Pseudoconvex Fully Nonlinear Partial Differential Operators. Strong Comparison Theorems

We introduce a new class of curvature PDO's describing relevant properties of real hypersurfaces of $C^{n+1}.$ In our setting the pseudoconvexity and the Levi form play the same role as the convexity and the real Hessian matrix play in the real Euclidean one. Our curvature operators are second order fully nonlinear PDO's not elliptic at any point. However, when computed on generalized $s$-pseudoconvex functions, we shall show that their characteristic form is nonnegative definite with kernel of dimension one. Moreover, we shall show that the missing ellipticity direction can be recovered by taking into account the CR structure of the hypersurfaces. These properties allow us to prove a strong comparison principle, leading to symmetry theorems for domains with constant curvatures and to identification results for domains with comparable curvatures.