Holomorphic Deformations of Real-Analytic CR Maps and Analytic Regularity of CR Mappings

2016 ◽  
Vol 27 (3) ◽  
pp. 1920-1939 ◽  
Author(s):  
Nordine Mir
Keyword(s):  
Cr Mappings ◽  
Real Analytic ◽  
Analytic Regularity

scholarly journals Convergence of formal invertible CR mappings between minimal holomorphically nondegenerate real analytic hypersurfaces

2001 ◽  
Vol 26 (5) ◽  
pp. 281-302 ◽  
Author(s):  
Joël Merker
Keyword(s):  
Forthcoming Paper ◽  
Holomorphic Mappings ◽  
Optimal Regularity ◽  
Cr Manifolds ◽  
Cr Mappings ◽  
Algebraic Sets ◽  
Riemann Geometry ◽  
Recent Advances ◽  
Real Analytic ◽  
Real Analytic Hypersurfaces

Recent advances in CR (Cauchy-Riemann) geometry have raised interesting fine questions about the regularity of CR mappings between real analytic hypersurfaces. In analogy with the known optimal results about the algebraicity of holomorphic mappings between real algebraic sets, some statements about the optimal regularity of formal CR mappings between real analytic CR manifolds can be naturally conjectured. Concentrating on the hypersurface case, we show in this paper that a formal invertible CR mapping between two minimal holomorphically nondegenerate real analytic hypersurfaces inℂnis convergent. The necessity of holomorphic nondegeneracy was known previously. Our technique is an adaptation of the inductional study of the jets of formal CR maps which was discovered by Baouendi-Ebenfelt-Rothschild. However, as the manifolds we consider are far from being finitely nondegenerate, we must consider some newconjugate reflection identitieswhich appear to be crucial in the proof. The higher codimensional case will be studied in a forthcoming paper.

ANALYTIC HYPOELLIPTICITY FOR SUMS OF SQUARES IN THE PRESENCE OF SYMPLECTIC NON TREVES STRATA

2019 ◽  
Vol 19 (6) ◽  
pp. 1877-1888 ◽  
Author(s):  
Antonio Bove ◽  
Marco Mughetti
Keyword(s):  
Critical Points ◽  
Sums Of Squares ◽  
Characteristic Variety ◽  
The Real ◽  
Real Analytic ◽  
Funct Anal ◽  
Analytic Regularity ◽  
Analytic Hypoellipticity

In Albano, Bove and Mughetti [J. Funct. Anal. 274(10) (2018), 2725–2753]; Bove and Mughetti [Anal. PDE 10(7) (2017), 1613–1635] it was shown that Treves conjecture for the real analytic hypoellipticity of sums of squares operators does not hold. Models were proposed where the critical points causing a non-analytic regularity might be interpreted as strata. We stress that up to now there is no notion of stratum which could replace the original Treves stratum. In the proposed models such ‘strata’ were non-symplectic analytic submanifolds of the characteristic variety. In this note we modify one of those models in such a way that the critical points are a symplectic submanifold of the characteristic variety while still not being a Treves stratum. We show that the operator is analytic hypoelliptic.

scholarly journals Analytic regularity of CR-mappings

2002 ◽  
Vol 9 (1) ◽  
pp. 73-93 ◽  
Author(s):  
Francine Meylan ◽  
Nordine Mir ◽  
Dmitri Zaitsev
Keyword(s):  
Cr Mappings ◽  
Analytic Regularity

scholarly journals On the classification theorem for CR mappings

1998 ◽  
Vol 150 ◽  
pp. 95-104
Author(s):  
Atsushi Hayashimoto
Keyword(s):  
Classification Theorem ◽  
Type Condition ◽  
Cr Mappings ◽  
Real Analytic ◽  
The One

Abstract.Let f: M → M′ be a real analytic CR mapping between hyper-surfaces with f(p) = q, where p ∈ M and q ∈ M′. In this paper, the relation between the type at p and the one at q is considered. As a corollary of the type condition theorem (Theorem 1.1), a classification theorem, which states that under certain type condition, any real analytic CR mapping as above is constant, is proved.

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