General unconditional Picard’s problem on complete noncompact Kähler manifolds

Determine whether every meromorphic function on a complete noncompact Kähler manifold with nonnegative Ricci curvature is necessarily constant if it omits three distinct values, without imposing any additional hypotheses such as growth or non-parabolicity assumptions (the unconditional case of Picard’s problem).

Background

The classical Picard theorem asserts that a nonconstant meromorphic function on the complex plane can omit at most two values. Extending this phenomenon to higher-dimensional complex manifolds led to Picard’s problem for complete noncompact Kähler manifolds with nonnegative Ricci curvature: whether every meromorphic function that omits three distinct values must be constant.

Prior results addressed special cases under additional assumptions (e.g., homogeneity, bounded dilatation, or growth restrictions). The paper introduces Green function and heat kernel approaches to Nevanlinna theory, resolving the non-parabolic case unconditionally and establishing results for the parabolic case under a weak growth condition. However, the authors explicitly note that the general unconditional case remains open in the literature, motivating further investigation.