Abstract. A uniform upper bound for the Diederich-Fornaess index is given for weakly pseudoconvex domains whose Levi-form of the boundary vanishes in ℓ-directions everywhere.
We present new results concerning the solvability, or lack of thereof, in the Cauchy problem for the ∂ operator with initial values assigned on a weakly pseudoconvex hypersurface, and provide illustrative examples.
We consider weakly q-convex domains with smooth boundary and show that the ∂-equation is locally solvable with regularity up to the boundary for smooth forms of degree (p, s) for s ≥ q.
Let D be a sufficiently small open subset of a generic CR manifold in C n . We show that the cohomology groups H p,q (D) are either zero or infinite dimensional.Using Fredholm-operator theory or some L 2 -estimates, it is often easier to obtain finiteness theorems for the natural cohomology groups associated to a complex manifold than to actually show the solvability of the ∂-equation. Therefore one would like to have a criterion which permits to pass from finiteness theorems to vanishing theorems.It was proved by Laufer that for any open subset D of a Stein manifold, the Dolbeault cohomology groups H p,q (D) are either zero or infinite dimensional ([La]). The purpose of this note is to give a version of this theorem for embedded CR manifolds.Let us denote by M a smooth generic CR manifold embedded into C n , of real codimension k, and by H p,q (M ), 0 ≤ p ≤ n, 0 ≤ q ≤ n − k, the cohomology groups of the Cauchy-Riemann complexes obtained from the ∂ M -operator (see [Bo] for the definitions). We then obtain the following theorem.
Theorem. Let z be an arbitrary point of M . Then there exists an open neighborhood U z of z in M such that for every open subsetWe point out that this theorem holds without any hypothesis on the Levi form of M .Our result holds only for sufficiently small open subsets of CR manifolds. We also provide an example of a CR manifold, globally embedded into some
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