A comparison theorem for the first Dirichlet eigenvalue of a domain in a Kaehler submanifold

Abstract

AbstractWe give a sharp lower bound for the first eigenvalue of the Dirichlet eigenvalue problem on a domain of a complex submanifold of a Kaehler manifold with curvature bounded from above. The bound on the first eigenvalue is given as a function of the extrinsic outer radius and the bounds on the curvature, and it is attained only on geodesic spheres of a space of constant holomorphic sectional curvature embedded in the Kaehler manifold as a totally geodesic submanifold.