On Levi-flat hypersurfaces with prescribed boundary

@article{Dolbeault2009OnLH,
  title={On Levi-flat hypersurfaces with prescribed boundary},
  author={Pierre Dolbeault and Giuseppe Tomassini and Dmitri Zaitsev},
  journal={arXiv: Complex Variables},
  year={2009}
}
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 studied case $n=2$. We first establish necessary conditions on $S$ at both complex and CR points, needed for the existence of $M$. All CR points have to be nonminimal and all complex points have to be "flat". Then, adding a positivity condition at complex points, which is similar to the ellipticity… 
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