Nevanlinna Theory on Geodesic Balls of Complete K\"ahler Manifolds

TL;DR

This paper extends Nevanlinna theory to meromorphic mappings from geodesic balls in complete K"ahler manifolds with non-negative Ricci curvature, establishing new second main theorems and a Picard theorem using heat kernel methods.

Contribution

It introduces a heat kernel approach to Nevanlinna theory on K"ahler manifolds and generalizes classical results to broader geometric contexts.

Findings

Established a second main theorem for mappings into complex projective manifolds.

Proved a global second main theorem for non-compact source manifolds with Green functions.

Derived a Picard theorem for complete K"ahler manifolds with non-negative Ricci curvature.

Abstract

We study Nevanlinna theory of meromorphic mappings from a geodesic ball of a general complete K\"ahler 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 K\"ahler manifolds with non-negative Ricci curvature.