Generalized Ejiri's Rigidity Theorem for Submanifolds in Pinched Manifolds

Citation:

Hongwei XU,Li LEI,Juanru GU.Generalized Ejiri's Rigidity Theorem for Submanifolds in Pinched Manifolds[J].Chinese Annals of Mathematics B,2020,41(2):285~302
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Authors:

Hongwei XU; Li LEI;Juanru GU

Foundation:

This work was supported by the National Natural Science Foundation of China (Nos.11531012, 11371315, 11301476).
Abstract: Let $M^{n}(n\geq4)$ be an oriented compact submanifold with parallel mean curvature in an $(n+p)$-dimensional complete simply connected Riemannian manifold $N^{n+p}$. Then there exists a constant $\delta(n,p)\in(0,1)$ such that if the sectional curvature of $N$ satisfies $\ov{K}_{N}\in[\delta(n,p), 1]$, and if $M$ has a lower bound for Ricci curvature and an upper bound for scalar curvature, then $N$ is isometric to $S^{n+p}$. Moreover, $M$ is either a totally umbilic sphere $S^n\big(\frac{1}{\sqrt{1+H^2}}\big)$, a Clifford hypersurface $S^{m}\big(\frac{1}{\sqrt{2(1+H^2)}}\big)\times S^{m}\big(\frac{1}{\sqrt{2(1+H^2)}}\big)$ in the totally umbilic sphere $S^{n+1}\big(\frac{1}{\sqrt{1+H^2}}\big)$ with $n=2m$, or $\mathbb{C}P^{2}\big(\frac{4}{3}(1+H^2)\big)$ in $S^7\big(\frac{1}{\sqrt{1+H^2}}\big)$. This is a generalization of Ejiri's rigidity theorem.

Keywords:

Minimal submanifold, Ejiri rigidity theorem, Ricci curvature, Mean curvature

Classification:

53C40, 53C42
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