Index estimates for closed minimal submanifolds of the sphere

In this paper we are interested in comparing the spectra of two elliptic operators acting on a closed minimal submanifold of the Euclidean unit sphere. Using an approach introduced by Savo in [A Savo. Index Bounds for Minimal Hypersurfaces of the Sphere. Indiana Univ. Math. J. 59 (2010), 823-837.], we are able to compare the eigenvalues of the stability operator acting on sections of the normal bundle and the Hodge Laplacian operator acting on $1$ -forms. As a byproduct of the technique and under a suitable hypothesis on the Ricci curvature of the submanifold we obtain that its first Betti's number is bounded from above by a multiple of the Morse index, which provide evidence for a well-known conjecture of Schoen and Marques & Neves in the setting of higher codimension.

On Minimal Hypersurfaces of a Unit Sphere

Minimal compact hypersurface in the unit sphere Sn+1 having squared length of shape operator A2<n are totally geodesic and with A2=n are Clifford hypersurfaces. Therefore, classifying totally geodesic hypersurfaces and Clifford hypersurfaces has importance in geometry of compact minimal hypersurfaces in Sn+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 Sn+1. It is shown that a necessary and sufficient condition for a minimal compact hypersurface M in S5 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 Sn+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 Sn+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 Aw.

r-Stablity Index of r-Maximal Closed Hypersurfaces in de Sitter Spaces

It is well-known that some of minimal (or maximal) hypersurfaces are stable. However, there is growing recognition on unstable hypersurfaces by introducing the concept of index of stability for minimal ones. For instance, the index of stability for minimal hypersurefces in Euclidean n-sphere has been defined by J. Simons and followed by many people. Also, Barros and Sousa have studied a high order extention of index as the concept of r-index (i.e. index of r-stability) on r-minimal hypersurfaces of n-sphere. They gave low bonds for r-stability index of r-minimal hypersurfaces in Euclidean sphere. In this paper, we low bounds for the r-stability index of r-maximal closed spacelike hypersurfaces in the de Sitter space.