On Yau rigidity theorem for submanifolds in pinched manifolds

Juan-Ru Gu (Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou, China; and Center of Mathematical Sciences, Zhejiang University, Hangzhou, China)

Abstract

In this paper, we investigate Yau’s rigidity problem for compact submanifolds with parallel mean curvature in pinched Riemannian manifolds. Firstly, we prove that if $M^n$ is an oriented closed minimal submanifold in an $(n + p)$-dimensional complete simply connected Riemannian manifold $N^{n+p}$, then there exists a constant $\delta_0 (n, p) \in (0, 1)$ such that if the sectional curvature of $N$ satisfies $\overline{K}_N \in [ \delta_0 (n, p), 1]$, and if $M$ has a lower bound for the sectional curvature and an upper bound for the normalized scalar curvature, then $N$ is isometric to $S^{n+p}$. Moreover, $M$ is either a totally geodesic sphere, one of the Clifford minimal hypersurfaces $S^k (\sqrt{\frac{k}{n}}) \times S^{n-k} (\sqrt{\frac{n-k}{n}})$ in $S^{n+1}$ for $k = 1, \dotsc, n-1$, or the Veronese submanifold in $S^{n+d}$, where $d = \frac{1}{2} n (n + 1) - 1$. We then generalize the above theorem to the case where $M$ is a compact submanifold with parallel mean curvature in a pinched Riemannian manifold.

Keywords

submanifolds, rigidity theorem, sectional curvature, mean curvature, pinched Riemannian manifold

2010 Mathematics Subject Classification

53C40, 53C42

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The authors’ research was supported by the NSFC, grant nos. 11531012, 11371315, and 11301476.

Received 19 June 2017

Published 9 February 2018