On Minimal Hypersurfaces of a Unit Sphere

Minimal compact hypersurface in the unit sphere S n + 1 having squared length of shape operator A 2 < n are totally geodesic and with A 2 = n are Clifford hypersurfaces. Therefore, classifying totally geodesic hypersurfaces and Clifford hypersurfaces has importance in geometry of compact minimal hypersurfaces in S n + 1 . One finds a naturally induced vector field w called the associated vector field and a smooth function ρ called support function on the hypersurface M of S n + 1 . It is shown that a necessary and sufficient condition for a minimal compact hypersurface M in S 5 to be totally geodesic is that the support function ρ is a non-trivial solution of static perfect fluid equation. Additionally, this result holds for minimal compact hypersurfaces in S n + 1 , ( n > 2 ), provided the scalar curvature τ is a constant on integral curves of w . Yet other classification of totally geodesic hypersurfaces among minimal compact hypersurfaces in S n + 1 is obtained using the associated vector field w an eigenvector of rough Laplace operator. Finally, a characterization of Clifford hypersurfaces is found using an upper bound on the integral of Ricci curvature in the direction of the vector field A w .