On an $n$-manifold in $\mathbf{C}^n$ near an elliptic complex tangent

  title={On an \$n\$-manifold in \$\mathbf\{C\}^n\$ near an elliptic complex tangent},
  author={Xiaojun Huang},
  journal={Journal of the American Mathematical Society},
  • Xiaojun Huang
  • Published 1 July 1998
  • Mathematics
  • Journal of the American Mathematical Society
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 complex tangent, however, the consideration can be much more complicated and the manifold may acquire a nontrivial local hull of holomorphy and many other biholomorphic invariants. The study of such a problem was first carried out in a celebrated paper of E. Bishop [BIS] where, for each sufficiently… 
23 Citations
Asymptotics in a Riemann-Hilbert boundary problem for pseudoholomorphic curves
where we x a volume element on to achieve the last isomorphism. The section (3) vanishes precisely at the complex points 2 . Such a point is called elliptic if @F ^ @F is transversal to the zero
Real submanifolds of maximum complex tangent space at a CR singular point, I
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
Flattening of CR singular points and analyticity of the local hull of holomorphy I
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
Equivalence problem for Bishop surfaces
AbstractThe paper has two parts. We first briefly survey recent studies on the equivalence problem for real submanifolds in a complex space under the action of biholomorphic transformations. We will
CR singular images of generic submanifolds under holomorphic maps
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
Flattening a non-degenerate CR singular point of real codimension two
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
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
Holomorphic extension of CR functions, envelopes of holomorphy and removable singularities
This is an extensive (published) survey on CR geometry, whose major themes are: formal analytic reflection principle; generic properties of Systems of (CR) vector fields; pairs of foliations and
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 real surfaces in C2 near complex tangents and hyperbolic surface transformations
It is well known that the complex analytical properties of a real submanifold M in the complex space C n are most accessible through consideration of the complex tangents to M. The properties we have
On a problem of Moser
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
Geometric analysis in several complex variables
In this thesis, we answer several questions in the function theory of several complex variables which possess local geometric features but which need global non-linear analysis method for the
Complex tangents of real surfaces in complex surfaces
Introduction. In this paper we study the complex tangents of real surfaces in complex surfaces. More precisely, let M be a closed real surface, i.e., a smooth, compact, two-dimensional manifold
Analytic surfaces in C^2 and their local hull of holomorphy
a) We consider real analylic surfaces M (i.e. dim*M:2) in C2. Two such surfaces M, ft are called equivalent if they can be mapped into each olher by a mapping which is biholomorphic in the complex
Geometry of Low-dimensional Manifolds: Filling by holomorphic discs and its applications
The survey is devoted to application of the technique of filling by holomorphic discs to different symplectic and complex analytic problems. COMPLEX AND SYMPLECTIC RECOLLECTIONS J -Convexity Let X, J
Pseudo holomorphic curves in symplectic manifolds
Definitions. A parametrized (pseudo holomorphic) J-curve in an almost complex manifold (IS, J) is a holomorphic map of a Riemann surface into Is, say f : (S, J3 ~(V, J). The image C=f(S)C V is called
CR Manifolds and the Tangential Cauchy-Riemann Complex
PRELIMINARIES. Differential Forms and Stokes Theorem. Distributions and Curents. Fundamental Solutions for(these symbols are to be used in an equation -----? ? z ---- see note at bottom)and D. Edge
It is shown that, if a generating manifold does not contain proper submanifolds of the same CR dimension as , then all CR functions can be extended from into some wedge with edge . In particular,
Nonlinear functional analysis
This graduate-level text offers a survey of the main ideas, concepts, and methods that constitute nonlinear functional analysis. It features extensive commentary, many examples, and interesting,