Levi flat

Suppose Mn is at least a C2 hypersurface.


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 ρ=0. The vanishing of the Levi form is equivalent to the complex Hessian of ρ vanishing on all holomorphic vectors tangentPlanetmathPlanetmath to the hypersurface. Hence M is Levi-flat if and only if the complex bordered Hessian of ρ 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

rank[ρρzρz¯ρzz¯]=2    for all points on {ρ=0}

Here ρz is the row vector [ρz1,,ρzn], ρz¯ is the column vector [ρz1,,ρzn]T, and ρzz¯ is the complex Hessian [2ρziz¯j]ij.

Let TcM be the complex tangent space of M, that is at each point pM, define TpcM=J(TpM)TpM, where J is the complex structure. Since M is a hypersurface the dimension of TpcM is always 2n-2, and so TcM is a subbundle of TM. M is Levi-flat if and only if TcM is involutive. Since the leaves are graphs of functions that satisfy the Cauchy-Riemann equationsMathworldPlanetmath, the leaves are complex analyticPlanetmathPlanetmath. 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 n by the equation Imz1=0. 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.
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