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

TL;DR

This paper investigates conditions under which closed minimal hypersurfaces in the 6-sphere are isoparametric, removing previous scalar curvature restrictions and establishing rigidity results for hypersurfaces with specific principal curvature configurations.

Contribution

It introduces new assumptions on principal curvatures for minimal hypersurfaces in S^6 to be isoparametric, extending prior results by removing scalar curvature constraints.

Findings

Removed nonnegative scalar curvature assumption from previous results.

Identified conditions under which hypersurfaces are isoparametric.

Proved that hypersurfaces with a point having exactly two principal curvatures are Clifford tori.

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.