On the average of the scalar curvature of minimal hypersurfaces of spheres with low stability index
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.
scientific article; zbMATH DE number 2138873