Minimal Hypersurfaces with constant scalar curvature in ${\mathbf{S}}^6$

Abstract

In this paper, we propose certain assumptions on the principal curvatures for a closed minimal hypersurface $M^5$ in ${\mathbf{S}}^6$ to be isoparametric, provided that the functions $S,f_3,f_4$ are constants. Our result removes the nonnegative scalar curvature assumption as in Tang and Yan \cite{TY}. Finally, as a rigidity result, if $M^5\subset {\mathbf{S}}^6$ has a point with exactly two distinct principal curvatures, then it must be a Clifford torus.