Brown-York mass and positive scalar curvature I: First eigenvalue problem and its applications
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Abstract:
In this article, we investigate the connection between scalar curvature and first eigenfunctions via positive mass theorem for Brown-York mass. For compact manifolds with nice boundary, we show that a sharp inequality holds for first eigenfunctions when posing appropriate assumptions on scalar curvature and first eigenvalue. This inequality implies that for a compact $n$-dimensional manifold with boundary, its first eigenvalue is no less than $n$, if its scalar curvature is at least $n(n-1)$ with appropriate boundary conditions posed, where equality holds if and only the manifold is isometric to the canonical upper hemisphere. As an application, we derive an estimate for the area of event horizon in a vacuum static space with positive cosmological constant, which reveals an interesting connection between the area of event horizon and Brown-York mass. This estimate generalizes a similar result of Shen [Proc. Amer. Math. Soc. 125 (1997), pp. 901–905] for three dimensional vacuum static spaces and also improves the uniqueness result of de Sitter space-time due to Hizagi-Montiel-Raulot.References
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Bibliographic Information
- Wei Yuan
- Affiliation: Department of Mathematics, Sun Yat-sen University, Guangzhou, Guangdong 510275, People’s Republic of China
- MR Author ID: 1040481
- Email: yuanw9@mail.sysu.edu.cn
- Received by editor(s): July 4, 2021
- Received by editor(s) in revised form: April 10, 2022
- Published electronically: September 15, 2022
- Additional Notes: This work was supported by NSFC (Grant No. 12071489, No. 11601531, No. 11521101) and the Fundamental Research Funds for the Central Universities (Grant No. 2016-34000-31610258)
- Communicated by: Jiaping Wang
- © Copyright 2022 by Wei Yuan
- Journal: Proc. Amer. Math. Soc. 151 (2023), 313-326
- MSC (2020): Primary 53C20; Secondary 53C21, 53C24
- DOI: https://doi.org/10.1090/proc/16105
- MathSciNet review: 4504627