Integral formulas for a class of curvature PDE's and applications to isoperimetric inequalities and to symmetry problems

Abstract

We prove integral formulas for closed hypersurfaces in ℂ n +1 , which furnish a relation between elementary symmetric functions in the eigenvalues of the complex Hessian matrix of the defining function and the Levi curvatures of the hypersurface. Then we follow the Reilly approach to prove an isoperimetric inequality. As an application, we obtain the “Soap Bubble Theorem” for star-shaped domains with positive and constant Levi curvatures bounding the classical mean curvature from above.