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The ∂∂¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \overline \partial $$\end{document}-Bochner Formulas for Holomorphic Mappings between Hermitian Manifolds and Their Applications

Authors
  • Tang, Kai1
  • 1 Zhejiang Normal University, Jinhua, 321004, China , Jinhua (China)
Type
Published Article
Journal
Acta Mathematica Scientia
Publisher
Springer-Verlag
Publication Date
Jun 29, 2021
Volume
41
Issue
5
Pages
1659–1669
Identifiers
DOI: 10.1007/s10473-021-0515-4
Source
Springer Nature
Keywords
Disciplines
  • Article
License
Yellow

Abstract

In this paper, we derive some ∂∂¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \overline \partial $$\end{document}-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) ℓ-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.

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