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.

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 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 On holomorphic extension of functions on singular real hypersurfaces inℂn

2001 ◽  
Vol 26 (3) ◽  
pp. 173-178
Author(s):  
Tejinder S. Neelon
Keyword(s):  
Limit Point ◽  
Infinite Order ◽  
Holomorphic Extension ◽  
Real Hypersurfaces ◽  
Zero Set ◽  
Riemann Equation ◽  
Real Analytic ◽  
Cauchy Riemann ◽  
Extension Of Functions ◽  
Tangential Cauchy Riemann Equation

The holomorphic extension of functions defined on a class of real hypersurfaces inℂnwith singularities is investigated. Whenn=2, we prove the following: everyC1function onΣthat satisfies the tangential Cauchy-Riemann equation on boundary of{(z,w)∈ℂ2:|z|k<P(w)},P∈C1,P≥0andP≢0, extends holomorphically inside provided the zero setP(w)=0has a limit point orP(w)vanishes to infinite order. Furthermore, ifPis real analytic then the condition is also necessary.

scholarly journals Some Common Fixed-Point Theorems for Generalized-Contractive-Type Mappings on Complex-Valued Metric Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Chakkrid Klin-eam ◽  
Cholatis Suanoom
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Contractive Type ◽  
Complex Valued ◽  
Common Fixed Point Theorems

Fixed-point theory in complex valued metric spaces has greatly developed in recent times. In this paper, we prove certain common fixed-point theorems for two single-valued mappings in such spaces. The mappings we consider here are assumed to satisfy certain metric inequalities with generalized fixed-point theorems due to Rouzkard and Imdad (2012). This extends and subsumes many results of other authors which were obtained for mappings on complex-valued metric spaces.

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 On Complex-Valued Triple Controlled Metric Spaces and Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Nabil Mlaiki ◽  
Thabet Abdeljawad ◽  
Wasfi Shatanawi ◽  
Hassen Aydi ◽  
Yaé Ulrich Gaba
Keyword(s):  
Fixed Point ◽  
Integral Equations ◽  
Metric Spaces ◽  
Polynomial Equations ◽  
Degree Polynomial ◽  
Fredholm Type ◽  
Type Integral ◽  
Type Spaces ◽  
Complex Valued

In this manuscript, we introduce the concept of complex-valued triple controlled metric spaces as an extension of rectangular metric type spaces. To validate our hypotheses and to show the usability of the Banach and Kannan fixed point results discussed herein, we present an application on Fredholm-type integral equations and an application on higher degree polynomial equations.

scholarly journals Fejér monotonicity and fixed point theorems with applications to a nonlinear integral equation in complex valued Banach spaces

2020 ◽  
Vol 21 (1) ◽  
pp. 135
Author(s):  
Godwin Amechi Okeke ◽  
Mujahid Abbas
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Banach Spaces ◽  
Nonlinear Integral Equation ◽  
Metric Spaces ◽  
Banach Algebras ◽  
Cone Metric Spaces ◽  
Nonlinear Operators ◽  
Complex Valued ◽  
Cone Metric

It is our purpose in this paper to prove some fixed point results and Fej´er monotonicity of some faster fixed point iterative sequences generated by some nonlinear operators satisfying rational inequality in complex valued Banach spaces. We prove that results in complex valued Banach spaces are valid in cone metric spaces with Banach algebras. Furthermore, we apply our results in solving certain mixed type VolterraFredholm functional nonlinear integral equation in complex valued Banach spaces.

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