A comparison theorem for the mean exit time from a domain in a Kähler manifold (Q1205527)

The main purpose of this paper is to study a domain \(\Omega\) of a Kähler manifold with Ricci and antiholomorphic Ricci curvature bounded from below. The authors consider bounds on the mean and \(JN\)-mean curvature of its boundary. They prove a comparison theorem between the mean exit time function \(E(x)\) defined on \(\Omega\) with the mean exit time \(E^ \lambda_ R(\underline x)\) from the geodesic ball of radius \(R\) of the complex projective space \(CP^ n(\lambda)\). The equality \(E(x)=E^ \lambda_ R(\underline x)\) gives a characterization of the geodesic ball among the domains \(\Omega\). It is proved also a comparison theorem for the mean curvature of hypersurfaces parallel to the boundary of \(\Omega\) using the Index Lemma for submanifolds.