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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...

Cartan-Chern-Moser theory on algebraic hypersurfaces and an application to the study of automorphism groups of algebraic domains

Xiaojun Huang, Shanyu Ji (2002)

Annales de l’institut Fourier

For a strongly pseudoconvex domain D ⊂ ℂ n + 1 defined by a real polynomial of degree k 0 , we prove that the Lie group Aut ( D ) can be identified with a constructible Nash algebraic smooth variety in the CR structure bundle Y of ∂ D , and that the sum of its Betti numbers is bounded by a certain constant C n , k 0 depending only on n and k 0 . In case D is simply connected, we further give an explicit but quite rough bound in terms of the dimension and the degree of the defining polynomial. Our approach is to adapt the Cartan-Chern-Moser...

Completeness of the Bergman metric on non-smooth pseudoconvex domains

Bo-Yong Chen (1999)

Annales Polonici Mathematici

We prove that the Bergman metric on domains satisfying condition S is complete. This implies that any bounded pseudoconvex domain with Lipschitz boundary is complete with respect to the Bergman metric. We also show that bounded hyperconvex domains in the plane and convex domains in ℂ n are Bergman comlete.

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...

Contact geometry and CR-structures on spheres

John Bland, Tom Duchamp (1995)

Banach Center Publications

A normal form for small CR-deformations of the standard CR-structure on the (2n+1)-sphere is presented. The space of normal forms is parameterized by a single function on the sphere. For n>1, the normal form is used to obtain explicit embeddings into ℂ n + 1 . For n=1, the cohomological obstruction to embeddability is identified.

CR submanifolds of maximal CR dimension in complex manifolds

Mirjana Djorić, Masafumi Okumura (2002)

Banach Center Publications

The aim of this paper is to investigate n-dimensional real submanifolds of complex manifolds in the case when the maximal holomorphic tangent space is (n-1)-dimensional. In particular, we give some examples and we consider the Levi form on these submanifolds, especially when the ambient space is a complex space form. Moreover, we show that on some remarkable class of real hypersurfaces of complex space forms, the Levi form cannot vanish identically.

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