scholarly journals On Holomorphic Curves Tangent to Real Hypersurfaces of Infinite Type

2020 ◽  
Author(s):  
Joe Kamimoto
Keyword(s):  
Infinite Order ◽  
Singularity Theory ◽  
Holomorphic Curves ◽  
Real Hypersurfaces ◽  
Sufficient Condition ◽  
Holomorphic Curve ◽  
Geometric Properties ◽  
Important Concept ◽  
Infinite Type ◽  
Newton Polyhedra

AbstractThe purpose of this paper is to investigate the geometric properties of real hypersurfaces of D’Angelo infinite type in $${{\mathbb {C}}}^n$$ C n . In order to understand the situation of flatness of these hypersurfaces, it is natural to ask whether there exists a nonconstant holomorphic curve tangent to a given hypersurface to infinite order. A sufficient condition for this existence is given by using Newton polyhedra, which is an important concept in singularity theory. More precisely, equivalence conditions are given in the case of some model hypersurfaces.

scholarly journals A Note on Infinite Type Germs of a Real Hypersurface in

2019 ◽  
Vol 35 (2) ◽  
Author(s):  
Nguyen Thi Kim Son ◽  
Chu Van Tiep
Keyword(s):  
Vector Field ◽  
Automorphism Group ◽  
Real Hypersurface ◽  
Infinite Order ◽  
Holomorphic Vector Field ◽  
Key Words And Phrases ◽  
Holomorphic Curve ◽  
Infinite Type ◽  
Field Automorphism ◽  
Order Contact

 Abstract: The purpose of this article is to show that there exists a smooth real hypersurface germ  of D'Angelo infinite type in  such that it does not admit any (singular) holomorphic curve that has infinite order contact with  at . 2010 Mathematics Subject Classification. Primary 32T25; Secondary 32C25. Key words and phrases:  Holomorphic vector field, automorphism group, real hypersurface, infinite type point.  

scholarly journals Singularities of lightcone pedals of spacelike curves in Lorentz-Minkowski 3-space

2016 ◽  
Vol 14 (1) ◽  
pp. 889-896 ◽  
Author(s):  
Liang Chen
Keyword(s):  
Singularity Theory ◽  
Smooth Functions ◽  
Geometric Properties ◽  
Timelike Surface

AbstractIn this paper, geometric properties of spacelike curves on a timelike surface in Lorentz-Minkowski 3-space are investigated by applying the singularity theory of smooth functions from the contact viewpoint.

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 Compactness criteria for real algebraic sets and Newton polyhedra

2018 ◽  
Vol 30 (6) ◽  
pp. 1387-1395
Author(s):  
Phu Phat Pham ◽  
Tien Son Pham
Keyword(s):  
Newton Polyhedron ◽  
Sufficient Condition ◽  
Zero Set ◽  
Algebraic Sets ◽  
Newton Polyhedra ◽  
Necessary And Sufficient Criteria ◽  
Necessary And Sufficient

Abstract Let {f\colon\mathbb{R}^{n}\rightarrow\mathbb{R}} be a polynomial and {\mathcal{Z}(f)} its zero set. In this paper, in terms of the so-called Newton polyhedron of f, we present a necessary criterion and a sufficient condition for the compactness of {\mathcal{Z}(f)} . From this we derive necessary and sufficient criteria for the stable compactness of {\mathcal{Z}(f)} .

scholarly journals Applications of Mittag-Leffer Type Poisson Distribution to a Subclass of Analytic Functions Involving Conic-Type Regions

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Muhammad Ghaffar Khan ◽  
Bakhtiar Ahmad ◽  
Nazar Khan ◽  
Wali Khan Mashwani ◽  
Sama Arjika ◽  
...  
Keyword(s):  
Poisson Distribution ◽  
Analytic Functions ◽  
Convex Combination ◽  
Partial Sums ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Geometric Properties ◽  
Distortion Bounds ◽  
Janowski Functions ◽  
Necessary And Sufficient

In this article, we introduce a new subclass of analytic functions utilizing the idea of Mittag-Leffler type Poisson distribution associated with the Janowski functions. Further, we discuss some important geometric properties like necessary and sufficient condition, convex combination, growth and distortion bounds, Fekete-Szegö inequality, and partial sums for this newly defined class.

Hyperbolic worldsheets and worldlines of null Cartan curves in de Sitter 3-space

2021 ◽  
pp. 2150026
Author(s):  
Qingxin Zhou ◽  
Jingbo Xu ◽  
Zhigang Wang
Keyword(s):  
Singularity Theory ◽  
Close Relation ◽  
Dual Relationships ◽  
Normal Curve ◽  
De Sitter ◽  
Geometric Invariant ◽  
Geometric Properties ◽  
Hyperbolic Quadric ◽  
Cuspidal Edge

The hyperbolic worldsheets and the hyperbolic worldline generated by null Cartan curves are defined and their geometric properties are investigated. As applications of singularity theory, the singularities of the hyperbolic worldsheets and the hyperbolic worldline are classified by using the approach of the unfolding theory in singularity theory. It is shown that under appropriate conditions, the hyperbolic worldsheet is diffeomorphic to cuspidal edge or swallowtail type of singularity and the hyperbolic worldline is diffeomorphic to cusp. An important geometric invariant which has a close relation with the singularities of the hyperbolic worldsheets and worldlines is found such that the singularities of the hyperbolic worldsheets and worldlines can be characterized by the invariant. Meanwhile, the contact of the spacelike normal curve of a null Cartan curve with hyperbolic quadric or world hypersheet is discussed in detail. In addition, the dual relationships between the spacelike normal curve of a null Cartan curve and the hyperbolic worldsheet are described. Moreover, it is demonstrated that the spacelike normal curve of a null Cartan curve and the hyperbolic worldsheet are [Formula: see text]-dual each other.

H1 boundedness of oscillatory singular integrals with degenerate phase functions

1994 ◽  
Vol 116 (2) ◽  
pp. 353-358
Author(s):  
Yibiao Pan
Keyword(s):  
Hardy Space ◽  
Singular Integral ◽  
Phase Function ◽  
Infinite Order ◽  
Singular Integrals ◽  
Integral Operators ◽  
Uniform Boundedness ◽  
Sufficient Condition ◽  
Oscillatory Singular Integrals ◽  
Phase Functions

AbstractIn this paper we study the uniform boundedness of oscillatory singular integral operators with degenerate phase functions on the Hardy space H1. The H1 boundedness was previously known when the phase function is nondegenerate. Here we obtain a sufficient condition for H1 boundedness which allows the phase function vanishing to infinite order.

scholarly journals A Lehto–Virtanen-type theorem and a rescaling principle for an isolated essential singularity of a holomorphic curve in a complex space

2015 ◽  
Vol 26 (06) ◽  
pp. 1541009
Author(s):  
Yûsuke Okuyama
Keyword(s):  
Complex Space ◽  
Type Theorem ◽  
Essential Singularity ◽  
Holomorphic Curves ◽  
Holomorphic Curve ◽  
Picard Type Theorem

We establish a Lehto–Virtanen-type theorem and a rescaling principle for an isolated essential singularity of a holomorphic curve in a complex space, which are useful for establishing a big Picard-type theorem and a big Brody-type one for holomorphic curves.

scholarly journals Propagation sets of holomorphic curves

2020 ◽  
pp. 2050126
Author(s):  
Jianhua Zheng ◽  
Qiming Yan
Keyword(s):  
Finite Order ◽  
Complex Plane ◽  
Linear Relation ◽  
Infinite Order ◽  
Holomorphic Curves

We consider a problem of whether a property of holomorphic curves on a subset [Formula: see text] of the complex plane can be extended to the whole complex plane. In this paper, the property we consider is the uniqueness of holomorphic curves. We introduce the propagation set. Simply speaking, [Formula: see text] is a propagation set if linear relation of holomorphic curves on the part of preimage of hyperplanes contained in [Formula: see text] can be extended to the whole complex plane. If the holomorphic curves are of infinite order, we prove the existence of a propagation set which is the union of a sequence of disks. (In fact, the method applies to the case of finite order.) For a general case, the union of a sequence of annuli will be a propagation set. The classic five-value theorem and four-value theorem of Nevanlinna are established in such propagation sets.

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