Holomorphic functions on complete Hermitian manifolds with flat Chern connection, II
Abstract
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 show 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. We also construct examples of complete Chern-flat manifolds with torsion of exponential growth, but the underlying complex manifolds do not admit any complex Lie group structures. Next, we obtain sharp upper bounds on the dimensions of holomorphic functions of polynomial growth on complete Chern-flat manifolds, without any 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, where we assume that the torsion has sublinear growth. Our approach combines Cauchy estimates with nilpotent analysis along the orbits of holomorphic flows generated by parallel unitary frames on such manifolds.
Related
- Holomorphic functions on a class of complete Hermitian manifolds
- Hermitian manifolds with constant holomorphic sectional curvature or real bisectional curvature
- Jacobi polynomials on the simplex of Hermitian matrices and zonal spherical functions on partial flag manifolds
- Wronskian curvature positivity is equivalent to the existence of a holomorphic projective connection
- A numerically flat rank-two bundle without a holomorphic connection on a ∂∂̅-threefold
BibTeX
@misc{li2026holomorphic,
title = {{Holomorphic functions on complete Hermitian manifolds with flat Chern connection, II}},
author = {Duoxin Li and Hanyu Wu and Bo Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10159},
eprint = {2609.10159},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10159},
}
RIS
TY - UNPB
TI - Holomorphic functions on complete Hermitian manifolds with flat Chern connection, II
AU - Duoxin Li
AU - Hanyu Wu
AU - Bo Yang
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10159
UR - https://arxiv.org/abs/2609.10159
AB - 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 show 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. We also construct examples of complete Chern-flat manifolds with torsion of exponential growth, but the underlying complex manifolds do not admit any complex Lie group structures. Next, we obtain sharp upper bounds on the dimensions of holomorphic functions of polynomial growth on complete Chern-flat manifolds, without any 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, where we assume that the torsion has sublinear growth. Our approach combines Cauchy estimates with nilpotent analysis along the orbits of holomorphic flows generated by parallel unitary frames on such manifolds.
ER -