数学物理学报(英文版) ›› 2021, Vol. 41 ›› Issue (5): 1659-1669.doi: 10.1007/s10473-021-0515-4

• 论文 • 上一篇    下一篇

THE $\partial\overline{\partial}$-BOCHNER FORMULAS FOR HOLOMORPHIC MAPPINGS BETWEEN HERMITIAN MANIFOLDS AND THEIR APPLICATIONS

汤凯   

  1. College of Mathematics and Computer Science, Zhejiang Normal University, Jinhua 321004, China
  • 收稿日期:2020-01-03 修回日期:2021-04-16 出版日期:2021-10-25 发布日期:2021-10-21
  • 作者简介:Kai TANG,E-mail:kaitang001@zjnu.edu.cn
  • 基金资助:
    The author was supported by National Natural Science Foundation of China (12001490) and Natural Science Foundation of Zhejiang Province (LQ20A010005).

THE $\partial\overline{\partial}$-BOCHNER FORMULAS FOR HOLOMORPHIC MAPPINGS BETWEEN HERMITIAN MANIFOLDS AND THEIR APPLICATIONS

Kai TANG   

  1. College of Mathematics and Computer Science, Zhejiang Normal University, Jinhua 321004, China
  • Received:2020-01-03 Revised:2021-04-16 Online:2021-10-25 Published:2021-10-21
  • Supported by:
    The author was supported by National Natural Science Foundation of China (12001490) and Natural Science Foundation of Zhejiang Province (LQ20A010005).

摘要: In this paper, we derive some $\partial\overline{\partial}$-Bochner formulas for holomorphic maps between Hermitian manifolds. As applications, we prove some Schwarz lemma type estimates, and some rigidity and degeneracy theorems. For instance, we show that there is no non-constant holomorphic map from a compact Hermitian manifold with positive (resp. non-negative) $\ell$-second Ricci curvature to a Hermitian manifold with non-positive (resp. negative) real bisectional curvature. These theorems generalize the results[5, 6] proved recently by L. Ni on Kähler manifolds to Hermitian manifolds. We also derive an integral inequality for a holomorphic map between Hermitian manifolds.

关键词: Schwarz lemmas, Bochner formulas, holomorphic map, Hermitian manifolds, $\ell$-second Ricci curvature

Abstract: In this paper, we derive some $\partial\overline{\partial}$-Bochner formulas for holomorphic maps between Hermitian manifolds. As applications, we prove some Schwarz lemma type estimates, and some rigidity and degeneracy theorems. For instance, we show that there is no non-constant holomorphic map from a compact Hermitian manifold with positive (resp. non-negative) $\ell$-second Ricci curvature to a Hermitian manifold with non-positive (resp. negative) real bisectional curvature. These theorems generalize the results[5, 6] proved recently by L. Ni on Kähler manifolds to Hermitian manifolds. We also derive an integral inequality for a holomorphic map between Hermitian manifolds.

Key words: Schwarz lemmas, Bochner formulas, holomorphic map, Hermitian manifolds, $\ell$-second Ricci curvature

中图分类号: 

  • 53C55