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# Pure and Applied Mathematics Quarterly

## Volume 12 (2016)

### Number 2

### On Yau rigidity theorem for submanifolds in pinched manifolds

Pages: 301 – 333

DOI: https://dx.doi.org/10.4310/PAMQ.2016.v12.n2.a6

#### Authors

#### 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

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

Received 19 June 2017

Published 9 February 2018