On Picard's Problem via Nevanlinna Theory

Xianjing Dong

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arXiv · 2405.09659

On Picard's Problem via Nevanlinna Theory

Xianjing Dong

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Abstract

We consider the classical Picard's problem for non-parabolic complete K\"ahler manifolds with non-negative Ricci curvature. Based on the global Green function approach, we give a positive answer to Picard's problem under certain condition by developing Nevanlinna theory. That is, we prove that every meromorphic function on such a manifold reduces to a constant if it omits three distinct values, provided that the manifold satisfies a volume growth condition; and prove that every meromorphic function of non-polynomial type growth on such a manifold can avoid 2 distinct values at most.

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Xianjing Dong. 2024-05-15. On Picard's Problem via Nevanlinna Theory. https://arxiv.org/abs/2405.09659

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