journal of functional analysis 138, 188212 (1996) Geometric Properties of Solutions of the Levi Curvature Equation in C 2 Zbignew Slodkowski Department of Mathematics, University of Illinois at Chicago, Chicago, Illinois 60681 and Giuseppe Tomassini* Scuola Normale Superiore, Piazza dei Cavalieri 7, Pisa, Italy Received April 1995 0. Introduction Let M=[u =0] be a smooth hypersurface of a domain 0 in C 2 . As it is well known M is Levi flat if and only if 0 u z 1 u z 2 L( u )=&det \ u z 1 u z 1 z 1 u z 1 z 2 + =0. u z 2 u z 2 z 1 u z 2 z 2 In general, for a given M we introduce the function k L ( M)=L( u) 3  |u |, the ``Levi curvature'' of M(|u | 2 =| u z 1 | 2 +| u z 2 | 2 ). | k L ( M)| depends only on M and k L ( M)0 means that locally on [u =0], [u <0 ] is pseudoconvex [12]. L( u), viewed as a differential operator acting on u is called the Levi operator (for non Cartesian hypersurfaces); L( u ) is an elliptic degenerate quasi-linear operator. In this paper we study for L( u ) the Dirichlet problem ( C): L( u )=k |u | 3 in 0, u = g on b0 where 0 is bounded and strictly pseudoconvex, k=k( z, t ) is a real function on 0_R and g : b0  R. The geometric counter part of this problem is the following: given a bounded domain 0 in C 2 and a family of hypersurfaces # c =[g =c] of b0 find a family of level sets M c =[u =c] such that bM c =# c and k L ( M c )=k(}, c ). In particular, when k=0 then the level sets M c form a family of Levi flat hypersurfaces with article no. 0061 188 0022-123696 18.00 Copyright  1996 by Academic Press, Inc. All rights of reproduction in any form reserved. * The second author was 400 supported by the project M.U.R.S.T. ``Geometria reale e complessa.''