2024/03/07 by Óscar Perdomo, Perdomo, Oscar · 2 citations
Mathematics · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2403.04223
Let M⊂ Sn+1 be the hypersurface generated by rotating a hypersurface M0 contained in the interior of the unit ball of ℝn-k+1. More precisely, M=\(√(1-|m|2) y, m):y∈ Sk, m∈ M0\. We derive the equation for the mean curvature of M in terms of the principal curvatures of M0. For the particular case when M0 is a surface of revolution in ℝ3, we provide a method for finding the eigenvalues of the Laplace and stability operators. To illustrate this method, we consider an example of a minimal embedded hypersurface in S6 and numerically compute all the eigenvalues of the Laplace operator less than 12, as well as all non-positive eigenvalues of the stability operators. For this example, we show that the stability index (the number of negative eigenvalues of the stability operator, counted with multiplicity) is 77, and the nullity (the multiplicity of the eigenvalue λ=0 of the stability operator) is 14. Similar results are found in the case where M0 is a hypersurface in ℝl+2 of the form (f2(u)z, f1(u)), with z in the l-dimensional unit sphere Sl. Carlotto and Schulz have found examples of embedded minimal hypersurfaces in the case where M0=Sk× S1.