A Real-analytic Nonpolynomially Convex Isotropic Torus with no Attached Discs

2018 ◽  
Vol 61 (2) ◽  
pp. 289-291
Author(s):  
Purvi Gupta
Keyword(s):  
Polynomial Convexity ◽  
Analytic Case ◽  
Real Analytic ◽  
Holomorphic Discs ◽  
Isotropic Torus

AbstractWe showbymeans of an example in that Gromov’s theoremon the presence of attached holomorphic discs for compact Lagrangianmanifolds is not true in the subcritical real-analytic case, even in the absence of an obvious obstruction, i.e., polynomial convexity.

scholarly journals A Question of Norton–Sullivan in the Analytic Case

2020 ◽  
Author(s):  
Jian Wang ◽  
Hui Yang
Keyword(s):  
Negative Answer ◽  
Smoothness Condition ◽  
Continuous Map ◽  
Geometric Conditions ◽  
Analytic Case ◽  
Real Analytic

Abstract In 1996, A. Norton and D. Sullivan asked the following question: If $f:\mathbb{T}^2\rightarrow \mathbb{T}^2$ is a diffeomorphism, $h:\mathbb{T}^2\rightarrow \mathbb{T}^2$ is a continuous map homotopic to the identity, and $h f=T_{\rho } h$, where $\rho \in \mathbb{R}^2$ is a totally irrational vector and $T_{\rho }:\mathbb{T}^2\rightarrow \mathbb{T}^2,\, z\mapsto z+\rho $ is a translation, are there natural geometric conditions (e.g., smoothness) on $f$ that force $h$ to be a homeomorphism? In [ 22], the 1st author and Z. Zhang gave a negative answer to the above question in the $C^{\infty }$ category: in general, not even the infinite smoothness condition can force $h$ to be a homeomorphism. In this article, we give a negative answer in the $C^{\omega }$ category (see also [ 22, Question 3]): we construct a real analytic conservative and minimal totally irrational pseudo-rotation of $\mathbb{T}^2$ that is semi-conjugate to a translation but not conjugate to a translation.

Sheets of Real Analytic Varieties

1960 ◽  
Vol 12 ◽  
pp. 51-67
Author(s):  
Andrew H. Wallace
Keyword(s):  
General Idea ◽  
Algebraic Varieties ◽  
Analytic Variety ◽  
Analytic Case ◽  
Real Algebraic Varieties ◽  
Analytic Varieties ◽  
Local Properties ◽  
Real Analytic ◽  
Real Analytic Variety

In a previous paper (4) the author worked out some results on the analytic connectivity properties of real algebraic varieties, that is to say, properties associated with the joining of points of the variety by analytic arcs lying on the variety. It is natural to ask whether these properties can be carried over to analytic varieties, since the proofs in the algebraic case depend mainly on local properties. But although this generalization can be carried out to a large extent, there are, nevertheless, difficulties in the analytic case, owing mainly to the fact (cf. 2, § 11) that a real analytic variety may not be definable by means of a set of global equations. Thus, although the general idea of the treatment given here is the same as in (4), some variation in the details of the method has proved to be necessary, and some of the final results are slightly weaker in form.

scholarly journals From a differentiable to a real analytic perturbation theory, applications to the Kupka Smale theorems

1986 ◽  
Vol 6 (3) ◽  
pp. 345-362 ◽  
Author(s):  
H. W. Broer ◽  
F. M. Tangerman
Keyword(s):  
Perturbation Theory ◽  
Normal Form ◽  
Heat Operator ◽  
Analytic Perturbation Theory ◽  
The Real ◽  
Smooth Case ◽  
Analytic Perturbation ◽  
Topological Proof ◽  
Analytic Case ◽  
Real Analytic

AbstractKupka-Smale like theorems are proven in the real analytic case, using existing perturbation schemes for the smooth case and the heat operator. As a consequence, a topological proof is obtained of Siegel's theorem on the generic divergence of normal form transformations.

scholarly journals J-holomorphic discs and real analytic hypersurfaces

2014 ◽  
Vol 64 (5) ◽  
pp. 2223-2250
Author(s):  
William Alexandre ◽  
Emmanuel Mazzilli
Keyword(s):  
Real Analytic ◽  
Real Analytic Hypersurfaces ◽  
Holomorphic Discs

scholarly journals Errata Corrige. Decomposition of CR-manifolds and splitting of CR-maps

2009 ◽  
pp. 233-233
Author(s):  
Atsushi Hayashimoto ◽  
Sung-Yeon Kim ◽  
Dmitri Zaitsev
Keyword(s):  
Finite Type ◽  
Dense Subset ◽  
Direct Products ◽  
Cr Manifolds ◽  
Splitting Property ◽  
The Real ◽  
Cancellation Problem ◽  
Analytic Case ◽  
Real Analytic ◽  
Infinitesimal Automorphisms

We show the uniqueness of local and global decompositions of abstract CR-manifolds into direct products of irreducible factors, and a splitting property for their CR-diffeomorphisms into direct products with respect to these decompositions. The assumptions on the manifolds are finite non-degeneracy and finite-type on a dense subset. In the real-analytic case, these are the standard assumptions that appear in many other questions. In the smooth case, the assumptions cannot be weakened by replacing “dense” with “open” as is demonstrated by an example. An application to the cancellation problem is also given. The proof is based on the development of methods of [BER99b], [BRZ00], [KZ01] and the use of “approximate infinitesimal automorphisms” introduced in this paper.

scholarly journals On the Local Theory of Continuous Infinite Pseudo Groups II

1961 ◽  
Vol 19 ◽  
pp. 55-91 ◽  
Author(s):  
Masatake Kuranishi
Keyword(s):  
Local Theory ◽  
Differential Systems ◽  
Exterior Differential Systems ◽  
The Real ◽  
Exterior Differential ◽  
Detailed Proof ◽  
Analytic Case ◽  
Real Analytic

In this chapter, we shall formulate, without proof, the theorys of exterior differential systems and of their prolongations using the language in the theory of jets developed by C. Ehresman, because such formulation seems to be most convenient in order to apply the theory to the theory of continuous infinite pseudo-groups, which we shall discuss in the next chapter. Since important theorems in our theory hold only in the real analytic case, we shall exclusively consider the real analytic case. So, we shall omit the adjective “analytic”, unless explicitly stated otherwise. As for the fundamental notions in the theory of jets and of differential systems, we refer to [5] and [6], respectively. Detailed proof of the contents of this Chapter will be published in [7].

scholarly journals On local hulls of Levi-flat hypersurfaces

2021 ◽  
pp. 2150050
Author(s):  
Rasul Shafikov ◽  
Alexandre Sukhov
Keyword(s):  
Singular Points ◽  
Polynomial Convexity ◽  
Local Polynomial ◽  
Real Analytic

We discuss local polynomial convexity of real analytic Levi-flat hypersurfaces in [Formula: see text], [Formula: see text], near singular points.

The Moments of Real Analytic Funtions

2020 ◽  
pp. 112-118 ◽  
Author(s):  
Ricardo Estrada
Keyword(s):  
Real Analytic

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