Flattening a non-degenerate CR singular point of real codimension two

@article{Fang2017FlatteningAN,
title={Flattening a non-degenerate CR singular point of real codimension two},
author={Hanlong Fang and Xiaojun Huang},
journal={Geometric and Functional Analysis},
year={2017},
volume={28},
pages={289-333}
}
• Published 27 March 2017
• Mathematics
• Geometric and Functional Analysis
This paper continues the previous studies in two papers of Huang–Yin [HY16,HY17] on the flattening problem of a CR singular point of real codimension two sitting in a submanifold in $${{\mathbb {C}}^{n+1}}$$Cn+1 with n + 1 ≥ 3, whose CR points are non-minimal. Partially based on the geometric approach initiated in [HY16] and a formal theory approach used in [HY17], we are able to provide more or less a complete solution to the flattening problem for a non-degenerate CR singular point along the…
11 Citations
A CR singular analogue of Severi’s theorem
• Mathematics
• 2019
Real-analytic CR functions on real-analytic CR singular submanifolds are not in general restrictions of holomorphic functions, unlike in the CR nonsingular case. We give a simple condition that
On the Levi-flat Plateau problem
• Mathematics
Complex Analysis and its Synergies
• 2020
We solve the Levi-flat Plateau problem in the following case. Let $$M \subset {\mathbb {C}}^{n+1}$$ M ⊂ C n + 1 , $$n \ge 2$$ n ≥ 2 , be a connected compact real-analytic codimension-two submanifold
On normal forms of complex points of small $\mathcal{C}^{2}$-perturbations of real $4$-manifolds embedded in a complex $3$-manifold
This paper is a sequel to our previous paper [24]. We extend the result on the behavior of the quadratic part of normal forms up to the quadratic part of small C2-perturbations of real 4-manifolds
On normal forms of complex points of small $\mathcal{C}^2$-perturbations of real $4$-manifolds embedded in a complex $3$-manifold -- II
This paper is a sequel to our previous paper \cite{TS2}. We extend the result on the behavior of the quadratic part of normal forms up to the quadratic part of small $\mathcal{C}^{2}$-perturbations
On CR singular CR images
• Mathematics
International Journal of Mathematics
• 2021
We say that a CR singular submanifold [Formula: see text] has a removable CR singularity if the CR structure at the CR points of [Formula: see text] extends through the singularity as an abstract CR
Stability of the hull(s) of an $n$-sphere in $\mathbb{C}^n$
• Mathematics
• 2020
We study the (global) Bishop problem for small perturbations of $\mathbf{S}^n$ --- the unit sphere of $\mathbb{C}\times\mathbb{R}^{n-1}$ --- in $\mathbb{C}^n$. We show that if $S\subset\mathbb{C}^n$
On Lewy extension for smooth hypersurfaces in ℂⁿ×ℝ
• Computer Science
Transactions of the American Mathematical Society
• 2018
An analogue of the Lewy extension theorem for a real dimension is proved and the theorem implies that if if if, then n times double-struck upper R.
Stability of the hull(s) of an n-sphere in Cn
• Mathematics
• 2021

References

SHOWING 1-10 OF 39 REFERENCES
Flattening of CR singular points and analyticity of the local hull of holomorphy I
• Mathematics
• 2012
This is the first article of the two papers in which we investigate the holomorphic and formal flattening problem for a codimension two real submanifold in $${\mathbb C}^n$$Cn with $$n\ge 3$$n≥3 near
Real submanifolds of maximum complex tangent space at a CR singular point, I
• Mathematics
• 2016
We study a germ of real analytic n-dimensional submanifold of $${\mathbf {C}}^n$$Cn that has a complex tangent space of maximal dimension at a CR singularity. Under some assumptions, we show its
Geometric and analytic problems for a real submanifold in ℂn with CR singularities
In this survey article, we present some known results and also propose some open questions related to the analytic and geometric aspects of Bishop submanifolds in a complex space. We mainly focus on
NORMAL FORMS FOR CR SINGULAR CODIMENSION TWO LEVI-FLAT SUBMANIFOLDS
• Mathematics
• 2015
Real-analytic Levi-flat codimension two CR singular submanifolds are a natural generalization to C m , m > 2, of Bishop surfaces in C 2 . Such submanifolds for example arise as zero sets of
On an $n$-manifold in $\mathbf{C}^n$ near an elliptic complex tangent
In this paper, we will be concerned with the local biholomorphic properties of a real n-manifold M in C. At a generic point, such a manifold basically has the nature of the standard R in C. Near a
Boundary problem for Levi flat graphs
• Mathematics
• 2009
In an earlier paper the authors provided general conditions on a real codimension 2 submanifold $S\subset C^{n}$, $n\ge 3$, such that there exists a possibly singular Levi-flat hypersurface $M$
CR singular images of generic submanifolds under holomorphic maps
• Mathematics
• 2014
The purpose of this paper is to organize some results on the local geometry of CR singular real-analytic manifolds that are images of CR manifolds via a CR map that is a diffeomorphism onto its
On a problem of Moser
• Mathematics
• 1995
1 0 Introduction This paper studies the analytic structure of the local hull of holomorphy of a 2-dimensional, real analytic manifold that is embedded in C 2. Our specific purpose is to solve a