On complete hypersurfaces with constant scalar curvature n(n-1) in the unit sphere

Abstract

Let Mⁿ be an n-dimensional complete and locally conformally flat hypersurface in the unit sphere Sⁿ⁺¹ with constant scalar curvature n(n-1). We show that if the total curvature (∫_M|H|ⁿ dv)^1/n of M is sufficiently small, then Mⁿ is totally geodesic.