An integral formula in Kahler geometry with applications

TL;DR

This paper develops an integral formula in Kahler geometry and applies it to derive geometric inequalities and characterize hypersurfaces of constant mean curvature in complex space forms.

Contribution

It introduces a new integral formula in Kahler geometry and uses it to prove isoperimetric inequalities and characterize Hopf hypersurfaces of constant mean curvature.

Findings

Established an integral formula in Kahler geometry.

Proved an isoperimetric inequality based on Hermitian curvature.

Characterized Hopf hypersurfaces of constant mean curvature as geodesic spheres.

Abstract

We establish an integral formula on a smooth, precompact domain in a Kahler manifold. We apply this formula to study holomorphic extension of CR functions. Using this formula we prove an isoperimetric inequality in terms of a positive lower bound for the Hermitian curvature of the boundary. Combining with a Minkowski type formula on the complex hyperbolic space we prove that any closed, embedded hypersurface of constant mean curvature must be a geodesic sphere, provided the hypersurface is Hopf. A similar result is established on the complex projective space.