Brown–York mass and positive scalar curvature II: Besse’s conjecture and related problems
The Besse’s conjecture was posed on the well-known book Einstein manifolds by Arthur L. Besse, which describes critical points of Hilbert–Einstein functional with constraint of unit volume and constant scalar curvature. In this article, we show that there is an interesting connection between Besse’s conjecture and positive mass theorem for Brown–York mass. With the aid of positive mass theorem, we investigate the geometric structure of CPE manifolds and this provides us further understandings about Besse’s conjecture. As a related topic, we also discuss corresponding results for V-static metrics.
KeywordsBesse’s conjecture Brown–York mass Positive mass theorem Scalar curvature V-static metric
We would like to express our appreciations to Professor Qing Jie for his constant support and encouragements. We would like to thank Professor Li Tong-Zhu for inspiring discussions back in University of California, Santa Cruz, and also the referee for his/her valuable comments.
- 16.Lohkamp, J.: The higher dimensional positive mass theorem I (2016). arXiv:math/0608795v2
- 17.Lohkamp, J.: The higher dimensional positive mass theorem II (2016). arXiv:1612.07505
- 27.Schoen, R.-M., Yau, S.-T.: Positive scalar curvature and minimal hypersurface singularities (2017). arXiv:1704.05490v1
- 29.Yuan, W.: Volume comparison with respect to scalar curvature (2016). arXiv:1609.08849
- 30.Yuan, W.: Brown–York mass and positive scalar curvature I-first eigenvalue problem and its applications (2018). arXiv:1806.07798