scholarly journals Non-degenerate real hypersurfaces in complex manifolds admitting large groups of pseudo-conformal transformations. I

1976 ◽  
Vol 62 ◽  
pp. 55-96 ◽  
Author(s):  
Keizo Yamaguchi
Keyword(s):  
Complex Manifold ◽  
Real Hypersurface ◽  
Levi Form ◽  
Real Hypersurfaces ◽  
Conformal Transformations ◽  
Complex Manifolds ◽  
Real Analytic ◽  
Large Groups

Let S (resp. S′) be a (real) hypersurface (i.e. a real analytic sub-manifold of codimension 1) of an n-dimensional complex manifold M (resp. M′). A homeomorphism f of S onto S′ is called a pseudo-conformal homeomorphism if it can be extended to a holomorphic homeomorphism of a neighborhood of S in M onto a neighborhood of S′ in M. In case such an f exists, we say that S and S′ are pseudo-conformally equivalent. A hypersurface S is called non-degenerate (index r) if its Levi-form is non-degenerate (and its index is equal to r) at each point of S.

scholarly journals Non-degenerate real hypersurfaces in complex manifolds admitting large groups of pseudo-conformal transformations II

1978 ◽  
Vol 69 ◽  
pp. 9-31
Author(s):  
Keizo Yamaguchi
Keyword(s):  
Real Hypersurfaces ◽  
Conformal Transformations ◽  
Complex Manifolds ◽  
Homogeneity Assumption ◽  
Large Groups

This is the continuation of our previous paper [3], and will complete, without homogeneity assumption, the classification of non-degenerate real hypersurfaces S of complex manifolds M for which the groups A(S) of pseudo-conformal transformations of S have either the largest dimension n2 + 2n or the second largest dimension.

scholarly journals Convergent normal form for real hypersurfaces at a generic Levi-degeneracy

2019 ◽  
Vol 2019 (749) ◽  
pp. 201-225
Author(s):  
Ilya Kossovskiy ◽  
Dmitri Zaitsev
Keyword(s):  
Normal Form ◽  
Moduli Space ◽  
Real Hypersurface ◽  
Convergence Result ◽  
Explicit Description ◽  
Real Hypersurfaces ◽  
Real Analytic ◽  
Real Analytic Hypersurfaces

Abstract We construct a complete convergent normal form for a real hypersurface in {\mathbb{C}^{N}} , {N\geq 2} , at a generic Levi-degeneracy. This seems to be the first convergent normal form for a Levi-degenerate hypersurface. As an application of the convergence result, we obtain an explicit description of the moduli space of germs of real-analytic hypersurfaces with a generic Levi-degeneracy. As another application, we obtain, in the spirit of the work of Chern and Moser [6], distinguished curves inside the Levi-degeneracy set that we call degenerate chains.

scholarly journals Nonexistence of real analytic Levi flat hypersurfaces in ℙ2

2000 ◽  
Vol 158 ◽  
pp. 95-98 ◽  
Author(s):  
Takeo Ohsawa
Keyword(s):  
Complex Manifold ◽  
Real Hypersurface ◽  
Real Analytic

AbstractA real hypersurface M in a complex manifold X is said to be Levi flat if it separates X locally into two Stein pieces. It is proved that there exist no real analytic Levi flat hypersurfaces in ℙ2.

scholarly journals Complete system of finite order for the embeddings of pseudo-hermitian manifolds into ℂN+1

1999 ◽  
Vol 155 ◽  
pp. 189-205 ◽  
Author(s):  
Sung-Yeon Kim
Keyword(s):  
Finite Order ◽  
Real Hypersurface ◽  
Complete System ◽  
Levi Form ◽  
Hermitian Manifold ◽  
Hermitian Manifolds ◽  
Real Analytic ◽  
Cauchy Riemann

AbstractLet (M, ν, θ) be a real analytic (2n+1)-dimensional pseudo-hermitian manifold with nondegenerate Levi form and F be a pseudo-hermitian embedding into ℂn+1. We show under certain generic conditions that F satisfies a complete system of finite order. We use a method of prolongation of the tangential Cauchy-Riemann equations and pseudo-hermitian embedding equation. Thus if F ∈ Ck(M) for sufficiently large k, F is real analytic. As a corollary, if M is a real hypersurface in ℂn+1, then F extends holomorphically to a neighborhood of M provided that F is sufficiently smooth.

Holomorphic Immersions of Bi-Disks into 9D Real Hypersurfaces with Levi Signature (2,2)

2020 ◽  
Author(s):  
Wei Guo Foo ◽  
Joël Merker
Keyword(s):  
Fixed Point ◽  
Real Hypersurface ◽  
Real Hypersurfaces ◽  
Zero Set ◽  
Cr Manifolds ◽  
Real Analytic ◽  
Unit Disks ◽  
Complex Valued

Abstract Inspired by an article of R. Bryant on holomorphic immersions of unit disks into Lorentzian CR manifolds, we discuss the application of Cartan’s method to the question of the existence of bi-disk $\mathbb{D}^{2}$ in a smooth $9$D real-analytic real hypersurface $M^{9}\subset \mathbb{C}^{5}$ with Levi signature $(2,2)$ passing through a fixed point. The result is that the lift to $M^{9}\times U(2)$ of the image of the bi-disk in $M^{9}$ must lie in the zero set of two complex-valued functions in $M^{9}\times U(2)$. We then provide an example where one of the functions does not identically vanish, thus obstructing holomorphic immersions.

Generalized -Einstein Real Hypersurfaces in and

2020 ◽  
Vol 63 (4) ◽  
pp. 909-920
Author(s):  
Yaning Wang
Keyword(s):  
Scalar Curvature ◽  
Mean Curvature ◽  
Constant Coefficient ◽  
Real Hypersurface ◽  
Constant Mean Curvature ◽  
Constant Scalar Curvature ◽  
Real Hypersurfaces

AbstractIn this paper we obtain some new characterizations of pseudo-Einstein real hypersurfaces in $\mathbb{C}P^{2}$ and $\mathbb{C}H^{2}$. More precisely, we prove that a real hypersurface in $\mathbb{C}P^{2}$ or $\mathbb{C}H^{2}$ with constant mean curvature is generalized ${\mathcal{D}}$-Einstein with constant coefficient if and only if it is pseudo-Einstein. We prove that a real hypersurface in $\mathbb{C}P^{2}$ with constant scalar curvature is generalized ${\mathcal{D}}$-Einstein with constant coefficient if and only if it is pseudo-Einstein.

scholarly journals Tameness of Complex Dimension in a Real Analytic Set

2013 ◽  
Vol 65 (4) ◽  
pp. 721-739
Author(s):  
Janusz Adamus ◽  
Serge Randriambololona ◽  
Rasul Shafikov
Keyword(s):  
Positive Integer ◽  
Complex Manifold ◽  
Analytic Set ◽  
Complex Dimension ◽  
Real Analytic ◽  
Set Of Points

AbstractGiven a real analytic set X in a complex manifold and a positive integer d, denote by Ad the set of points p in X at which there exists a germ of a complex analytic set of dimension d contained in X. It is proved that Ad is a closed semianalytic subset of X.

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