具有平坦Chern联络的完备Hermitian流形上的全纯函数，II
Holomorphic functions on complete Hermitian manifolds with flat Chern connection, II
- Huaqiao University（华侨大学）
- Xiamen University（厦门大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究完备Chern-flat流形上的全纯函数，证明其椭圆性、在挠率多项式增长下双全纯等价于复Lie群，并给出多项式增长全纯函数的尖锐维数估计及与复欧氏空间的双全纯等价条件，推广了先前结果。
AI中文摘要：
在本文中，我们证明了关于具有消失Chern曲率的完备Hermitian流形（我们称之为完备Chern-flat流形）的函数论的若干结果。首先，我们证明任何完备单连通的Chern-flat流形在Gromov意义下是椭圆的。此外，如果挠率具有多项式增长，则该流形双全纯等价于一个复Lie群。其次，我们在不对挠率作增长假设的情况下，获得了完备Chern-flat流形上多项式增长全纯函数的尖锐维数估计。最后，我们证明任何完备Chern-flat流形，若其允许在某点形成局部全纯坐标系的、具有多项式增长的全纯函数，则该流形双全纯等价于复欧几里得空间。这些结果推广了我们之前的工作，其中假设了挠率的次线性增长。新的分析工具是沿此类流形上由平行酉标架生成的全纯流轨道的Cauchy估计的一个版本。
英文摘要：
In this paper, we prove several results concerning the function theory of complete Hermitian manifolds with vanishing Chern curvature, which we refer to as complete Chern-flat manifolds. First, we prove that any complete simply connected Chern-flat manifold is elliptic in the sense of Gromov. Moreover, it is biholomorphic to a complex Lie group if the torsion has polynomial growth. Next, we obtain sharp dimension estimates for holomorphic functions of polynomial growth on complete Chern-flat manifolds, with no growth assumption on the torsion. Finally, we prove that any complete Chern-flat manifold admitting polynomial-growth holomorphic functions that form a local holomorphic coordinate system at some point is biholomorphic to complex Euclidean space. These results generalize our previous work, in which sublinear growth of the torsion was assumed. The new analytic tool is a version of Cauchy estimates along orbits of holomorphic flows generated by parallel unitary frames on such manifolds.
