Boundary problem for Levi flat graphs

@article{Dolbeault2009BoundaryPF,
  title={Boundary problem for Levi flat graphs},
  author={Pierre Dolbeault and Giuseppe Tomassini and Dmitri Zaitsev},
  journal={arXiv: Complex Variables},
  year={2009}
}
In an earlier paper the authors provided general conditions on a real codimension 2 submanifold $S\subset C^{n}$, $n\ge 3$, such that there exists a possibly singular Levi-flat hypersurface $M$ bounded by $S$. In this paper we consider the case when $S$ is a graph of a smooth function over the boundary of a bounded strongly convex domain $\Omega\subset C^{n-1}\times R$ and show that in this case $M$ is necessarily a graph of a smooth function over $\Omega$. In particular, $M$ is non-singular. 
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References

SHOWING 1-10 OF 20 REFERENCES
On Levi-flat hypersurfaces with prescribed boundary
We address the problem of existence and uniqueness of a Levi-flat hypersurface $M$ in $C^n$ with prescribed compact boundary $S$ for $n\ge3$. The situation for $n\ge3$ differs sharply from the well
A note on polynomial hulls
Some properties of polynomially convex hulls of compact sets in Cn are deduced by simple topological arguments applied to known topological facts about polynomially convex sets. Introduction. It is
Pseudoconvexity of rigid domains and foliations of hulls of graphs
L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions
On boundaries of complex analytic varieties
A method for economically manufacturing high-grade pellets for metallurgical use in a blast furnace from several grades of iron ores, which method can produce high-strength green pellets and save
...
...