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Rigidity of area-minimizing $ 2$-spheres in $ n$-manifolds with positive scalar curvature


Author: Jintian Zhu
Journal: Proc. Amer. Math. Soc. 148 (2020), 3479-3489
MSC (2010): Primary 53C24; Secondary 53C42
DOI: https://doi.org/10.1090/proc/15033
Published electronically: April 22, 2020
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Abstract: We prove that the least area of noncontractible immersed spheres is no greater than $ 4\pi $ in any oriented manifold with dimension $ n+2\leq 7$ which satisfies $ R\geq 2$ and admits a continuous map to $ \mathbf S^2\times T^n$ with nonzero degree. We also prove a rigidity result for the equality case. This can be viewed as a generalization of the result in [Comm. Anal. Geom. 18 (2010), pp. 821-830] to higher dimensions.


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Additional Information

Jintian Zhu
Affiliation: Key Laboratory of Pure and Applied Mathematics, School of Mathematical Sciences, Peking University, Beijing, 100871, People’s Repblic of China
Email: zhujt@pku.edu.cn

DOI: https://doi.org/10.1090/proc/15033
Keywords: Rigidity, area-minimizing $2$-spheres, scalar curvature
Received by editor(s): May 10, 2019
Published electronically: April 22, 2020
Additional Notes: This research was partially supported by the NSFC grants No. 11671015 and 11731001.
Communicated by: Jiaping Wang
Article copyright: © Copyright 2020 American Mathematical Society