# Levi flat

Suppose $M\subset {\u2102}^{n}$ is at least a ${C}^{2}$ hypersurface.

###### Definition.

$M$ is *Levi-flat* if it is pseudoconvex from both sides, or equivalently if and only if the Levi form of $M$ vanishes identically.

Suppose $M$ is locally defined by $\rho =0$.
The vanishing of the Levi form is equivalent to the complex
Hessian of $\rho $ vanishing on all holomorphic vectors tangent^{} to the hypersurface.
Hence $M$
is Levi-flat if and only if the complex bordered Hessian of $\rho $
is of rank two on the hypersurface. In other words, it is not hard to see that
$M$ is Levi-flat if and only if

$$\mathrm{rank}\left[\begin{array}{cc}\hfill \rho \hfill & \hfill {\rho}_{z}\hfill \\ \hfill {\rho}_{\overline{z}}\hfill & \hfill {\rho}_{z\overline{z}}\hfill \end{array}\right]=2\mathit{\hspace{1em}\hspace{1em}\u2006}\text{for all points on}\{\rho =0\}\text{.}$$ |

Here ${\rho}_{z}$ is the row vector $[\frac{\partial \rho}{\partial {z}_{1}},\mathrm{\dots},\frac{\partial \rho}{\partial {z}_{n}}],$ ${\rho}_{\overline{z}}$ is the column vector ${[\frac{\partial \rho}{\partial {z}_{1}},\mathrm{\dots},\frac{\partial \rho}{\partial {z}_{n}}]}^{T},$ and ${\rho}_{z\overline{z}}$ is the complex Hessian ${\left[\frac{{\partial}^{2}\rho}{\partial {z}_{i}\partial {\overline{z}}_{j}}\right]}_{ij}.$

Let ${T}^{c}M$ be the complex tangent space of $M,$ that is at each point $p\in M,$
define
${T}_{p}^{c}M=J({T}_{p}M)\cap {T}_{p}M,$
where $J$ is the complex structure.
Since $M$ is a hypersurface the dimension
of ${T}_{p}^{c}M$ is always $2n-2,$ and so ${T}^{c}M$ is a subbundle of $TM.$ $M$ is Levi-flat
if and only if ${T}^{c}M$ is involutive. Since the leaves are graphs of functions that satisfy
the Cauchy-Riemann equations^{}, the leaves are complex analytic^{}. Hence, $M$ is Levi-flat, if and only if it is foliated by complex hypersurfaces.

The cannonical example of a Levi-flat hypersurface is the hypersurface defined in ${\u2102}^{n}$ by the equation $\mathrm{Im}{z}_{1}=0$. In fact, locally, all real analytic Levi-flat hypersurfaces are biholomorphic to this example.

## References

- 1 M. Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. , Princeton University Press, Princeton, New Jersey, 1999. Lars Hörmander. , North-Holland Publishing Company, New York, New York, 1973.
- 2 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title | Levi flat |
---|---|

Canonical name | LeviFlat |

Date of creation | 2013-03-22 17:39:41 |

Last modified on | 2013-03-22 17:39:41 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 4 |

Author | jirka (4157) |

Entry type | Definition |

Classification | msc 32V05 |

Synonym | Levi-flat |