On Picard's Problem via Nevanlinna Theory II
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915913613180928 |
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| author | Dong, Xianjing |
| author_facet | Dong, Xianjing |
| contents | This work continues the author's earlier work (2026, Studia Mathematica) on Picard's problem: is every meromorphic function on a complete noncompact Kähler manifold with nonnegative Ricci curvature necessarily a constant, if it avoids 3 distinct values? In that prior work, a positive answer was obtained under a growth condition for non-parabolic manifolds. In this paper, we give a full solution to the non-parabolic case by removing this growth condition via a global Green function approach. For the parabolic case, to overcome the obstacle arising from the absence of a positive global Green function, we introduce a heat kernel approach to Nevanlinna theory. Based on it, we develop a Carlson-Griffiths theory, which gives the first systematic result in Nevanlinna theory for parabolic Kähler manifolds. As a direct application, we confirm the parabolic case of Picard's problem under a weak growth condition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_03223 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Picard's Problem via Nevanlinna Theory II Dong, Xianjing Complex Variables 32H30, 32H25, 32A22 This work continues the author's earlier work (2026, Studia Mathematica) on Picard's problem: is every meromorphic function on a complete noncompact Kähler manifold with nonnegative Ricci curvature necessarily a constant, if it avoids 3 distinct values? In that prior work, a positive answer was obtained under a growth condition for non-parabolic manifolds. In this paper, we give a full solution to the non-parabolic case by removing this growth condition via a global Green function approach. For the parabolic case, to overcome the obstacle arising from the absence of a positive global Green function, we introduce a heat kernel approach to Nevanlinna theory. Based on it, we develop a Carlson-Griffiths theory, which gives the first systematic result in Nevanlinna theory for parabolic Kähler manifolds. As a direct application, we confirm the parabolic case of Picard's problem under a weak growth condition. |
| title | On Picard's Problem via Nevanlinna Theory II |
| topic | Complex Variables 32H30, 32H25, 32A22 |
| url | https://arxiv.org/abs/2604.03223 |