Value distribution of meromorphic mappings on complete Kähler connected sums with non-parabolic ends
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866917893338300416 |
|---|---|
| author | Dong, Xianjing |
| author_facet | Dong, Xianjing |
| contents | All harmonic functions on $\mathbb C^m$ possess Liouville's property, which is well-known as the Liouville's theorem. In 1979, Kuz'menko and Molchanov discovered a phenomenon that the Liouville's property is not rigid for some harmonic functions on the connected sum $\mathbb C^m\#\mathbb C^m,$ where there exist a large number of non-constant bounded harmonic functions. This discovery motivates us to explore conditions under which harmonic functions possess Liouville's property. In this paper, we discuss the value distribution of meromorphic mappings from complete Kähler connected sums with non-parabolic ends into complex projective manifolds. Under a geometric condition, we establish a second main theorem in Nevanlinna theory. As a consequence, we prove that the Cauchy-Riemann equation ensures the rigidity of Liouville's property for harmonic functions if such connected sums satisfy a volume growth condition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_10243 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Value distribution of meromorphic mappings on complete Kähler connected sums with non-parabolic ends Dong, Xianjing Complex Variables 32H30, 32A22, 32H25 All harmonic functions on $\mathbb C^m$ possess Liouville's property, which is well-known as the Liouville's theorem. In 1979, Kuz'menko and Molchanov discovered a phenomenon that the Liouville's property is not rigid for some harmonic functions on the connected sum $\mathbb C^m\#\mathbb C^m,$ where there exist a large number of non-constant bounded harmonic functions. This discovery motivates us to explore conditions under which harmonic functions possess Liouville's property. In this paper, we discuss the value distribution of meromorphic mappings from complete Kähler connected sums with non-parabolic ends into complex projective manifolds. Under a geometric condition, we establish a second main theorem in Nevanlinna theory. As a consequence, we prove that the Cauchy-Riemann equation ensures the rigidity of Liouville's property for harmonic functions if such connected sums satisfy a volume growth condition. |
| title | Value distribution of meromorphic mappings on complete Kähler connected sums with non-parabolic ends |
| topic | Complex Variables 32H30, 32A22, 32H25 |
| url | https://arxiv.org/abs/2409.10243 |