On the regularity of solutions of the prescribed Levi curvature equation in several complex variables

Annamaria Montanari University of Bologna, Italy

                      e-mail montanar@dm.unibo.it

We define the Levi curvature of a real hypersurface M = {z ∈ C^{n+1} : ρ(z) = 0} of class C^{2,α} at z ∈ M as

k(z) = ∏_{l=1}^{n} λ_{l},

with λ_{l} the eigenvalues of the Levi form of M at z.

In this talk we present the following result: If M is the boundary of a strictly pseudoconvex domain and

z ⟼ k(z)

is of class C^{∞}, then M is of class C^{∞}.

This equation naturally arises in the study of domains of holomorphy. Formally it is the analogous of the prescribed Gauss curvature equation for the complex structure. However, it is a fully nonlinear degenerate elliptic also if restricted to graphs of strictly pseudoconvex functions.

The existence of C^{2,α} solutions of the prescribed Levi curvature equation is an open problem for n ≥ 2, while it has been recently solved by Citti, Lanconelli and Montanari for n = 1.