# 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.

## 17 Citations

Boundaries of Levi-flat hypersurfaces: special hyperbolic points

- Mathematics
- 2012

Let $S\subset \C^n$, $n\geq 3$ 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.…

On the Levi-flat Plateau problem

- MathematicsComplex Analysis and its Synergies
- 2020

We solve the Levi-flat Plateau problem in the following case. Let $$M \subset {\mathbb {C}}^{n+1}$$ M ⊂ C n + 1 , $$n \ge 2$$ n ≥ 2 , be a connected compact real-analytic codimension-two submanifold…

Flattening a non-degenerate CR singular point of real codimension two

- Mathematics
- 2017

This paper continues the previous studies in two papers of Huang–Yin [HY16,HY17] on the flattening problem of a CR singular point of real codimension two sitting in a submanifold in $${{\mathbb…

Codimension Two CR Singular Submanifolds and Extensions of CR Functions

- Mathematics
- 2016

Let $$M \subset {\mathbb {C}}^{n+1}$$M⊂Cn+1, $$n \ge 2$$n≥2, be a real codimension two CR singular real analytic submanifold that is nondegenerate and holomorphically flat. We prove that every real…

Flattening of CR singular points and analyticity of the local hull of holomorphy I

- Mathematics
- 2012

This is the first article of the two papers in which we investigate the holomorphic and formal flattening problem for a codimension two real submanifold in $${\mathbb C}^n$$Cn with $$n\ge 3$$n≥3 near…

Geometric and analytic problems for a real submanifold in ℂn with CR singularities

- Mathematics
- 2017

In this survey article, we present some known results and also propose some open questions related to the analytic and geometric aspects of Bishop submanifolds in a complex space. We mainly focus on…

A normal form for a real 2-codimensional submanifold in $\mathbb{C}^{N+1}$ near a CR singularity

- Mathematics
- 2011

A CR singular analogue of Severi’s theorem

- Mathematics
- 2019

Real-analytic CR functions on real-analytic CR singular submanifolds are not in general restrictions of holomorphic functions, unlike in the CR nonsingular case. We give a simple condition that…

A note on a family of analytic discs attached to a real submanifold M ⊂ C N + 1 with a CR singularity

- Mathematics
- 2011

In this note we construct a family of analytic discs attached to a real submanifold M ⊂ CN+1 of codimension 2 with a CR singularity. These discs are mutually disjoint and form a smooth hypersurface…

On a family of analytic discs attached to a real submanifold M\subset\mathbb{C}^{N+1}

- Mathematics
- 2012

We construct a family of analytic discs attached to a real submanifold M \subset $\mathbb{C}^{N+1}$ of codimension $2$ defined near a CR singularity.

## References

SHOWING 1-10 OF 20 REFERENCES

On Levi-flat hypersurfaces with prescribed boundary

- Mathematics
- 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…

A note on polynomial hulls

- Mathematics
- 1972

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

- Mathematics
- 1995

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

- Materials Science
- 1977

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…