Average of the mean curvature integral in complex space forms

Abstract

Abstract In real space forms, given a regular hypersurface S, it is known that the integral over the space of s-planes of the mean curvature integral of intersections with S is a multiple of the mean curvature integral of S. We study the corresponding expression in a complex space form. The role of the s-planes is played by the complex s-planes. We prove that the same property does not hold but another term appears. We express this term as the integral of the normal curvature in the direction obtained from applying the complex structure to the normal direction. As an application, we give the measure of the set of complex lines meeting a compact domain in a complex space form, and we characterize the reproductive continuous invariant valuations of degree 2n – 2 in the standard Hermitian space.