常黎曼数量曲率Hermitian结构中的临界Hermitian度量研究

doi:10.13371/j.cnki.chin.q.j.m.2026.01.001

数学季刊 ›› 2026, Vol. 41 ›› Issue (1): 1-14.doi: 10.13371/j.cnki.chin.q.j.m.2026.01.001

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常黎曼数量曲率Hermitian结构中的临界Hermitian度量研究

1. School of Mathematics and Statistics, Henan University, Kaifeng 475004, China; 2. Department of Mathematics, University of California at Riverside, CA 92521, U.S.A.

收稿日期:2025-04-07 出版日期:2026-03-30 发布日期:2026-03-30
作者简介:GUAN Daniel (1962-), male, native of Fuzhou, Fujian, professor of Henan University, Ph.D supervisor, Ph.D, engages in complex geometry; YAN Xiao-feng (1996-), male, native of Kaifeng, Henan, Ph.D student of Henan University.
基金资助:
Supported by National Natural Science Foundation of China (Grant No. 12171140).

On the Critical Hermitian Metrics in the Hermitian Structures with Constant Riemann Scalar Curvatures

1. School of Mathematics and Statistics, Henan University, Kaifeng 475004, China; 2. Department of Mathematics, University of California at Riverside, CA 92521, U.S.A.

Received:2025-04-07 Online:2026-03-30 Published:2026-03-30
About author:GUAN Daniel (1962-), male, native of Fuzhou, Fujian, professor of Henan University, Ph.D supervisor, Ph.D, engages in complex geometry; YAN Xiao-feng (1996-), male, native of Kaifeng, Henan, Ph.D student of Henan University.
Supported by:
Supported by National Natural Science Foundation of China (Grant No. 12171140).

摘要/Abstract

摘要： It is well known that critical points of the total scalar curvature functional S on the space of all smooth Riemannian structures of volume 1 on a compact manifold M are exactly the Einstein metrics. When the domain of S is restricted to the space of constant scalar curvature metrics, there has been a conjecture that a critical point is also
Einstein or isometric to a standard sphere. In the Riemannian case, it’s tangent space satisfies a decomposition. In this paper, we prove that if we only consider the Hermitian metrics, it also have a decomposition. Then we obtain the equation of the critical points among the Hermitian metrics.

关键词: Hermitian metric, Critical points equation, Scalar curvature, Conformal Hermitian structure

Abstract: It is well known that critical points of the total scalar curvature functional S on the space of all smooth Riemannian structures of volume 1 on a compact manifold M are exactly the Einstein metrics. When the domain of S is restricted to the space of constant scalar curvature metrics, there has been a conjecture that a critical point is also
Einstein or isometric to a standard sphere. In the Riemannian case, it’s tangent space satisfies a decomposition. In this paper, we prove that if we only consider the Hermitian metrics, it also have a decomposition. Then we obtain the equation of the critical points among the Hermitian metrics.

Key words: Hermitian metric, Critical points equation, Scalar curvature, Conformal , Hermitian structure

中图分类号:

关庄丹, 闫笑峰. 常黎曼数量曲率Hermitian结构中的临界Hermitian度量研究[J]. 数学季刊, 2026, 41(1): 1-14.

GUAN Daniel , YAN Xiao-feng. On the Critical Hermitian Metrics in the Hermitian Structures with Constant Riemann Scalar Curvatures[J]. Chinese Quarterly Journal of Mathematics, 2026, 41(1): 1-14.

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常黎曼数量曲率Hermitian结构中的临界Hermitian度量研究

On the Critical Hermitian Metrics in the Hermitian Structures with Constant Riemann Scalar Curvatures

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