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# Mathematical Research Letters

## Volume 24 (2017)

### Number 2

### A characterization of Clifford hypersurfaces among embedded constant mean curvature hypersurfaces in a unit sphere

Pages: 503 – 534

DOI: https://dx.doi.org/10.4310/MRL.2017.v24.n2.a12

#### Authors

#### Abstract

Let $\Sigma$ be an $n(\geq 3)$-dimensional compact embedded hypersurface in a unit sphere with constant mean curvature $H \geq 0$ and with two distinct principal curvatures $\lambda$ and $\mu$ of multiplicity $n-1$ and $1$, respectively. It is known that if $\lambda \gt \mu$, there exist many compact embedded constant mean curvature hypersurfaces [26]. In this paper, we prove that if $\mu \gt \lambda$, then $\Sigma$ is congruent to a Clifford hypersurface. The proof is based on the arguments used by Brendle [10].

#### Keywords

Clifford hypersurface, Simons-type identity, constant mean curvature, embedded hypersurface

#### 2010 Mathematics Subject Classification

53C40, 53C42

Received 8 June 2015

Accepted 19 July 2015

Published 24 July 2017