Suppose is at least a hypersurface.
Suppose is locally defined by . The vanishing of the Levi form is equivalent to the complex Hessian of vanishing on all holomorphic vectors tangent to the hypersurface. Hence is Levi-flat if and only if the complex bordered Hessian of is of rank two on the hypersurface. In other words, it is not hard to see that is Levi-flat if and only if
Here is the row vector is the column vector and is the complex Hessian
Let be the complex tangent space of that is at each point define where is the complex structure. Since is a hypersurface the dimension of is always and so is a subbundle of is Levi-flat if and only if is involutive. Since the leaves are graphs of functions that satisfy the Cauchy-Riemann equations, the leaves are complex analytic. Hence, 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 by the equation . In fact, locally, all real analytic Levi-flat hypersurfaces are biholomorphic to this example.
- 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.