On a variational theorem of Gauduchon and torsion-critical manifolds
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- by Dongmei Zhang and Fangyang Zheng;
- Proc. Amer. Math. Soc. 151 (2023), 1749-1762
- DOI: https://doi.org/10.1090/proc/16236
- Published electronically: January 13, 2023
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Abstract:
In 1984, Gauduchon [Math. Ann. 267 (1984), pp. 495–518] considered the functional of $L^2$-norm of his torsion $1$-form on a compact Hermitian manifold. He obtained the Euler-Lagrange equation for this functional, and showed that in dimension $2$ the critical metrics must be balanced (namely with vanishing torsion $1$-form). In this note we extend his result to higher dimensions, and show that critical metrics are balanced in all dimensions. We also consider the $L^2$-norm of the full Chern torsion, and show by examples that there are critical points of this functional that are not Kähler.References
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Bibliographic Information
- Dongmei Zhang
- Affiliation: School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, People’s Republic of China
- Email: {2250825921@qq.com}
- Fangyang Zheng
- Affiliation: School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, People’s Republic of China
- MR Author ID: 272367
- Email: 20190045@cqnu.edu.cn; franciszheng@yahoo.com
- Received by editor(s): May 27, 2022
- Received by editor(s) in revised form: July 9, 2022
- Published electronically: January 13, 2023
- Additional Notes: The first author was partially supported by National Natural Science Foundations of China with the grant No.12071050 and 12141101, Chongqing grant cstc2021ycjh-bgzxm0139, and is supported by the 111 Project D21024.
- Communicated by: Lu Wang
- © Copyright 2023 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 151 (2023), 1749-1762
- MSC (2020): Primary 53C55; Secondary 53C05
- DOI: https://doi.org/10.1090/proc/16236
- MathSciNet review: 4550367