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Analytic torsions on contact manifolds

Michel Rumin, Neil Seshadri (2012)

Annales de l’institut Fourier

We propose a definition for analytic torsion of the contact complex on contact manifolds. We show it coincides with Ray–Singer torsion on any 3 -dimensional CR Seifert manifold equipped with a unitary representation. In this particular case we compute it and relate it to dynamical properties of the Reeb flow. In fact the whole spectral torsion function we consider may be interpreted on CR Seifert manifolds as a purely dynamical function through Selberg-like trace formulae, that hold also in variable...

Automorphism groups of minimal real-analytic CR manifolds

Robert Juhlin, Bernhard Lamel (2013)

Journal of the European Mathematical Society

We show that the local automorphism group of a minimal real-analytic CR manifold M is a finite dimensional Lie group if (and only if) M is holomorphically nondegenerate by constructing a jet parametrization.

Canonical contact forms on spherical CR manifolds

Wei Wang (2003)

Journal of the European Mathematical Society

We construct the CR invariant canonical contact form can ( J ) on scalar positive spherical CR manifold ( M , J ) , which is the CR analogue of canonical metric on locally conformally flat manifold constructed by Habermann and Jost. We also construct another canonical contact form on the Kleinian manifold Ω ( Γ ) / Γ , where Γ is a convex cocompact subgroup of Aut C R S 2 n + 1 = P U ( n + 1 , 1 ) and Ω ( Γ ) is the discontinuity domain of Γ . This contact form can be used to prove that Ω ( Γ ) / Γ is scalar positive (respectively, scalar negative, or scalar vanishing) if and...

Complex vector fields and hypoelliptic partial differential operators

Andrea Altomani, C. Denson Hill, Mauro Nacinovich, Egmont Porten (2010)

Annales de l’institut Fourier

We prove a subelliptic estimate for systems of complex vector fields under some assumptions that generalize the essential pseudoconcavity for CR manifolds, that was first introduced by two of the authors, and the Hörmander’s bracket condition for real vector fields.Applications are given to prove the hypoellipticity of first order systems and second order partial differential operators.Finally we describe a class of compact homogeneous CR manifolds for which the distribution of ( 0 , 1 ) vector fields satisfies...

Free CR distributions

Gerd Schmalz, Jan Slovák (2012)

Open Mathematics

There are only some exceptional CR dimensions and codimensions such that the geometries enjoy a discrete classification of the pointwise types of the homogeneous models. The cases of CR dimensions n and codimensions n 2 are among the very few possibilities of the so-called parabolic geometries. Indeed, the homogeneous model turns out to be PSU(n+1,n)/P with a suitable parabolic subgroup P. We study the geometric properties of such real (2n+n 2)-dimensional submanifolds in n + n 2 for all n > 1. In...

Hörmander systems and harmonic morphisms

Elisabetta Barletta (2003)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Given a Hörmander system X = { X 1 , , X m } on a domain Ω 𝐑 n we show that any subelliptic harmonic morphism φ from Ω into a ν -dimensional riemannian manifold N is a (smooth) subelliptic harmonic map (in the sense of J. Jost & C-J. Xu, [9]). Also φ is a submersion provided that ν m and X has rank m . If Ω = 𝐇 n (the Heisenberg group) and X = 1 2 L α + L α ¯ , 1 2 i L α - L α ¯ , where L α ¯ = / z ¯ α - i z α / t is the Lewy operator, then a smooth map φ : Ω N is a subelliptic harmonic morphism if and only if φ π : ( C ( 𝐇 n ) , F θ 0 ) N is a harmonic morphism, where S 1 C ( 𝐇 n ) π 𝐇 n is the canonical circle bundle and F θ 0 is the Fefferman...

Lie group structures on groups of diffeomorphisms and applications to CR manifolds

M. Salah Baouendi, Linda Preiss Rothschild, Jörg Winkelmann, Dimitri Zaitsev (2004)

Annales de l’institut Fourier

We give general sufficient conditions to guarantee that a given subgroup of the group of diffeomorphisms of a smooth or real-analytic manifold has a compatible Lie group structure. These results, together with recent work concerning jet parametrization and complete systems for CR automorphisms, are then applied to determine when the global CR automorphism group of a CR manifold is a Lie group in an appropriate topology.

Minimal surfaces in pseudohermitian geometry

Jih-Hsin Cheng, Jenn-Fang Hwang, Andrea Malchiodi, Paul Yang (2005)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We consider surfaces immersed in three-dimensional pseudohermitian manifolds. We define the notion of (p-)mean curvature and of the associated (p-)minimal surfaces, extending some concepts previously given for the (flat) Heisenberg group. We interpret the p-mean curvature not only as the tangential sublaplacian of a defining function, but also as the curvature of a characteristic curve, and as a quantity in terms of calibration geometry. As a differential equation, the p-minimal surface equation...

On local CR-transformations of Levi-degenerate group orbits in compact Hermitian symmetric spaces

Wilhelm Kaup, Dmitri Zaitsev (2006)

Journal of the European Mathematical Society

We present a large class of homogeneous 2-nondegenerate CR-manifolds M , both of hypersurface type and of arbitrarily high CR-codimension, with the following property: Every CR-equivalence between domains U , V in M extends to a global real-analytic CR-automorphism of M . We show that this class contains G -orbits in Hermitian symmetric spaces Z of compact type, where G is a real form of the complex Lie group Aut ( Z ) 0 and G has an open orbit that is a bounded symmetric domain of tube type.

On some rigidity properties of mappings between CR-submanifolds in complex space

Francine Meylan, Nordine Mir, Dimitri Zaitsev (2003)

Journées équations aux dérivées partielles

We survey some recent results on holomorphic or formal mappings sending real submanifolds in complex space into each other. More specifically, the approximation and convergence properties of formal CR-mappings between real-analytic CR-submanifolds will be discussed, as well as the corresponding questions in the category of real-algebraic CR-submanifolds.

Prolongement d’un courant positif quasi-plurisurharmonique

Noureddine Ghiloufi, Khalifa Dabbek (2009)

Annales mathématiques Blaise Pascal

Le but de cet article est de montrer un résultat de prolongement d’un courant positif, défini en dehors d’un obstacle fermé, dont le d d c est dominé par un courant positif fermé de masse localement finie. On étudie divers types d’obstacles  : soit un ensemble fermé pluripolaire complet, soit l’ensemble des zéros d’une fonction strictement k -convexe positive. Dans la troisième partie, sous des conditions sur la dimension de Hausdorff de l’obstacle, on démontre le prolongement d’un tel courant. On termine...

Rigidity of CR morphisms between compact strongly pseudoconvex CR manifolds

Stephen S.-T. Yau (2011)

Journal of the European Mathematical Society

Let X 1 and X 2 be two compact strongly pseudoconvex CR manifolds of dimension 2 n - 1 5 which bound complex varieties V 1 and V 2 with only isolated normal singularities in N 1 and N 2 respectively. Let S 1 and S 2 be the singular sets of V 1 and V 2 respectively and S 2 is nonempty. If 2 n - N 2 - 1 1 and the cardinality of S 1 is less than 2 times the cardinality of S 2 , then we prove that any non-constant CR morphism from X 1 to X 2 is necessarily a CR biholomorphism. On the other hand, let X be a compact strongly pseudoconvex CR manifold of...

Semi-global solutions of ∂b with Lp (1 ≤ p ≤ ∞) bounds on strongly pseudoconvex real hypersurfaces in Cn (n ≥ 3).

C. H. Chang, H. P. Lee (1999)

Publicacions Matemàtiques

Let M be an open subset of a compact strongly pseudoconvex hypersurface {ρ = 0} defined by M = D × Cn-m ∩ {ρ = 0}, where 1 ≤ m ≤ n-2, D = {σ(z1, ..., zm) < 0} ⊂ Cm is strongly pseudoconvex in Cm. For ∂b closed (0, q) forms f on M, we prove the semi-global existence theorem for ∂b if 1 ≤ q ≤ n-m-2, or if q = n - m - 1 and f satisfies an additional “moment condition”. Most importantly, the solution operator satisfies Lp estimates for 1 ≤ p ≤ ∞ with p = 1 and ∞ included.

Tangential Cauchy-Riemann equations on quadratic manifolds

Marco M. Peloso, Fulvio Ricci (2002)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We study the tangential Cauchy-Riemann equations ¯ b u = ω for 0 , q -forms on quadratic C R manifolds. We discuss solvability for data ω in the Schwartz class and describe the range of the tangential Cauchy-Riemann operator in terms of the signatures of the scalar components of the Levi form.

The Poincaré lemma and local embeddability

Judith Brinkschulte, C. Denson Hill, Mauro Nacinovich (2003)

Bollettino dell'Unione Matematica Italiana

For pseudocomplex abstract C R manifolds, the validity of the Poincaré Lemma for 0 , 1 forms implies local embeddability in C N . The two properties are equivalent for hypersurfaces of real dimension 5 . As a corollary we obtain a criterion for the non validity of the Poicaré Lemma for 0 , 1 forms for a large class of abstract C R manifolds of C R codimension larger than one.

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