On the average of the scalar curvature of minimal hypersurfaces of spheres with low stability index (Q1765978)

This paper deals with minimal hypersurfaces in the sphere with low stability index. Let \(M\subset S^n\) be a non totally geodesic compact minimal hypersurface embedded in the \(n\)-dimensional unit sphere \(S^n\). Denote by \(A\) the second fundamental form of \(M\) in \(S^n\). The author proves that if the stability index of \(M\) is equal to \(n+2\) then the average \(\int_M |A|^2\) of the length square of \(A\) is not larger than \(n-1\), where equality holds if and only if \(M\) is isometric to a Clifford minimal hypersurface. Finally, the author proposes two conjectures.