Abstract
In this paper, we consider the rigidity for an \(n(\ge 4)\)-dimensional submanifold \(M^n\) with parallel mean curvature in the space form \({\mathbb {M}}^{n+p}_{c}\) when the integral Ricci curvature of M has some bound. We prove that, if \(c+H^2>0\) and \(\Vert {\mathrm{Ric}_{-}^{\lambda }}\Vert _{n/2}< \epsilon (n,c, \lambda , H)\) for \(\lambda \) satisfying \( \tfrac{n-2}{n-1} (c+H^2) < \lambda \le c+H^2\), then M is the totally umbilical sphere \(\mathbb S^n\left( \tfrac{1}{\sqrt{c+H^2}}\right) \). Here H is the norm of the parallel mean curvature of M, and \(\epsilon (n,c,\lambda , H)\) is a positive constant depending only on \(n, c,\lambda \) and H. This extends part of the earlier work of Xu and Gu (Geom Funct Anal 23(5):1684–1703, 2013) from pointwise Ricci curvature lower bound to integral Ricci curvature lower bound.
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Acknowledgements
This work was done while the first author was visiting UCSB during 2017–2018. He would like to thank UCSB Math Department for the hospitality, and he also would like to acknowledge financial support from China Scholarship Council (CSC No. 201706295018) and Top International University Visiting Program for Outstanding Young scholars of Northwestern Polytechnical University. The authors would like to thank Professor Haizhong Li for bringing the question to our attention, Professor Hongwei Xu for his interest and the anonymous referee for very help comments and suggestions. H. Chen was supported by NSFC Grant No. 11601426, Natural Science Foundation of Shaanxi Province Grant No. 2020JQ-101, and the Fundamental Research Funds for the Central Universities Grant No. 310201911cx013. G. Wei was partially supported by NSF DMS 1506393.
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Chen, H., Wei, G. Rigidity of Minimal Submanifolds in Space Forms. J Geom Anal 31, 4923–4933 (2021). https://doi.org/10.1007/s12220-020-00462-7
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DOI: https://doi.org/10.1007/s12220-020-00462-7