We study the tangential Cauchy-Riemann equations for -forms on quadratic manifolds. We discuss solvability for data 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.
In questo lavoro si calcola la caratteristica di Eulero delle varietà CR standard.
We show that a CR function of class , 0 ≤ k < ∞, on a tube submanifold of holomorphically extends to the convex hull of the submanifold. The extension and all its derivatives through order k are shown to have nontangential pointwise boundary values on the original tube submanifold. The -norm of the extension is shown to be no bigger than the -norm of the original CR function.
This article is dedicated to the centenary of the local CR equivalence problem, formulated by Henri Poincaré in 1907. The first part gives an account of Poincaré’s heuristic counting arguments, suggesting existence of infinitely many local CR invariants. Then we sketch the beautiful completion of Poincaré’s approach to the problem in the work of Chern and Moser on Levi nondegenerate hypersurfaces. The last part is an overview of recent progress in solving the problem on Levi degenerate manifolds....
For pseudocomplex abstract manifolds, the validity of the Poincaré Lemma for forms implies local embeddability in . The two properties are equivalent for hypersurfaces of real dimension . As a corollary we obtain a criterion for the non validity of the Poicaré Lemma for forms for a large class of abstract manifolds of codimension larger than one.
Necessary topological conditions are given for the closed CR embedding of a CR manifold into a Stein manifold or into a complex projective space.
The Chern-Moser (CM) normal form of a real hypersurface in can be obtained by considering automorphisms whose derivative acts as the identity on the complex tangent space. However, the CM normal form is also invariant under a larger group (pseudo-unitary linear transformations) and it is this property that makes the CM normal form special. Without this additional restriction, various types of normal forms occur. One of them helps to give a simple proof of a (previously complicated) theorem about...