Levi flat

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

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 dimensionof $T_p^{c}M$ is always $2n-2,$ and so $T^{c}M$ is a subbundle of $TM.$ $M$ is Levi-flatif and only if $T^{c}M$ is involutive. Since the leaves are graphs of functions that satisfythe 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 ${ℂ}^n$ bythe equation $\mathrm{Im}{z}_1=0$. In fact, locally, all real analytic Levi-flat hypersurfacesare biholomorphic to this example.