Nevanlinna Theory on Geodesic Balls of Complete Kähler Manifolds

Abstract

We study Nevanlinna theory of meromorphic mappings from a geodesic ball of a general complete Kähler manifold with non-negative Ricci curvature into a complex projective manifold by introducing a heat kernel method. When dimension of a target manifold is not greater than one of a source manifold, we establish a second main theorem which is a generalization of the classical second main theorem for a ball of ${\mathbb{C}}^m.$ If a source manifold is non-compact and it carries a positive global Green function, then we establish a global second main theorem for the source manifold. As a result, we obtain a Picard's theorem for complete Kähler manifolds with non-negative Ricci curvature.

Some errors appeared in the article that seem difficult to correct. For example, the Green function $G_R(o,x)$ for the geodesic ball $B(R)$ was misunderstood to satisfy the Dirichlet boundary condition on the geodesic sphere $\partial B(R)$, however, this is not the case