Flexible and inflexible CRCR submanifolds

In this paper we prove new embedding results for compactly supported deformations of $CR$ submanifolds of $\mathbb{C}^{n+d}$: We show that if $M$ is a $2$-pseudoconcave $CR$ submanifold of type $(n,d)$ in $\mathbb{C}^{n+d}$, then any compactly supported $CR$ deformation stays in the space of globally $CR$ embeddable in $\mathbb{C}^{n+d}$ manifolds. This improves an earlier result, where $M$ was assumed to be a quadratic $2$-pseudoconcave $CR$ submanifold of $\mathbb{C}^{n+d}$. We also give examples of weakly $2$-pseudoconcave $CR$ manifolds admitting compactly supported $CR$ deformations that are not even locally $CR$ embeddable.Comment: arXiv admin note: text overlap with arXiv:1611.0542

Non locally trivializable CRCR line bundles over compact Lorentzian CRCR manifolds

We consider compact $CR$ manifolds of arbitrary $CR$ codimension that satisfy certain geometric conditions in terms of their Levi form. Over these compact $CR$ manifolds, we construct a deformation of the trivial $CR$ line bundle over $M$ which is topologically trivial over $M$ but fails to be even locally $CR$ trivializable over any open subset of $M$. In particular, our results apply to compact Lorentzian $CR$ manifolds of hypersurface type.Comment: to appear in Annales de l'Institut Fourie

Remarks on weakly pseudoconvex boundaries: Erratum

We make two tiny corrections to our previous paper with the same title, and also obtain, as a bonus, something new

The Cauchy-Riemann equation with support conditions on domains with Levi-degenerate boundaries

In einem ersten Teil betrachten wir ein relativ kompaktes Gebiet Omega einer n-dimensionalen Kähler-Mannigfaltigkeit, mit Lipschitz-Rand, welches eine gewisse "log delta"-Pseudokonvexität besitzt. Wir zeigen, dass die Cauchy-Riemann Gleichung mit exaktem Träger in Omega für alle Bigrade (p,q) mit 0< q< n-1 eine Lösung besitzt. Ausserdem ist das Bild des Cauchy-Riemann Operators auf glatten (p,n-1)-Formen mit exaktem Träger in Omega abgeschlossen. Wir geben Anwendungen für die Lösbarkeit der tangentialen Cauchy-Riemann Gleichungen für glatte Formen und Ströme auf Rändern von schwach pseudokonvexen Gebieten Steinscher Mannigfaltigkeiten und für die Lösbarkeit der tangentialen Cauchy-Riemann Gleichungen für Ströme auf Levi-flachen CR Mannigfaltigkeiten beliebiger Kodimension. In einem zweiten Teil untersuchen wir die Cauchy-Riemann Gleichung mit Randbedingung Null entlang einer Hyperfläche mit konstanter Signatur. Wir geben Anwendungen für die Lösbarkeit der tangentialen Cauchy-Riemann Gleichung für glatte Formen mit kompaktem Träger und für Ströme auf der Hyperfläche. Wir zeigen auch, dass das Hartogs-Phänomen in schwach 2-konvex-konkaven Hyperflächen mit konstanter Signatur Steinscher Mannigfaltigkeiten gilt.In a first part, we consider a domain Omega with Lipschitz boundary, which is relatively compact in an n-dimensional Kaehler manifold and satisfies some "log delta-pseudoconvexity" condition. We show that the Cauchy-Riemann equation with exact support in Omega admits a solution in bidegrees (p,q), 1 < q < n. Moreover, the range of the Cauchy-Riemann operator acting on smooth (p,n-1)-forms with exact support in Omega is closed. Applications are given to the solvability of the tangential Cauchy-Riemann equations for smooth forms and currents for all intermediate bidegrees on boundaries of weakly pseudoconvex domains in Stein manifolds and to the solvability of the tangential Cauchy-Riemann equations for currents on Levi-flat CR manifolds of arbitrary codimension. In a second part, we study the Cauchy-Riemann equation with zero Cauchy data along a hypersurface with constant signature. Applications to the solvability of the tangential Cauchy-Riemann equations for smooth forms with compact support and currents on the hypersurface are given. We also prove that the Hartogs phenomenon holds in weakly 2-convex-concave hypersurfaces with constant signature of Stein manifolds