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



Sung-Hong Min (Department of Mathematics, Chungnam National University, Daejeon, Korea)

Keomkyo Seo (Department of Mathematics, Sookmyung Women’s University, Seoul, Korea)


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].


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