Levi flat

Suppose $M\subset{\mathbb{C}}^{n}$ 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

\operatorname{rank}\left[\begin{matrix}\rho&\rho_{z}\\ \rho_{\bar{z}}&\rho_{z\bar{z}}\end{matrix}\right]=2\ \ \ \text{ for all points% on $\{\rho=0\}$. }

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

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 $T M .$ $M$ 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 ${\mathbb{C}}^{n}$ by the equation $\operatorname{Im}z_{1}=0$ . 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