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PreprintarXiv licencearXiv

Holomorphic functions on a class of complete Hermitian manifolds

  • Zeqing Miao
  • Hanyu Wu
  • Bo Yang
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Abstract

In this paper, we prove a gradient estimate for holomorphic functions on complete Hermitian manifolds with nonnegative second Ricci curvature. As an application, we establish several Liouville-type theorems for holomorphic functions and mappings between complete Hermitian manifolds under suitable curvature assumptions. In particular, on a complete Hermitian manifold with nonnegative second Ricci curvature and Chern torsion of linear growth, any bounded holomorphic function must be constant. Moreover, we construct examples to show that both the gradient estimate and this Liouville theorem are sharp in a suitable sense. In the second part of the paper, we study holomorphic functions on simply connected complete Hermitian manifolds with vanishing Chern curvature and torsion norm of sublinear growth. Combining the gradient estimate with methods from sub-Riemannian geometry, we obtain a complete characterization of holomorphic functions of polynomial growth on such manifolds.

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Cite

BibTeXRIS
BibTeX
@misc{miao2026holomorphic,
  title = {{Holomorphic functions on a class of complete Hermitian manifolds}},
  author = {Zeqing Miao and Hanyu Wu and Bo Yang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.01594},
  eprint = {2608.01594},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.01594},
}
RIS
TY  - UNPB
TI  - Holomorphic functions on a class of complete Hermitian manifolds
AU  - Zeqing Miao
AU  - Hanyu Wu
AU  - Bo Yang
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2608.01594
UR  - https://arxiv.org/abs/2608.01594
AB  - In this paper, we prove a gradient estimate for holomorphic functions on complete Hermitian manifolds with nonnegative second Ricci curvature. As an application, we establish several Liouville-type theorems for holomorphic functions and mappings between complete Hermitian manifolds under suitable curvature assumptions. In particular, on a complete Hermitian manifold with nonnegative second Ricci curvature and Chern torsion of linear growth, any bounded holomorphic function must be constant. Moreover, we construct examples to show that both the gradient estimate and this Liouville theorem are sharp in a suitable sense. In the second part of the paper, we study holomorphic functions on simply connected complete Hermitian manifolds with vanishing Chern curvature and torsion norm of sublinear growth. Combining the gradient estimate with methods from sub-Riemannian geometry, we obtain a complete characterization of holomorphic functions of polynomial growth on such manifolds.
ER  -