Ilya Kossovskiy, Bernhard Lamel, Ming Xiao
We provide regularity results for CR-maps between real hypersurfaces in complex spaces of different dimension with a Levi-degenerate target. We address both the real-analytic and the smooth case. Our results allow immediate applications to the study of proper holomorphic maps between Bounded Symmetric Domains.
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Singular Levi-flat hypersurfaces and codimension one foliations
研究了具有奇点的Levi平坦实解析超曲面。我们证明了超曲面正则部分上的Levi叶状结构在适当的意义下可以全纯扩张到奇点的邻域。
We discuss local polynomial convexity of real analytic Levi-flat hypersurfaces in $\mathbb C^n$, $n>1$, near singular points.
We prove the non-existence of real analytic Levi flat hypersurface whose complement is 1-convex and Levi foliation is transversely affine in compact Kahler surfaces.
Germs of singular Levi-flat hypersurfaces and holomorphic foliations
证明了实解析Levi-平坦超曲面的Levi叶状结构在非临界奇点附近可扩张到$d$ -网，并允许一个多值亚纯首次积分.
On real-analytic Levi-flat hypersurfaces and holomorphic webs
本文研究紧复曲面上与实解析Levi平坦超曲面相切的全纯网，证明其在特定条件下具有多值亚纯首次积分，并证明由Grassmannian中不可约实解析曲线诱导的Levi叶状结构可延拓为复射影空间上的代数网。
On Normal Forms for Levi-flat hypersurfaces with an Isolated Line Singularity
证明了具有孤立线奇点的局部实解析Levi-平坦超曲面的规范形的存在性。给出了以复直线为奇点的Levi平坦超曲面是C中实解析曲线通过全纯函数拉回的充分条件.
On Levi-flat hypersurfaces tangent to holomorphic webs
研究了与余维为1的全纯网芽相切的实解析Levi-平坦超曲面的芽。我们引入局部网的首次积分的概念。特别地，我们证明了一个$k$ -网具有通过与Levi-平坦超曲面相切的原点的多个不变解析子簇，它具有全纯的首次积分.
On the displacement rigidity of Levi flat hypersurfaces - The case of boundaries of disc bundles over compact Riemann surfaces
Levi平坦超曲面的不存在定理在文献中引起了极大的兴趣。接下来要问的问题是，现有的Levi平坦超曲面在变形下至少是刚性的。在这里，被认为是某些紧凑的黎曼曲面上的盘丛的边界的情况下。
Deformations of Levi flat hypersurfaces in complex manifolds
首先给出余维为1的可积分布的变形理论。我们定义了一个参数化的家庭的光滑超曲面附近的Levi平坦的超曲面L，使Levi平坦的变形给出的解决方案的Maurer-Cartan方程在DGLA相关的Levi叶理。我们说L是无穷小刚性的，如果L的Levi平坦变形的模空间的原点切锥是平凡的。证明了紧致复流形中紧致横截可平行Levi平坦超曲面的无穷小刚性，并给出了Kahler流形中无穷小刚性的充分条件.作为应用，我们证明了一类包含复射影平面的流形中横截可平行的Levi平坦超曲面的不存在性.
A Morse-Bott normal form for real analytic Levi-flat hypersurfaces
本文证明了由全纯函数实部消失定义的实解析Levi-flat超曲面的Morse-Bott标准型存在性，恢复了Burns-Gong标准型并为二次超曲面提供新标准型。
On the dimension of the stability group for a Levi non-degenerate hypersurface
本文对复空间中局部定义的非球面实解析超曲面进行了分类，这些超曲面的Levi形式具有不超过一个的负特征值，并且局部CR-自同构群的维数具有第二大值。
We study holomorphic foliations tangent to singular real-analytic Levi-flat hypersurfaces in compact complex manifolds of complex dimension two. We give some hypotheses to guarantee the existence of dicritical singularities of these objects. As consequence, we give some applications to holomorphic foliations tangent to real-analytic Levi-flat hypersurfaces with singularities in $\mathbb{P}^2$.
We study normal forms of germs of singular real-analytic Levi-flat hypersurfaces. We prove the existence of rigid normal forms for singular Levi-flat hypersurfaces which are defined by the vanishing of the real part of complex quasi-homogeneous polynomials with isolated singularity. This result generalizes previous results of Burns-Gong and Fernández-Pérez. Furthermore, we prove the existence of two new rigid normal forms for singular real-analytic Levi-flat hypersurfaces which are preserved by a change of isochore coordinates, that is, a change of coordinates that preserves volume.
We study singular real analytic Levi-flat subsets invariant by singular holomorphic foliations in complex projective spaces. We give sufficient conditions for a real analytic Levi-flat subset to be the pull-back of a semianalytic Levi-flat hypersurface in a complex projective surface under a rational map or to be the pull-back of a real algebraic curve under a meromorphic function. In particular, we give an application to the case of a singular real analytic Levi-flat hypersurface. Our results improve previous ones due to Lebl and Bretas -- Fernández-Pérez -- Mol.
On the dynamics of foliations in $\mathbb{P}^n$ tangent to Levi-flat hypersurfaces
设$\mathcal{F}$是$\mathbb{P}^n$中的余维1的全纯叶理，$n\geq 2$，其不变式为具有正则部分$M^{*}$的实解析Levi平坦超曲面$M$.然后，$\overline{M^{*}}$外的每个$\mathcal{F}$叶在$\overline{M^{*}}$中积累。
We prove that the normal bundle to the Levi foliation of a smooth Levi flat hypersurface does not admit a Hermitian metric with positive curvature along the leaves in compact Kähler manifolds of dimension at least 3. This represents an answer to a conjecture of Marco Brunella.
Stein compacts in Levi-flat hypersurfaces
我们探讨了Levi平坦超曲面M的Levi叶状结构的几何性质与M中紧集的全纯凸性之间的联系，或部分有界于M。应用包括Cauchy-黎曼函数的可拓性、dibar_b-方程的可解性、全纯函数和CR函数的可逼近性以及dibar-Neumann算子的全局正则性。
New invariant tensors in CR structures and a normal form for real hypersurfaces at a generic Levi degeneracy
我们在CR结构中引入了新的不变张量，它可以被看作是高阶Levi形式。利用二阶和三阶张量，我们给出了实超曲面在一般Levi退化下的完全形式正规形（在Chern-Moser意义下）。(We假设M在p处有一般的Levi退化，如果Levi行列式在p处为零，但它的微分不为零，并且M的Levi退化点的集合横截于p处的Levi零空间。）通过应用作者Baouendi和Rothschild关于形式映射的收敛定理，我们得出结论：上述形式规范形为实解析超曲面在一般Levi退化下提供了一个完整的双全纯不变量集.
Higher order invariants of Levi degenerate hypersurfaces
本文第一部分研究三维有限型超曲面的高阶CR不变量。利用完全规范形给出了具有平凡局部自同构群的超曲面的一个完整刻画，并对有限群给出了类似的结果。第二部分考虑有限Catlin多型超曲面和高维中的Kohn-Nirenberg现象。给出了$\mathbb C^{n+1}$中一类拟凸超曲面局部可凸的必要条件。
Local equivalence problem for Levi flat hypersurfaces
本文考虑光滑Levi平坦超曲面的芽，在以下局部等价的概念下：S_1 ~ S_2，如果它们的单侧邻域允许一个直到边界光滑的双全纯。我们引入了一个简单的不变量，这个关系，它允许证明一些平凡的特征（即等价于超平面）。然后，我们使用相同的不变量来构造无穷多个非平凡类，包括一个无限族的不等价的超曲面是几乎处处解析。
Affine rigidity of Levi degenerate tube hypersurfaces
设C（2，1）是一类连通的5维CR-超曲面，它们是2-非退化的且秩为1的一致Levi退化的.在最近的一篇文章中，我们证明了C（2，1）中的CR结构可约化为so（3，2）值绝对平行性。在本文中，我们应用这一结果研究了C^3中属于C（2，1）且CR-曲率全同为零的管超曲面。每一个这样的超曲面被证明是仿射等价于一个开放的子集管在未来的光锥。
Boundaries of Levi-flat hypersurfaces: special hyperbolic points
设$S\subset \C^n$，$n\geq 3$是紧致连通2-余维子流形，具有以下性质：存在Levi平坦超曲面，其边界为$S$，可能为流.我们的目标是得到这样的$S$包含至少一个特殊的1-双曲点的例子：球与两个角;基本模型和他们的胶合。还描述了图的特殊情况。
A contractible Levi-flat hypersurface in C^2 which is a determining set for pluriharmonic functions
我们在C^2中的椭球B的邻域中构造一个实解析列维平坦超曲面M，使得M的列维叶理的每一片叶子都是一个复圆盘，M与B的边界横交，并且M与B的交A具有以下性质：（i）A的闭包与R^3中的球微分同胚，并且具有C^2中的Stein邻域的基，（ii）A上的任何实解析函数，如果在每个Levi叶上是常数，则它也是常数，（iii）在A的连通开邻域中且在A上为零的任何多重调和函数恒为零。
Polynomial convexity of compacts that lies in certain Levi-flat hypersurfaces in $\mathbb{C}^2$
本文证明了位于某些 Levi 平坦超曲面上的全实圆盘的多项式凸性，并给出了 Hartog 三角形边界上全实圆盘多项式凸性的充要条件及一般紧子集的充分条件。
We give a complete classification of polynomial models for smooth real hypersurfaces of finite Catlin multitype in $\mathbb C^3$, which admit nonlinear infinitesimal CR automorphisms. As a consequence, we obtain a sharp 1-jet determination result for any smooth hypersurface with such model. The results also prove a conjecture of the first author about the origin of such nonlinear automorphisms (AIM list of problems, 2010). As another consequence, we describe all possible dimensions of the Lie algebra of infinitesimal CR automorphisms, which leads to a new "secondary" gap phenomenon.
Singular set of a Levi-flat hypersurface is Levi-flat
研究了奇异Levi平坦实解析超曲面的奇异集。我们证明了这样的超曲面的奇异集在适当的意义下是列维平坦的。我们还证明了，如果奇异集是足够小的，那么Levi-叶展延伸到一个奇异余维1全纯叶展的超曲面的邻域。
Explicit absolute parallelism for $2$-nondegenerate real hypersurfaces $M^5 \subset \mathbb{C}^3$ of constant Levi rank $1$
研究了$\mathbb{C}^3$的具有$2$ -非退化常Levi秩$1$的5维实超曲面$M^5$在双全纯映射下的局部等价问题.我们找到了两个不变量，$J$和$W$，它们用$M$的图形函数$F$明确地表示出来，它的环化给出了$M$局部双全纯于模型超曲面（光锥上的管）的充分必要条件.若$J$或$W$不为零，则证明了在$M$上的等价问题可归结为$\{e \}$ -结构之间的等价问题，即在$M$上构造了一个绝对并行.
Optimal lifting of Levi-degenerate hypersurfaces and applications to the Cauchy--Szegö projection
本文针对C^2中Levi退化有限型超曲面，通过优化Rothschild-Stein的提升方法将其构造性地提升到分层Lie群，得到水平向量场的显式Taylor展开，并据此建立了Cauchy--Szegö投影交换子的Schatten类估计。
On Normal Forms of Singular Levi-Flat Real Analytic Hypersurfaces
设$F(z)=\mathcal{R}e(P(z)) + h.o.t$是这样的，$M=(F=0)$定义了在$0\in\mathbb{C}^{n}$的实解析列维平坦的芽，$n\geq{2}$，其中$P(z)$是在$0\in\mathbb{C}^{n}$和Milnor数$μ$处具有孤立奇点的次数$k$的齐次多项式。证明了存在坐标$ϕ$的全纯变换，使得$ϕ(M)=(\mathcal{R}e(h)=0)$，其中$h(z)$是$μ+1$和$j^{k}_{0}(h)=P$的多项式.
We classify all (locally) homogeneous Levi non-degenerate real hypersurfaces in $\mathbb{C}^3$ with symmetry algebra of dimension $\geq 6$.
Convergent normal form for real hypersurfaces at generic Levi degeneracy
本文在$\CC{N},\,N\geq 2$中构造了实超曲面在一般Levi退化下的完全收敛规范形。这似乎是列维退化超曲面的第一个收敛正规形。特别是，我们得到的精神，在工作的陈和莫泽\cite{chern}，杰出的曲线在列维退化集，我们称之为\it退化链。
In our recent article (to appear in the Journal of Differential Geometry in 2016) we studied tube hypersurfaces in ${\mathbb C}^3$ that are 2-nondegenerate and uniformly Levi degenerate of rank 1. In particular, we discovered that for the CR-curvature of such a hypersurface to vanish it suffices to require that only two coefficients (called $Θ^2_{21}$ and $Θ^2_{10}$) in the expansion of a single component of the CR-curvature form be identically zero. In this paper, we show that, surprisingly, the vanishing of the entire CR-curvature is in fact implied by the vanishing of a single quantity deri…
The d-bar-Cauchy problem and nonexistence of Lipschitz Levi-flat hypersurfaces in CP^n with n>= 3
Lipschitz超曲面是局部为Lipschitz函数的图的超曲面。一个Lipschitz（或C^1）超曲面被称为列维平坦的，如果它被复维数（n-1）的复流形局部叶状。我们将证明CP^n中不存在Lipschitz Levi-平坦超曲面，其中n >= 3。我们对d-杆-柯西问题的新估计不同于早先的Siu积分核方法。
We study nonlinear automorphisms of Levi degenerate hypersurfaces of finite multitype. By recent results of Kolar, Meylan and Zaitsev, the Lie algebra of infinitesimal CR automorphisms may contain a graded component consisting of nonlinear vector fields of arbitrarily high degree, which has no analog in the classical Levi nondegenerate case, or in the case of finite type hypersurfaces in $\mathbb C^2$. We analyze this phenomenon for hypersurfaces of finite Catlin multitype in complex dimension three. The results provide a complete classification of such manifolds. As a consequence, we show on …
We establish an effective criterion for a dicritical singularity of a real analytic Levi-flat hypersurface. The criterion is stated in terms of the Segre varieties. As an application, we obtain a structure theorem for some class of currents in the nondicritical case.
In this work we study analytic Levi-flat hypersurfaces in complex algebraic surfaces. First, we show that if this foliation admits chaotic dynamics (i.e. if it does not admit a transverse invariant measure), then the connected components of the complement of the hypersurface are modifications of Stein domains. This allows us to extend the CR foliation to a singular algebraic foliation on the ambient complex surface. We apply this result to prove, by contradiction, that analytic Levi-flat hypersurfaces admitting a transverse affine structure in a complex algebraic surface have a transverse inva…
Extension of Levi-flat hypersurfaces past CR boundaries
研究了$C^\infty$ Levi-平坦超曲面边界上的局部条件，当边界是一般子流形时.对于非平凡的实解析边界，我们得到了一个扩张和唯一性结果，它迫使超曲面是实解析的。这使我们能够分类的所有真正的解析一般边界的列维平坦超曲面的法坐标。对于一般实解析边界的剩余情形，我们得到一个较弱的扩张定理。我们找到的例子表明，这两个扩展结果是最佳的。进一步，我们发现了一类无处极小的实解析子流形，它甚至不是$C^2$ Levi-平坦超曲面的边界.
Stationary holomorphic discs glued to a Levi non-degenerate hypersurface
我们得到了粘在某些Levi非退化超曲面上的固定圆盘的一个显式参数化。这些圆盘形成一个在双全纯作用下不变的族。我们使用这个参数化来构造这些超曲面的局部圆形表示。作为推论，我们得到了给定1-喷注的双全纯在某些方便点的唯一性。
Algebraic Levi-flat hypervarieties in complex projective space
研究了复射影空间中奇异实解析Levi平坦超曲面.我们定义了代数Levi平坦超曲面的秩，并研究了秩、度与奇点的类型和大小之间的关系。特别地，我们研究了代数Levi-平坦超曲面的退化奇点。然后给出了Levi平坦超曲面是$\mathbb{C}$中实解析曲线通过亚纯函数拉回的充要条件.在其他的例子中，我们构造了一个非代数半解析列维平坦超曲面与紧凑的叶子，是一个扰动的代数列维平坦品种。
In this article we provide a version of Chow's theorem for real analytic Levi-flat hypersurfaces in the complex projective space $\mathbb{P}^{n}$, $n \geq 2$. More specifically, we prove that a real analytic Levi-flat hypersurface $M \subset \mathbb{P}^{n}$, with singular set of real dimension at most $2n-4$ and whose Levi leaves are contained in algebraic hypersurfaces, is tangent to the levels of a rational function in $\mathbb{P}^{n}$. As a consequence, $M$ is a semialgebraic set. We also prove that a Levi foliation on $\mathbb{P}^{n}$ - a singular real analytic foliation whose leaves are i…
In our earlier articles we studied tube hypersurfaces in ${\mathbb C}^3$ that are 2-nondegenerate and uniformly Levi degenerate of rank 1. In particular, we showed that the vanishing of the CR-curvature of such a hypersurface is equivalent to the Monge equation with respect to one of the variables. In the present paper we provide an alternative shorter derivation of this equation by utilizing two invariants discovered by S. Pocchiola. We also investigate Pocchiola's invariants in the rigid case and give a partial classification of rigid 2-nondegenerate uniformly Levi degenerate of rank 1 hyper…
Linearization of transition functions along a certain class of Levi-flat hypersurfaces
本文研究通过悬挂构造的 Levi-平坦超曲面上转移函数的规范化问题，借助圆周动力学与线性化定理给出获得规范形的充分条件，形成 Arnol'd 定理的几何类比。
Holomorphically homogeneous CR real hypersurfaces $M^3 \subset \mathbb{C}^2$ were classified by Élie Cartan in 1932. In the next dimension, we complete the classification of simply-transitive Levi non-degenerate hypersurfaces $M^5 \subset \mathbb{C}^3$ using a novel Lie algebraic approach independent of any earlier classifications of abstract Lie algebras. Central to our approach is a new coordinate-free formula for the fundamental (complexified) quartic tensor. Our final result has a unique (Levi-indefinite) non-tubular model, for which we demonstrate geometric relations to planar equi-affine…
In this expository paper, we review a recent progress of the study of the Diederich--Fornaess index of complex domains with emphasis on the case of domains with Levi-flat boundary. It is exhibited that for any compact Levi-flat real hypersurface, the norm of its infinitesimal holonomy must exceed the curvature of its normal bundle at a point.
A complete normal form for everywhere Levi degenerate hypersurfaces in $\mathbb C^{3}$
本文为三维复空间中处处 2-非退化实超曲面构造了完备收敛正规形，发展了 Chern-Moser 同调方法，并给出了局部球性判据和模空间的显式描述。
We show the existence of a complex K3 surface $X$ which is not a Kummer surface and has a one-parameter family of Levi-flat hypersurfaces in which all the leaves are dense. We construct such $X$ by patching two open complex surfaces obtained as the complements of tubular neighborhoods of elliptic curves embedded in blow-ups of the projective planes at general nine points.
Microlocalization and nonexistence of $C^2$ Levi-flat hypersurfaces in $\Bbb CP^2
为谨慎起见，由于公式（2.5）中的错误（缺少复共轭的符号），作者撤回了该论文。我们非常感谢Marco Brunella指出了这个错误。
We investigate the construction of real analytic Levi-flat hypersurfaces in K3 surfaces. By taking images of real hyperplanes, one can construct such hypersurfaces in two-dimensional complex tori. We show that "almost every" K3 surfaces contains infinitely many Levi-flat hypersurfaces of this type. The proof relies mainly on a recent construction of Koike and Uehara, ideas of Verbitsky on ergodic complex structures, as well as an argument due to Ghys in the context of the study of the topology of generic leaves. -- On s'intéresse à la construction d'hypersurfaces Levi-plates analytiques rélles…
We study curvature restrictions of Levi-flat real hypersurfaces in complex projective planes, whose existence is in question. We focus on its totally real Ricci curvature, the Ricci curvature of the real hypersurface in the direction of the Reeb vector field, and show that it cannot be greater than -4 along a Levi-flat real hypersurface. We rely on a finiteness theorem for the space of square integrable holomorphic 2-forms on the complement of the Levi-flat real hypersurface, where the curvature plays the role of the size of the infinitesimal holonomy of its Levi foliation.
Nonexistence of smooth Levi-flat hypersurfaces in complex projective spaces of dimension >= 3
在本文中，我们证明了以下定理。主要定理设n ≥ 3，m ≥ 3 n/2 + 7。则P_n中不存在C^mLevi平坦的实超曲面M。M是Levi平坦的条件是：当M由C^m实值函数f的消失局部定义时，在M的每一点d d-杆f对M的复切空间的限制恒为零。P_2中C^ \infty Levi平坦实超曲面不存在的情形是由P_2中的动力系统中的问题所激发的。
Let $X$ be a connected complex manifold of dimension $\geq 3$ and $M$ a smooth compact Levi-flat real hypersurface in $X$. We show that the normal bundle to the Levi foliation does not admit a Hermitian metric with positive curvature along the leaves. This generalizes a result obtained by Brunella.
Normal forms and biholomorphic equivalence of real hypersurfaces in C^3
考虑了实解析超曲面$M$在特殊点$p_0\in M$的局部双全纯等价类的描述问题，给出了这类对象的规范形.为了使规范型携带关于双全纯等价类的有用信息，我们要求到规范型的变换对某个有限维群是唯一的。Chern-Moser的一个经典结果给出了Levi非退化超曲面的一个标准形.本文的主要结果涉及$\Bbb C^3$中的实解析超曲面$M$在某些Levi退化点$p_0\in M$，即$M$是2-非退化的点.我们给出了所有这类$(M,p_0)$的部分规范形，即与2-非退化相关的数据的规范形。在Levi型在$p_0$上秩为1的条件下，我们还给出了这种$(M,p_0)$的完全形式正规型.这个结果与作者M. S. Baouendi和L. P. Rothschild指出实解析非退化超曲面之间的形式等价收敛，给出了$\Bbb C^3$中实解析超曲面在Levi形式秩为1的2-非退化点处的双全纯等价类的描述.
J-holomorphic discs and real analytic hypersurfaces
在\mathbb{R}^6中，我们给出了一个实解析殆复结构J，一个实解析超曲面M和一个在M的Levi零集0处的向量v，使得M中不存在满足f（0）=0和\frac{\partial f}{\partial x}（0）=v的J-全纯圆盘f的芽，尽管M的Levi形式具有常秩.然后对任意超曲面M和任意复结构J，给出了存在这种圆盘芽的必要条件。
Nowhere minimal CR submanifolds and Levi-flat hypersurfaces
研究了高余维实解析无处极小CR子流形上全纯函数的局部唯一性。给出并研究了几乎极小性的一个充分条件。研究并刻划了一个较弱的必要条件，即包含一个可能奇异的实解析Levi平坦超曲面.对于余维为2的代数子流形，完全解决了这个问题，并给出了非代数子流形不包含的一个充分条件。作为结果，一个例子的子流形的余维2，而不是双全纯等价的代数之一，给出。我们还研究了Levi-平坦超曲面的奇点结构。
We continue our study, initiated in an earlier article, of a class of rigid hypersurfaces in ${\mathbb C}^3$ that are 2-nondegenerate and uniformly Levi degenerate of rank 1, having zero CR-curvature. We drop the restrictive assumptions of the earlier paper and give a complete description of the class. Surprisingly, the answer is expressed in terms of solutions of several well-known differential equations, in particular, the equation characterizing conformal metrics with constant negative curvature and a nonlinear $\bar\partial$-equation.
We prove that for every $n\geq 3$ the sharp upper bound for the dimension of the symmetry groups of homogeneous, 2-nondegenerate, $(2n+1)$-dimensional CR manifolds of hypersurface type with a $1$-dimensional Levi kernel is equal to $n^2+7$, and simultaneously establish the same result for a more general class of structures characterized by weakening the homogeneity condition. This supports Beloshapka's conjecture stating that hypersurface models with a maximal finite dimensional group of symmetries for a given dimension of the underlying manifold are Levi nondegenerate.
Yet another proof is given for a global estimate of the Diederich--Fornaess index of relatively compact domains with Levi-flat boundary, namely, the index must be smaller than or equal to the reciprocal of the dimension of the ambient space. This proof reveals that this kind of estimate makes sense and holds also for abstract compact Levi-flat CR manifolds.
We solve the Levi-flat Plateau problem in the following case. Let $M \subset {\mathbb C}^{n+1}$, $n \geq 2$, be a connected compact real-analytic codimension-two submanifold with only nondegenerate CR singularities. Suppose $M$ is a diffeomorphic image via a real-analytic CR map of a real-analytic hypersurface in ${\mathbb C}^n \times {\mathbb R}$ with only nondegenerate CR singularities. Then there exists a unique compact real-analytic Levi-flat hypersurface, nonsingular except possibly for self-intersections, with boundary $M$. We also study boundary regularity of CR automorphisms of domains…
We prove smooth and analytic versions of the classical Schwarz reflection principle for transversal CR mappings between two CR manifolds of hypersurface type.
Complete normal forms for real hypersurfaces in $\mathbb C^3$ at 2-nondegenerate points of Levi non-uniform rank zero
本文为 $\mathbb C^3$ 中在原点 2-非退化且 Levi 非一致秩为零的 $5$ 维实超曲面构造了完全正规型，利用等变移动标架理论解决了文献中缺失的正规型及双全纯等价问题。
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