Chinese Quarterly Journal of Mathematics >

2026 , Vol. 41 >Issue 1: 1 - 14

DOI: https://doi.org/10.13371/j.cnki.chin.q.j.m.2026.01.001

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

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  • 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.
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. 

Received date: 2025-04-07

  Online published: 2026-03-30

Supported by

Supported by National Natural Science Foundation of China (Grant No. 12171140).

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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

Cite this article

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

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