Bo Berndtsson
We give an estimate for the volume of an analytic variety (or more generally the mass of a positive closed current) close to a real submanifold $M$. Applications are given to the Hausdorff measure of the intersection of the variety with $M$ and the exponential integrability of plurisubharmonic functions on $M$.
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We study boundary properties of plurisubharmonic functions near real submanifolds of almost complex manifolds.
Plurisubharmonic Functions in Calibrated Geometries
本文引入并研究了标定几何中的多重次调和函数的概念。这些函数推广了复几何中经典的多重次调和函数，并具有许多重要的性质。此外，它们大量存在，而相应的多谐波通常相当稀少。将复分析中通过多重次调和函数建立的一些结果推广到标定流形上。本文深入研究的问题：伪凸性和核心，积极的φ电流，Duval-Sibony对偶，和边界的φ子流形，所有的上下文中的一般校准流形（X，φ）。全实子流形的类似物被用来构造具有莫尔斯理论所允许的每一种拓扑型的严格φ-凸空间的巨大族。具体的校准在全文中用作示例。类比的霍奇猜想在校准几何被认为是。
Maximal plurisubharmonic models
一个维数为n且中心为V的解析对是一个对（V，M），其中M是一个（复）维数为n的复流形，V是一个n维全实闭解析子流形。对于一个解析对（V，M），我们将M中的函数u的类与M中的一个正有界区间相关联，使得对于V中的每个p，u（p）= 0。如果这个类允许一个极大函数u，则三元组（V，M，u）称为极大多重次调和模型。在解析对（V，M）的中心V上定义了一个伪度量E（V，M）之后，我们证明了（见定理4.1，定理5.1）极大多重次调和模型提供了由Lempert和Szoke在[16]中引入的Monge-Ampere模型的自然推广。
Let $M$ be a pseudoconvex, oriented, bounded and closed CR submanifold of $\mathbb{C}^{n}$ of hypersurface type. Our main result says that when a certain $1$-form on $M$ is exact on the null space of the Levi form, then the complex Green operator on $M$ satisfies Sobolev estimates. This happens in particular when $M$ admits a set of plurisubharmonic defining functions or when $M$ is strictly pseudoconvex except for the points on a simply connected complex submanifold.
The trace operator of quasi-plurisubharmonic functions on compact Kähler manifolds
本文研究了准亚调和函数在紧致凯勒流形上的迹算子，提出新的L^2扩展定理并应用于大线丛的限制体积。
The Bergman-Shilov boundary for subfamilies of $q$-plurisubharmonic functions
本文为紧致Hausdorff空间上某些上半连续函数子类引入Bergman-Shilov边界，证明其存在唯一性，建立与峰值点及Bishop型定理的联系，推广Bychkov定理，并证明光滑伪凸域上$q$-多次调和函数Shilov边界的部分具有$q$维复子流形叶状结构。
Let $K$ be a compact subset of a totally-real manifold $M$, where $M$ is either a $\mathcal{C}^2$-smooth graph in $\mathbb{C}^{2n}$ over $\mathbb{C}^n$, or $M=u^{-1}\{0\}$ for a $\mathcal{C}^2$-smooth submersion $u$ from $\mathbb{C}^n$ to $\mathbb{R}^{2n-k}$, $k\leq n$. In this case we show that $K$ is polynomially convex if and only if for a fixed neighbourhood $U$, defined in terms of the defining functions of $M$, there exists a plurisubharmonic function $Ψ$ on $\mathbb{C}^n$ such that $K\subset \{Ψ<0\}\subset U$.
Stein neighborhood bases for complex--totally real unions
本文研究闭复与全实子流形并集的Stein邻域基，证明特定线性构型具有Stein邻域基，并给出无Stein邻域基的反例。
Duality of Positive Currents and Plurisubharmonic Functions in Calibrated Geometry
最近，作者证明，有一个强大的位势论附加到任何校准的流形（X，X）。特别地，在X上存在n-多重次调和函数、n-凸区域、n-凸边界等，所有这些都是相互关联的，并且具有许多良好的性质。在本文中，我们表明，在很强的意义上，多重次调和函数是极极流的极流子流形，或更一般地说，极流的原始文件中研究的校准。特别地，我们建立了一个类似的Duval-Sibony对偶，它用X上的正格林流和K上的Jensen测度刻画了X中紧集K的正凸包中的点。我们还完全用π-多重次调和函数刻画了π-电流的边界.具体的校准在全文中用作示例。类比的霍奇猜想在校准几何被认为是。
On the extension of Kähler currents on compact complex manifolds
本文解决了Coman-Guedj-Zeriahi提出的关于紧复流形上多重次调和函数限制的问题，在紧厄米特流形上给出了肯定答案。
We compute the extremal plurisubharmonic function of the real torus viewed as a compact subset of its natural algebraic complexification.
An Introduction to Potential Theory in Calibrated Geometry
在校准几何中引入并研究了多重次调和函数的概念。这些函数是复几何中经典多重次调和函数的推广，具有重要的性质。此外，它们大量存在，而相应的多谐波通常相当稀少。将复分析中通过多重次调和函数建立的一些结果推广到标定流形上。特别地，引入并研究了标定流形（X，X）的伪凸性概念。全实子流形的类似物也被引入并用于构造具有莫尔斯理论所允许的每一种拓扑型的严格凸空间的巨大族。具体的校准在全文中用作示例。
A complex surface admitting a strongly plurisubharmonic function but no holomorphic functions
为了解决一个老问题，我们在复射影平面CP ^2中找到了一个区域X，它允许一个强多重次调和函数，但使得X上的每个全纯函数都是常数。定义域X可以被选择为同构于2-球面上的定向实2-平面丛。
We observe that a recent result by Gardiner and Sjödin, solving a problem of Král on subharmonic functions, can be easily generalized to yield a somewhat stronger result. This can be combined with a viscosity technique of ours, which we slightly improve, to obtain Radó- type theorems for plurisubharmonic functions. Finally, we study the Borel complexity of the critical set and the set where the gradient does not exist finitely of subharmonic functions and general real valued functions.
Extremal omega-plurisubharmonic functions as envelopes of disc functionals - Generalization and applications to the local theory
本文将复流形$X$上函数$\sup \{u \in \PSH(X,ω) ; u\leq ϕ\}$的Poletsky圆盘包络公式推广到实（1，1）流$ω=ω_1-ω_2$是两个正闭（1，1）流之差，$ϕ$是$ω_1$ -上连续函数与多重次调和函数之差的情形.
Dinh--Sibony theory of tangent and density currents is a recent but powerful tool to study positive closed currents. Over twenty years ago, Alessandrini and Bassanelli initiated the theory of the Lelong number of a positive plurisubharmonic current in $\mathbb{C}^k$ along a linear subspace. Although the latter theory is intriguing, it has not yet been explored in-depth since then. Introducing the concept of the generalized Lelong numbers and studying these new numerical values, we extend both theories to a more general class of positive plurisubharmonic currents and in a more general context o…
Valuative analysis of planar plurisubharmonic functions
证明了2维全纯芽环R上的赋值可以自然地在多重次调和函数上赋值，从而得到Demailly意义下的广义Lelong数.因此，任何多重次调和函数都定义了R上赋值集V上的一个实值函数，并且通过一个根据V上的树结构定义的自然拉普拉斯算子，定义了V上的一个正测度。在较弱的正则性假设下，给出了两个多重次调和函数的混合蒙格-安培质量的精确公式。因此，任何广义的Lelong数都可以解释为估值的平均值。使用我们的机器，我们还表明，任何正的封闭（1，1）电流T的奇异性可以在以下意义上衰减：存在一个有限的组合物的爆破，使拉回的T分解成两部分，第一个与一个除数与正规交叉支持，第二个具有小Lelong数。
In this note we investigate the regularity of geodesics in the space of convex and plurisubharmonic functions. In the real setting we prove (optimal) local C^{1,1} regularity. We construct examples which prove that the global C^{1,1} regularity fails both in the real and complex case in contrast to the Kähler manifold setting. Finally we show a necessary and sufficient conditions for existence of a smooth geodesic between two smooth strictly convex functions.
We show that the monodromy theorem holds on arbitrary connected free sets for noncommutative free analytic functions. Applications are numerous-- pluriharmonic free functions have globally defined pluriharmonic conjugates, locally invertible functions are globally invertible, and there is no nontrivial cohomology theory arising from analytic continuation on connected free sets. We describe why the Baker-Campbell-Hausdorff formula has finite radius of convergence in terms of monodromy, and solve a related problem of Martin-Shamovich. We generalize the Dym-Helton-Klep-McCullough-Volcic theorem--…
On rigid $q$-plurisubharmonic functions and $q$-pseudoconvex tube domains in $\mathbb{C}^n$
本文证明管状域上的 $q$-多次调和函数等价于基域上的实 $q$-凸函数，并由此刻画 $q$-伪凸管状域，应用于仿射线性映射图的补集和 Reinhardt 域。
Regularity of geodesics in the spaces of convex and plurisubharmonic functions II
本研究延续凸与多重次调和函数空间测地线的探究，证明光滑严格凸函数间测地线的最优正则性，拓展$C^{1,α}$范畴正则性理论，并探讨环面不变严格多重次调和函数间测地线的正则性。
Polya's inequalities, global uniform integrability and the size of plurisubharmonic lemniscates
首先证明了一个新的不等式，该不等式一致地比较了Borel子集关于任意给定的复欧氏球$\B \sub \C^n$的相对体积与它在$\C^n$中关于同一球$\B$的相对对数容量.证明了$\C^n$的任意一般实子空间的欧氏球Borel子集的一个类似比较不等式。然后，我们给出了这些不等式的几个有趣的应用。首先我们得到了\psh上无穷远对数奇点的\psh函数的Lelong类以及超凸域$\W \Sub \C^n.$上有界Monge-Ampère质量的\psh函数的Cegrell类的\psh双纽线的相对大小的一致估计，然后我们还推出了\psh函数的Lelong类和Cegrell类的全局性态的新结果.
Continuous approximation of quasi-plurisubharmonic functions
设$X$是紧致Kähler流形，$θ$是表示一个大上同调类$α\in H^{1,1}(X,\R)$的光滑闭$(1,1)$ -实形式.本说明的目的是显示，使用多能和粘性技术，任何$θ$ -pluisubharmonic函数$\f$可以近似从上面的一个递减序列的连续$θ$ -多重次调和函数具有最小的奇异性，假设存在一个这样的功能。
A weakly complete space is a complex space admitting a (smooth) plurisubharmonic exhaustion function. In this paper, we classify those weakly complete complex surfaces for which such exhaustion function can be chosen real analytic: they can be modifications of Stein spaces or proper over a non compact (possibly singular) complex curve or foliated with real analytic Levi-flat hypersurfaces which in turn are foliated by dense complex leaves (these we call surfaces of Grauert type). In the last case, we also show that such Levi-flat hypersurfaces are in fact level sets of a global proper plurihar…
A noncommutative (nc) function in $x_1,\dots,x_g,x_1^*,\dots,x_g$ is called plurisubharmonic (plush) if its nc complex Hessian takes only positive semidefinite values on an nc neighborhood of 0. The main result of this paper shows that an nc rational function is plush if and only if it is a composite of a convex rational function with an analytic (no $x_j^*$) rational function. The proof is entirely constructive. Further, a simple computable necessary and sufficient condition for an nc rational function to be plush is given in terms of its minimal realization.
We study the minimum sets of plurisubharmonic functions with strictly positive Monge-Ampère densities. We investigate the relationship between their Hausdorff dimension and the regularity of the function. Under suitable assumptions we prove that the minimum set cannot contain analytic subvarieties of large dimension. In the planar case we analyze the influence on the regularity of the right hand side and consider the corresponding free boundary problem with irregular data. We provide sharp examples for the Hausdorff dimension of the minimum set and the related free boundary. We also draw sever…
Mappings between real submanifolds in complex space
本文综述了复空间中实子流形之间形式映射的有限决定、收敛和逼近的一些最新结果。文中还给出了一些说明。
Equivalences of real submanifolds in complex space
证明了对C^N中的任意实解析子流形M，存在包含在M中的真解析子流形V，使得对M中的任意点p，\V中的任意实解析子流形M'，以及M'中的任意点p'，子流形M和M'分别在p和p'处的芽形式等价当且仅当它们是双全纯等价的.更一般的结果k-等价也陈述和证明。
A reflection principle for real-analytic submanifolds of complex spaces
本文给出了不同维一般子流形之间CR-映射的不变非退化条件，并利用它证明了这类映射的一个反射原理。
Algebraicity of local holomorphisms between real-algebraic submanifolds of complex spaces
我们证明了$C^n$和$C^{n'}$之间的全纯映射$M\subset C^n$的芽是代数的，只要$M'$不包含复解析圆盘，$M'\subset C^{n'}$是一般的和最小的。我们还提出了一个算法，寻找复解析盘在一个真正的子流形。
Symplectic geometry and the uniqueness of Grauert tubes
紧致实解析黎曼流形M有一个正则复化，其多重次调和耗尽函数满足齐次复Monge-Ampere方程，称为Grauert管.从复分析的观点出发，一些作者考虑了给定的复流形是否可以从这种构造中以多种方式产生。我们表明，给定一个紧凑的M和有限的排气，基本的黎曼结构是唯一的。证明使用的技术全纯磁盘跨越两个确切的拉格朗日子流形的余切丛的M，和施瓦茨反射。
Holomorphic maps of real submanifolds in complex spaces of different dimensions
定义了一个新的与三元组（M，M '，H）相关联的局部不变量（称为退化），其中M和M'分别是C^N和C^N '的实子流形，H：M->M'是全纯映射、形式全纯映射或光滑CR-映射.我们使用这个不变量找到充分条件下，有限喷流的依赖性，收敛性和代数结果举行。
In case M is Levi non-degenerate in the sense Tumanov, we construct stationary discs for $M$. If furthermore M satisfies an additional non-degeneracy condition, we apply the method of stationary discs to obtain 2-jet determination of CR automorphisms of M.
Let M be a real analytic Riemannian manifold. An adapted complex structure on $TM$ is a complex structure on a neighborhood of the zero section such that the leaves of the Riemann foliation are complex submanifolds. This structure is called entire if it may be extended to the whole of $TM$. We prove here that the only real analytic Zoll metric on the $n$-sphere with an entire adapted complex structure on $TM$ is the round sphere. Using similar ideas, we answer a special case of an algebraization question raised by the first author, characterizing some Stein manifolds as affine algebraic in ter…
Projection on Segre varieties and determination of holomorphic mappings between real submanifolds
证明了将一个有限型实解析一般子流形变换成另一个实解析一般子流形的全纯映射的芽由它在目标流形的Segre簇上的投影决定.给出了一个映射到目标流形的Segre簇的芽是一个全纯映射将源流形投影到目标流形的充要条件.给出了一个应用于双全纯等价问题的例子。
Let $(X,ω)$ be a compact $n$-dimensional Kähler manifold on which the integral of $ω^n$ is $1$. Let $K$ be an immersed real $\mathcal{C}^3$ submanifold of $X$ such that the tangent space at any point of $K$ is not contained in any complex hyperplane of the (real) tangent space at that point of $X.$ Let $μ$ be a probability measure compactly supported on $K$ with $L^p$ density for some $p>1.$ We prove that the complex Monge-Ampère equation $(dd^c φ+ ω)^n=μ$ has a Hölder continuous solution.
Reflection ideals and mappings between generic submanifolds in complex space
本文研究了多维复空间中光滑一般子流形之间的形式映射，建立了有限决定、收敛性、局部双全纯和代数等价等结果。我们的有限决定性结果给出了充分条件，以保证一个形式映射是由它的喷流在一个预定阶点上唯一决定的。对于实解析的一般子流形，在适当的假设下证明了形式映射的收敛性，并给出了自然几何条件，保证了如果这类子流形的两个芽形式等价，则它们必双全纯等价.此外，如果子流形是实代数的，我们解决的问题，决定当双全纯等价蕴涵代数等价。特别地，我们证明了如果$\C^N$中的两个实代数超曲面是双全纯等价的，那么它们实际上是代数等价的。所有的结果首先证明在更一般的背景下的“反射理想”相关联的正式映射之间的正式以及真正的分析和真正的代数流形。
Pseudoholomorphic discs attached to CR-submanifolds of almost complex spaces
设E是殆复流形的一般实子流形.利用E的Levi形式研究了E上的Bishop圆的几何性质。
On complex points of codimension 2 submanifolds
本文研究复$n$维流形中余维2实子流形的复点结构。我们表明，一个复杂的点的局部结构的合痕只取决于它们的类型（椭圆或双曲）。我们还证明了任何这样的子流形都可以光滑地同构到具有2-严格伪凸邻域基的子流形。
The Bremermann-Dirichlet problem for unbounded domains of $C^n$
给定一个无界强伪凸区域D和一个定义在bD上的连续实值函数h，研究了D上的（极大）多重次调和函数u使得u=h在bD上的存在性.
On the local algebraizability of real analytic generic submanifolds of C^n
我们证明了稳定C^n中极小的、有限非退化的实代数子流形的双全纯局部（伪）群是实代数局部李群（SM的作品）Baouendi，P. Ebenfelt，L. P. Rothschild和D.扎伊采夫出版在这个方向限制，没有可理解的理由，各向同性组的一个固定的中心点）。给出了C^n中任意余维实解析刚性管局部可代数化的必要条件。在不使用Elie Cartan等价算法的情况下，我们解释到目前为止唯一已知的例子Im w = e^（|z^2|），由于X. Huang，S. Ji和S.S. Yau在2001年提出了C^2的不可代数化的Levi非退化实解析超曲面。这些基本准则为S.M. Baouendi，P. Ebenfelt和L.- P. Rothschild，在综述文章中：“复空间中实子流形的局部几何性质”，Bull. Amer. Math. Soc.（N.S.）37（2000），no.3，309- 336。第二个答案（充要条件）的地方algebraizability在齐次的情况下将随后出现。
Moduli spaces of model real submanifolds: two alternative approaches
本研究针对Beloshapka通用CR-模型的模空间构造问题，提出源自等价问题理论与李对称理论的两种替代方法，并借此计算出CR维数1、余维数4的通用CR-模型类对应的模空间M(1,4)。
Real-analytic Levi-flat codimension two CR singular submanifolds are a natural generalization to ${\mathbb{C}}^m$, $m > 2$, of Bishop surfaces in ${\mathbb{C}}^2$. Such submanifolds for example arise as zero sets of mixed-holomorphic equations with one variable antiholomorphic. We classify the codimension two Levi-flat CR singular quadrics, and we notice that new types of submanifolds arise in dimension 3 or greater. In fact, the nondegenerate submanifolds, i.e. higher order purturbations of $z_m=\bar{z}_1z_2+\bar{z}_1^2$, have no analogue in dimension 2. We prove that the Levi-foliation exten…
Plurisubharmonic functions and positive currents of type (1,1) over almost complex manifolds
若$(X,J)$是几乎复流形，则称一个函数$u$是$X$上的多重次调和函数，如果它是上半连续的，且它对每一局部伪全纯曲线的约束是次调和的.在复数情况下，证明了多次谐波性等价于$(1,1)$电流$i\partial_{_J}\bar{\partial}_{_J}u$为正的事实，（$(1,1)$电流$i\partial_{_J}\bar{\partial}_{_J}u$在这里不必闭合）.如果$u$是${\cal C}^2$类，则该猜想是平凡的.该结果在复可积情况下是初等的，因为算子$i\partial_{_J}\bar{\partial}_{_J}$可以写成复坐标中具有常系数的算子。因此，通过使用通常的卷积核进行正则化来保持电流的正性。这是不可能的，在几乎复杂的不可积的情况下，结果的证明需要更多的内在研究。本章证明了$(1,1)$ -电流$i\partial_{_J}\bar{\partial}_{_J}u$为正的必要性.我们还证明了一个上半连续函数$f$的特殊情况下，这是连续的补充奇异轨迹$f^{-1}(-\infty)$的积极性的充分性。为了证明的充分性的积极性在一般情况下的一个真正的分布$u$，我们建议一种方法依赖于一个相当微妙的正则化参数介绍Demailly。该方法是利用几乎复流形切丛上的Chern联络诱导的流来正则化函数$u$。
Let $(V,ω)$ be a compact Kähler manifold such that $V$ admits a cover by Zariski-open Stein sets with the property that $ω$ has a strictly plurisubharmonic exhaustive potential on each element of the cover. If $X\subset V$ is an analytic subvariety, we prove that any $ω|_X$-plurisubharmonic function on $X$ extends to a $ω$-plurisubharmonic function on $V$. This result generalizes a previous result of ours on the extension of singular metrics of ample line bundles. It allows one to show that any transcendental Kähler class in the real Neron-Severi space $NS_{\mathbb R}(V)$ has this extension pr…
Let $M \subset {\mathbb{C}}^{n+1}$, $n \geq 2$, be a real codimension two CR singular real-analytic submanifold that is nondegenerate and holomorphically flat. We prove that every real-analytic function on $M$ that is CR outside the CR singularities extends to a holomorphic function in a neighborhood of $M$. Our motivation is to prove the following analogue of the Hartogs-Bochner theorem. Let $Ω\subset {\mathbb{C}}^n \times {\mathbb{R}}$, $n \geq 2$, be a bounded domain with a connected real-analytic boundary such that $\partial Ω$ has only nondegenerate CR singularities. We prove that if $f \…
The Lefschetz theorem for CR manifolds and the nonexistence of real analytic Levi flat submanifolds
证明了Hermitian对称空间中CR子流形的Lefchetz定理。作为应用，我们证明了这类流形中不存在实解析Levi平坦子流形.
On formal maps between generic submanifolds in complex space
设H：（M，p）->（M '，p'）是C^N中具有非零Jacobian的实解析一般子流形芽之间的形式映射.设M在p处极小，M'在p'处全纯非退化，我们证明了映射H的收敛性。作为结果，我们得到了一个新的收敛性结果的任意形式映射之间的实解析超曲面时，目标不包含任何全纯曲线。在M和M'都是超曲面的情形下，我们还证明了当M仅为极小时，相应的反射函数的收敛性。这使我们能够推导出一个新的Artin型逼近定理的正式地图的一般满秩。
CR singularities of real fourfolds in $\mathbb{C}^3$
利用局部全纯坐标变换将实解析定义方程的二次系数变换为规范形，对复三维空间中实4子流形的CR奇点进行了分类。二次系数决定了一个交指数，它出现在紧致子流形CR奇点的整体计数公式中。
We study a germ of real analytic n-dimensional submanifold of C n that has a complex tangent space of maximal dimension at a CR singularity. Under some assumptions , we show its equivalence to a normal form under a local biholomorphism at the singularity. We also show that if a real submanifold is formally equivalent to a quadric, it is actually holomorphically equivalent to it, if a small divisors condition is satisfied. Finally, we investigate the existence of a complex submanifold of positive dimension in C n that intersects a real submanifold along two totally and real analytic submanifold…
Regularity and estimates for $J$-holomorphic discs attached to a maximal totally real submanifold
证明了极大全实子流形上的伪全纯圆盘的正则性继承自子流形的正则性和殆复结构的正则性。证明是基于一个明确的下界的计算Kobayashi度量在殆复流形，这也产生明确的估计，这样的光盘的Hölderian规范。
Real submanifolds of maximum complex tangent space at a CR singular point
研究了${\mathbf C}^n$的一个实解析$n$ -维子流形的芽，它在CR奇点处具有一个最大维数的复切空间。在其复化允许最大甲板变换数的条件下，我们研究了它在奇点处的局部（可能是形式）双全纯作用下到规范型的变换。我们首先正式共轭其相关的可逆映射$σ$适当的正规形式，并表明所有这些正规形式可以发散。如果奇点是{ \it阿贝尔}，我们表明，在某些假设下的线性部分的$σ$在奇点，实子流形是全纯等价的解析规范形。我们还表明，如果一个真正的子流形是形式上等价于一个二次曲面，它实际上是全纯等价的，如果一个小因子条件得到满足。最后证明了在一般情况下，${\mathbf C}^n$中存在一个正维数的复子流形，它沿着两个全实解析子流形与实子流形相交，这两个全实解析子流形在{ \it复型}的CR奇点处横截相交.
A survey on normal forms of real submanifolds with CR singularity
本文综述了从Bishop和Moser-Webster开创性工作到近期进展的实子流形正规形研究，聚焦CR奇点问题的核心方法与贡献。
Dynamics of the Segre varieties of a real submanifold in complex space
复空间中实解析一般子流形的迭代Segre映射是研究这类流形之间的全纯映射、形式映射和CR映射的重要工具。本文提出了光滑或形式子流形的迭代Segre映射理论。特别地，我们用迭代Segre映射的秩刻画了有限型.这里给出的方法是新的，简化了证明，并扩展了以前已知的真实分析的情况下的结果。
Let $w_0$ be a bounded, $C^3$, strictly plurisubharmonic function defined on $B_1\subset \mathbb{C}^n$. Then $w_0$ has a neighborhood in $L^{\infty}(B_1)$ with the following property: for any continuous, plurisubharmonic function $u$ in this neighborhood solving $1-\eps \le MA(u)\le 1+\eps$, one has $u\in W^{2,p}(B_{\frac{1}{2}})$, as long as $\eps>0$ is small enough depending only on $n$ and $p$. This partially generalizes Caffarelli's interior $W^{2,p}$ estimates for real Monge-Ampere to the complex version.
We construct a formal normal form for a real 2-codimensional submanifold $M\subset\mathbb{C}^{N+1}$ near a CR singularity approximating the sphere. This result gives a higher dimensional extension of Huang-Yin's normal form in $\mathbb{C}^{2}$.
We construct a family of analytic discs attached to a real submanifold M \subset $\mathbb{C}^{N+1}$ of codimension $2$ defined near a CR singularity.
We prove that the Bergman kernel function associated to a smooth measure supported on a piecewise-smooth maximally totally real submanifold K in C^n is of polynomial growth (e.g, in dimension one, K is a finite union of transverse Jordan arcs in C). Our bounds are sharp when K is smooth. We give an application to equidistribution of zeros of random polynomials extending a result of Shiffman-Zelditch to the higher dimensional setting.
Smooth submanifolds intersecting any analytic curve in a discrete set
我们构造例子$C^\infty$光滑子流形在${\Bbb C}^n$和${\Bbb R}^n$的余维2和1，相交于每一个复杂的，分别是真正的，解析曲线在离散集。这些例子被实现为紧环面或适当嵌入的欧几里得空间，并且是拟解析函数的图形。在复的情况下，这些子流形包含实$n$维环面或欧几里得空间，它们不是多极的，而与任何复解析圆盘的交是极的。
We prove that any embedded Maslov index two analytic disc attached to a totally real torus in the complex two-dimensional affine space extends to an analytic filling provided that the torus is contained in a regular level set of a strictly plurisubharmonic function.
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