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

TL;DR

This paper derives integral formulas relating complex Hessian eigenvalues and Levi curvatures of hypersurfaces, leading to isoperimetric inequalities and a soap bubble theorem for star-shaped domains with constant Levi curvatures.

Contribution

It introduces new integral formulas for hypersurfaces in complex space and applies them to establish geometric inequalities and symmetry results.

Findings

Established integral formulas linking eigenvalues and Levi curvatures.

Proved an isoperimetric inequality for hypersurfaces in complex space.

Derived a soap bubble theorem for star-shaped domains with constant Levi curvatures.

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.