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The problem of analytic representation of integrable CR functions on hypersurfaces with singularities is treated. The nature of singularities does not matter while the set of singularities has surface measure zero. For simple singularities like cuspidal points, edges, corners, etc., also the behaviour of representing analytic functions near singular points is studied. [ABSTRACT FROM AUTHOR]

In [12] Michel and Perotti have shown ${\cal C}^k$ estimates for solutions to the ${\overline \partial}$ -equation on piecewise strictly pseudoconvex domains. Our aim in this paper is to prove similar estimates on q-convex wedges. [ABSTRACT FROM AUTHOR]

In this paper a new explicit integral formula is derived for solutions of the tangential Cauchy-Riemann equations on CR q-concave manifolds and optimal estimates in the Lipschitz norms are obtained. [ABSTRACT FROM AUTHOR]

Let H be a C2 hypersurface in ℂn, n ≥ 3, and let M be a generic submanifold of H of real codimension one. We describe classes of compact removable singularities K for Lp-solutions of the tangential Cauchy-Riemann equations on H under the conditions K ⊂ M, 1 ≤ p ≤ ∞. Removability is understood here in the classical sense, but new effects occur based on compulsory analytic extension and envelopes of holomorphy. The classical theory gives results only in the case p > 1. But even for p > 1, removable singularities for Lp-solutions of the tangential Cauchy-Riemann equations may be metrically much more massive than the classical theory predicts. The results for p = 1 are close to corresponding results on removability in the sense of analytic extension justifying the name “removability” for the latter subject. No Levi-form condition on H is posed and the description is given intrinsically in terms of the CR-structure of M. This may be interesting in connection with generalizations, for example, to more general CR-manifolds instead of H. [ABSTRACT FROM AUTHOR]

Let Ω be a bounded strictly pseudoconvex domain in ℂn, n ≥ 3, with boundary ∂Ω, of class C2. A compact subset K is called removable if any analytic function in a suitable small neighborhood of ∂Ω K extends to an analytic function in Ω. We obtain sufficient conditions for removability in geometric terms under the condition that K is contained in a generic C2 -submanifold M of co-dimension one in ∂Ω. The result uses information on the global geometry of the decomposition of a CR-manifold into CR-orbits, which may be of some independent interest. The minimal obstructions for removability contained in M are compact sets K of two kinds. Either K is the boundary of a complex variety of co-dimension one in Ω or it is an exceptional minimal CR-invariant subset of M, which is a certain analog of exceptional minimal sets in co-dimension one foliations. It is shown by an example that the latter possibility may occur as a nonremovable singularity set. Further examples show that the germ of envelopes of holomorphy of neighborhoods of ∞Ω K for K ⊂ M may be multisheeted. A couple of open problems are discussed. [ABSTRACT FROM AUTHOR]

We give a new algebraic characterization of holomorphic nondegeneracy for embedded real algebraic hypersurfaces in $\mathcal{C}^{N+1}$ , $N\geq 1$ . We then use this criterion to prove the following result about real analyticity of smooth CR mappings: any smooth CR mapping H between a real analytic hypersurface and a rigid polynomial holomorphically nondegenerate hypersurface is real analytic, provided the map H is not totally degenerate in the sense of Baouendi and Rothschild. [ABSTRACT FROM AUTHOR]

Let U be a neighborhood of 0 in ℂ n , 1 ≤ s ≤ n − 1, m ∈ ℕ, M = {(z1,..., zn) ∈ U: Imz n = ¦z1¦2 + ...¦zs¦2 + (¦zs+1¦2 +...+ ¦zn−1¦2)m + some error} term be a hypersurface in U. In this paper, the author constructs homotopy operators for the local tangential Cauchy-Riemann operator on the neighborhoods M δ:= M ∩ {¦zn¦ < δ} of 0 in M. C 0 -estimates up to the boundary are proved for the operators. The constants in the estimates do not depend on δ. [ABSTRACT FROM AUTHOR]

Let M be a smooth CR-manifold embedded into ℂ. Let p be a point in M and let C be a small truncated cone in M (in suitable Euclidean coordinates on M) with vertex p which 'symmetry axis' is a real vector in the complex tangent space. Then one can deform M into a smooth CR-manifold M letting fixed all points outside C in such a way that p is a minimal point of M . This result is used to give a new proof of the fact that wedge extendability of continuous CR-functions propagates along the CR-orbits of a CR-manifold. It allows also to prove the following natural result which was conjectured by Trepreau. Let M be a smooth generic CR-manifold in ℂ. Suppose M consists of one single CR-orbit. Then each continuous CR-function on M is wedge extendable at any point of M. Uniqueness theorems for continuous CR-functions are derived. [ABSTRACT FROM AUTHOR]

The main result of this paper is that the identity component of the automorphism group of a compact, connected, strictly pseudoconvex CR manifold is compact unless the manifold is CR equivalent to the standard sphere. In dimensions greater than 3, it has been pointed out by D. Burns that this result follows from known results on biholomorphism groups of complex manifolds with boundary and the fact that any such CR manifold M can be realized as the boundary of an analytic variety. When M is 3-dimensional, Burns's proof breaks down because abstract CR 3-manifolds are generically not realizable as boundaries. This paper provides an intrinsic proof of compactness that works in any dimension. [ABSTRACT FROM AUTHOR]

Let D ⊂ C , N ≥ 2 be a bounded open set with C boundary and let L be an open connected set of affine complex hyperplanes in C containing a hyperplane that misses $$\bar D$$ . Let E = ∪Λ, Γ = E ∩ bD. Suppose that f ∈ C(Γ) and assume that for every Λ ∈ L that meets bD transversely and for every ( N −1, N −2)-form with constant coefficients. Then f satisfies the weak tangential Cauchy-Riemann equations on Γ. It is shown by an example that the condition that L contains a hyperplane that misses $$\bar D$$ cannot be dropped. [ABSTRACT FROM AUTHOR]

Let M ⊑ ℂ be a real-analytic, nonspherical hypersurface passing through the origin and having nondegenerate Levi form. Let Aut M be the stability group of 0. When n = 12 an example is constructed for which Aut M cannot be linearized. [ABSTRACT FROM AUTHOR]

We prove that there are compact strictly pseudoconvex CR manifolds, embedded into some Euclidean space, that admit small deformations that are also embeddable but their embeddings cannot be chosen close to the original embedding. [ABSTRACT FROM AUTHOR]

We study germs of smooth CR mappings between embedded real hypersurfaces in complex spaces of the same dimension. In particular, we are interested in the generic rank of such mappings. If H: M → M′ is a CR map between two hypersurfaces M and M′, we prove that if M′ does not contain any germ of a holomorphic manifold then either H is constant or the generic rank of H is odd. We also prove that if there is no formal holomorphic vector field tangent to M, then either H is constant or generically H is a local diffeomorphism. It follows, as a special case, that if M and M′ are of D-finite type (in the sense of D'Angelo) then H is either constant or is generically a local diffeomorphism. [ABSTRACT FROM AUTHOR]

Let g0 be a real Lie-algebra and g its complexification. The main result of the paper is the complete classification of CR-structures in two nontrivial cases: the solvable and the Levi-flat ones. These classifications are obtained via a new Lie-product ω (depending on J) on the linear subspace p giving the CR-structure. [ABSTRACT FROM AUTHOR]