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A sharp differentiable pinching theorem for submanifolds in space forms


Authors: Juan-Ru Gu and Hong-Wei Xu
Journal: Proc. Amer. Math. Soc. 144 (2016), 337-346
MSC (2010): Primary 53C20, 53C24, 53C40
DOI: https://doi.org/10.1090/proc/12908
Published electronically: September 11, 2015
MathSciNet review: 3415600
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Abstract: Let $ M$ be an $ n$-dimensional compact submanifold in the simply connected space form $ F^{n+p}(c)$ with $ c+H^2>0$. We verify that if the sectional curvature of $ M$ satisfies $ K_{M} > \frac {n-2}{n+2}c+\frac {n^2H^2}{8(n+2)},$ then $ M$ is diffeomorphic to a spherical space form. Moreover, we show that if $ M$ is an oriented compact submanifold in $ F^{n+p}(c)$ with $ c\ge 0$, and if $ n\neq 3,5$, $ K_{M} > \frac {n-2}{n+2}c+\frac {n^2H^2}{8(n+2)},$ then $ M$ is diffeomorphic to the standard $ n$-sphere. It should be emphasized that our results are optimal for $ n=4$.


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Additional Information

Juan-Ru Gu
Affiliation: Center of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, People’s Republic of China
Email: gujr@cms.zju.edu.cn

Hong-Wei Xu
Affiliation: Center of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, People’s Republic of China
Email: xuhw@cms.zju.edu.cn

DOI: https://doi.org/10.1090/proc/12908
Keywords: Submanifolds, sphere theorem, sectional curvature, Ricci flow, stable current
Received by editor(s): November 1, 2014
Published electronically: September 11, 2015
Additional Notes: This research was supported by the NSFC, Grant Nos. 11371315, 11301476, and 1153012; the trans5-CENTURY0 training Programme Foundation for Talents by the Ministry of Education of China; and the China Postdoctoral Science Foundation, Grant No. 2013T60582.
Communicated by: Lei Ni
Article copyright: © Copyright 2015 American Mathematical Society