Nevanlinna Theory on Geodesic Balls of Complete Kähler Manifolds

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 ${\mathbb{C}}^m\mathrm{.}$ 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.