The Second Gap for Minimal Hypersurfaces in the Unit Sphere with Constant Cubic Trace

Abstract

Let $M^n$, $n\ge 3$, be a closed oriented minimal hypersurface in the unit sphere with constant scalar curvature and constant $f_3=\tr(h^3)$, where $h$ is the second fundamental form. We prove that $S=\norm h^2>n$ implies $S\ge 2n$. Equality holds if and only if each connected component is locally congruent to a Cartan minimal isoparametric hypersurface with three distinct principal curvatures.

13 pages, Comments are welcome