Stable minimal cones in R8 and R9 with constant scalar curvature

Let M be an n-dimensional riemannian manifold. A natural problem in geometry is that of finding k-dimensional submanifolds N ⊂ M with the property that for any bounded open set U in M , the k-volume of N ∩ U is less than or equal to the volume of any other submanifold in M with boundary equal to ∂(N ∩ U). The submanifolds of M with the property above are called area-minimizing. Notice that when k = 1, area-minimizing submanifolds are geodesics. Locally, the problem reduces to the one of finding minimal submanifolds, manifolds which mean curvature vector vanishes, globally, the problem of finding complete area-minimizing submanifolds is a difficult one, even in the case when M is the euclidian space R. It is clear that planes in R3 are area-minimizing. In general, hyperplanes in R are area-minimizing hypersurfaces. A family of hypersurface of R which is important in the study of area-minimizing hypersurfaces are the cones: N ⊂ R is a cone, if for every a > 0, ap ∈ N any time p ∈ N . The study of cones is important for two reasons, the first one is that if p ∈ N is a singular point of a area-minimizing hypersurface S, then there is an area-minimizing tangent cone, which makes the role of tangent space at p, with the property that p ∈ S is a removable singularity if and only if this tangent cone is a hyperplane. The second reason area-minimizing cones are important is that if S is a complete area-minimizing hypersurface, then there is an area-minimizing cone that makes the role of tangent cone at infinity; this cone is a hyperplane if and only if S is a hyperplane. A giant step toward this problem of classifying area-minimizing hypersurfaces in euclidean spaces was made by James Simons in 1968 [S]. He showed that the only areaminimizing complete hypersurfaces in R, with n ≤ 7, are the hyperplanes. On the other hand Bombieri-De Giorgi-Guisti showed that the hypercone