An integral formula in Kahler geometry with applications

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.