### A characterization of totally real generic submanifolds of strictly pseudoconvex boundaries in Cn admitting a local foliation by interpolation submanifolds.

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Tuboids are tube-like domains which have a totally real edge and look asymptotically near the edge as a local tube over a convex cone. For such domains we state an analogue of Cartan’s theorem on the holomorphic convexity of totally real domains in ${\mathbb{R}}^{n}\subset {\u2102}^{n}$.

Let $M$ be a two dimensional totally real submanifold of class ${C}^{2}$ in ${\mathbf{C}}^{2}$. A continuous map $F:\stackrel{\u203e}{\Delta}\to {\mathbf{C}}^{2}$ of the closed unit disk $\stackrel{\u203e}{\Delta}\subset \mathbf{C}$ into ${\mathbf{C}}^{2}$ that is holomorphic on the open disk $\Delta $ and maps its boundary $b\Delta $ into $M$ is called an analytic disk with boundary in $M$. Given an initial immersed analytic disk ${F}^{0}$ with boundary in $M$, we describe the existence and behavior of analytic disks near ${F}^{0}$ with boundaries in small perturbations of $M$ in terms of the homology class of the closed curve ${F}^{0}\left(b\Delta \right)$ in $M$. We also prove a regularity theorem...

We show that the local automorphism group of a minimal real-analytic CR manifold $M$ is a finite dimensional Lie group if (and only if) $M$ is holomorphically nondegenerate by constructing a jet parametrization.

Let S ⊂ ℂⁿ, n ≥ 3, be a compact connected 2-codimensional submanifold having the following property: there exists a Levi-flat hypersurface whose boundary is S, possibly as a current. Our goal is to get examples of such S containing at least one special 1-hyperbolic point: a sphere with two horns, elementary models and their gluings. Some particular cases of S being a graph are also described.

For a strongly pseudoconvex domain $D\subset {\u2102}^{n+1}$ defined by a real polynomial of degree ${k}_{0}$, we prove that the Lie group $\mathrm{Aut}\left(D\right)$ can be identified with a constructible Nash algebraic smooth variety in the CR structure bundle $Y$ of $\partial D$, 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...

In 1958, H. Grauert proved: If D is a strongly pseudoconvex domain in a complex manifold, then D is holomorphically convex. In contrast, various cases occur if the Levi form of the boundary of D is everywhere zero, i.e. if ∂D is Levi flat. A review is given of the results on the domains with Levi flat boundaries in recent decades. Related results on the domains with divisorial boundaries and generically strongly pseudoconvex domains are also presented. As for the methods, it is explained how Hartogs...