The Levi Monge-Ampere equation: Smooth regularity of strictly Levi convex solutions

We prove smoothness of strictly Levi convex solutions to the Levi equation in several complex variables. This equation is fully non linear and naturally arises in the study of real hypersurfaces in Cn+1, for n greater than or equal to 2. For a particular choice of the right-hand side, our equation has the meaning of total Levi curvature of a real hypersurface Cn+1 and it is the analogous of the equation with prescribcd Gauss curvature for the complex structure. However, it is degenerate elliptic also if restricted to strictly Levi convex functions. This basic failure does not allow its to use elliptic techniques such in the classical real and complex Monge-Ampere, equations. By taking into account the natural geometry of the problem we prove that first order intrinsic derivatives of strictly Levi convex solutions satisfy a good equation. The smoothness of solutions is then achieved by mean of a bootstrap argument in tangent directions to the hypersurface.