Morse 指标

Morse index of free boundary disk for pseudoconvex domain

本文研究伪凸域或非负双截曲率 Kähler 流形中非全纯圆盘的 $\overline{\partial}$-能量 Morse 指标，证明能量极小化圆盘必为全纯，且非全纯临界圆盘的 Morse 指标至少为 $n-1$，并推广到 $k$-伪凸域。

Optimal $L^2$ extension of sections from subvarieties in weakly pseudoconvex manifolds

In this paper, we obtain optimal $L^2$ extension of holomorphic sections of a holomorphic vector bundle from subvarieties in weakly pseudoconvex Kähler manifolds. Moreover, in the case of line bundle the Hermitian metric is allowed to be singular.

Volume renormalization for complete Einstein--Kähler metrics

For a strictly pseudoconvex domain in a complex manifold we define a renormalized volume with respect to the approximately Einstein complete Kähler metric of Fefferman. We compute the conformal anomaly in complex dimension two and apply the result to derive a renormalized Chern--Gauss--Bonnet formula. Relations between renormalized volume and the CR $Q$-curvature are also investigated.

The geometry of domains with negatively pinched Kähler metrics

We study how the existence of a negatively pinched Kähler metric on a domain in complex Euclidean space restricts the geometry of its boundary. In particular, we show that if a convex domain admits a complete Kähler metric, with pinched negative holomorphic bisectional curvature outside a compact set, then the boundary of the domain does not contain any complex subvariety of positive domain. Moreover, if the boundary of the domain is smooth, then it is of finite type in the sense of D'Angelo. We also use curvature to provide a characterization of strong pseudoconvexity amongst convex domains. …

Holomorphic invariant strongly pseudoconvex complex Finsler metrics

Let $B_n$ and $P_n$ be the unit ball and the unit polydisk in $\mathbb{C}^n$ with $n\geq 2$ respectively. Denote $\mbox{Aut}(B_n)$ and $\mbox{Aut}(P_n)$ the holomorphic automorphism group of $B_n$ and $P_n$ respectively. In this paper, we prove that $B_n$ admits no $\mbox{Aut}(B_n)$-invariant strongly pseudoconvex complex Finsler metric other than a constant multiple of the Poincar$\acute{\mbox{e}}$-Bergman metric, while $P_n$ admits infinite many $\mbox{Aut}(P_n)$-invariant complete strongly convex complex Finsler metrics other than the Bergman metric. The $\mbox{Aut}(P_n)$-invariant complex …

Kähler 流形的 Wong-Rosay 型定理

The Wong-Rosay type theorem for Kähler manifolds

本文将 Wong-Rosay 定理从 $\mathbb{C}^n$ 的域推广至具有负截面曲率的单连通完备 Kähler 流形，并展示了具有全纯非不变度量流形的 Wong-Rosay 型定理。

Kähler-Einstein metrics and obstruction flatness of circle bundles

Obstruction flatness of a strongly pseudoconvex hypersurface $Σ$ in a complex manifold refers to the property that any (local) Kähler-Einstein metric on the pseudoconvex side of $Σ$, complete up to $Σ$, has a potential $-\log u$ such that $u$ is $C^\infty$-smooth up to $Σ$. In general, $u$ has only a finite degree of smoothness up to $Σ$. In this paper, we study obstruction flatness of hypersurfaces $Σ$ that arise as unit circle bundles $S(L)$ of negative Hermitian line bundles $(L, h)$ over Kähler manifolds $(M, g).$ We prove that if $(M,g)$ has constant Ricci eigenvalues, then $S(L)$ is obst…

Bergman核的局部化与Cheng-Yau猜想在实解析伪凸域上的研究

Localization of Bergman Kernels and the Cheng-Yau Conjecture on Real Analytic Pseudoconvex Domains

本文通过Bergman核的局部化理论，证明了实解析伪凸域的Bergman度量为Einstein度量当且仅当该域与单位球形域双全纯等价，从而推进了Cheng-Yau猜想的研究。

On the Steinness of a class of Kähler manifolds

Let $(M^n, g)$ be a complete non-compact Kähler manifold with non-negative and bounded holomorphic bisectional curvature. We prove that $M$ is holomorphically covered by a pseudoconvex domain in $\C^n$ which is homeomorphic to $\R^{2n}$, provided $(M^n, g)$ has uniform linear average quadratic curvature decay.

Deformation of Einstein metrics and $L^2$ cohomology on strictly pseudoconvex domains

We construct new complete Einstein metrics on smoothly bounded strictly pseudoconvex domains in Stein manifolds. This is done by deforming the Kähler-Einstein metric of Cheng and Yau, the approach that generalizes the works of Roth and Biquard on the deformations of the complex hyperbolic metric on the unit ball. Recasting the problem into the question of vanishing of an $L^2$ cohomology and taking advantage of the asymptotic complex hyperbolicity of the Cheng-Yau metric, we establish the possibility of such a deformation when the dimension of the domain is larger than or equal to three.

完备非紧凯勒流形谱的下界

Bottom of the spectrum of complete noncompact Kähler manifolds

综述了完备非紧凯勒流形上Hodge Laplacian谱的下界，重点讨论凯勒双曲流形和有界对称域，以及Ricci和双截曲率假设下的上界定理与刚性结果。

Bergman度量和Kähler Carathéodory度量的局部刚性

Local Rigidity of the Bergman Metric and of the Kähler Carathéodory Metric

论文证明了严格伪凸域在边界附近Carathéodory度量局部Kähler时，该域与单位球域双全纯等价，并建立了Bergman度量具有常全纯截面曲率的局部刚性定理，同时强调了其与Lu常数的关系。

非紧复流形上的非线性椭圆方程

Fully non-linear elliptic equations on noncompact complex manifolds

本文研究非紧复流形上非线性椭圆方程的先验估计与存在性，应用包括构造具有给定体积形式的凯勒度量及爱因斯坦度量。

完备非紧Kähler流形上经由Bergman核的不变度量等价性

Equivalence of Invariant metrics via Bergman kernel on complete noncompact Kähler manifolds

本文在完备非紧Kähler流形上，仅利用Bergman核与Bergman度量n次外积之比有界这一条件，证明了Bergman度量与完备Kähler–Einstein度量的等价性，并通过双参数拟凸域族验证了方法的有效性。

完全凯勒-爱因斯坦度量的迭代构造

An iterative construction of complete Kähler--Einstein metrics

本文提出了一种迭代方法，用于在非紧致凯勒流形上构造具有负标量曲率的凯勒-爱因斯坦度量，并探讨了其在相对canonical包上的半正曲率性质。

A Burns-Epstein invariant for ACHE 4-manifolds

We define a renormalized characteristic class for Einstein asymptotically complex hyperbolic (ACHE) manifolds of dimension 4: for any such manifold, the polynomial in the curvature associated to the characteristic class euler-3signature is shown to converge. This extends a work of Burns and Epstein in the Kahler-Einstein case. This extends a work of Burns and Epstein in the Kahler-Einstein case. We also define a new global invariant for any 3-dimensional pseudoconvex CR manifold, by a renormalization procedure of the eta invariant of a sequence of metrics which approximate the CR structure. Fi…

经典域上全纯不变强伪凸复Finsler度量的几何

Geometry of holomorphic invariant strongly pseudoconvex complex Finsler metrics on the classical domains

本文在I-IV型不可约经典域上通过对Bergman度量形变构造了一类全纯不变强伪凸复Finsler度量，证明了其完备性及与Bergman度量相似的负曲率性质。

Geometry and Topology of the space of plurisubharmonic functions

Let $Ω$ be a strongly pseudoconvex domain. We introduce the Mabuchi space of strongly plurisubharmonic functions in $Ω$. We study metric properties of this space using Mabuchi geodesics and establish regularity properties of the latter, especially in the ball. As an application we study the existence of local Kähler-Einstein metrics.

非同胚的完全周期接触形式与常Tanaka-Webster标量曲率

Non-homothetic complete periodic contact forms with constant Tanaka--Webster scalar curvature

研究非紧致严格伪凸CR流形上具有常Tanaka-Webster标量曲率的非同胚接触形式的存在性及应用。

完全凯勒度量谱底的下界来自有限质量 plurisubharmonic 排斥

Bottom of the Spectrum of Complete Kähler Metrics from Finite-Mass Plurisubharmonic Exhaustions

本文研究了通过有限质量 plurisubharmonic 排斥构造的完全凯勒度量的谱底，证明了其等于 n²，并应用 Cegrell 定理展示了其在超凸域上的应用。

关于复Monge-Ampère算子的一些变分问题

Some variational problems for the complex Monge--Amp{è}re operator

研究复Monge-Ampère方程在强伪凸Kähler流形上的Dirichlet问题，当右端项随解递减时，通过流方法建立平滑解的存在性。

On a characterization of the complex hyperbolic space

Consider a compact Kähler manifold $M^m$ with Ricci curvature lower bound $Ric_M\geq -2(m+1) .$ Assume that its universal cover $% \widetilde{M}$ has maximal bottom of spectrum $λ_1(\widetilde{M}%) =m^2.$ Then we prove that $\widetilde{M}$ is isometric to the complex hyperbolic space $\Bbb{CH}^m.$

扭曲微分形式的狄利克雷格林核估计与索伯列夫型不等式

Dirichlet Green kernel estimates and Sobolev-type inequalities for twisted differential forms

该研究针对凯勒流形上的微分形式，证明了一致格林核估计，导出带边界项的索伯列夫型不等式，还得到无需伪凸性的\bar\nabla可解性及适用于扭曲德拉姆复形的类似结果。

非紧致流形的伯格曼-托普利茨量子化

Berezin-Toeplitz Quantization of non-compact manifolds

本文研究非紧致复几何背景下伯格曼-托普利茨量子化的性质，证明了在满足特定谱间隙条件下的Bergman投影的衰减性和渐进行为，并给出了托普利茨算子的闭包代数和星积结构。