Stable minimal cones in ℝ8 and ℝ9 with constant scalar curvature

In this paper we prove that if M ⊏ ℝn , n = 8 or n = 9, is a n - 1 dimensional stable minimal complete cone such that its scalar curvature varies radially, then M must be either a hyperplane or a Clifford minimal cone. By Gauss' formula, the condition on the scalar curvature is equivalent to the condition that the function K1(m)2 + . + Kn-1 (m)2 varies radially. Here the Ki are the principal curvatures at m ∈ M. Under the same hypothesis, for M ⊏ ℝ10 we prove that if not only K1(m)2 + . + Kn-1 (m)2 varies radially but either K1(m)3 + . + Kn-1 (m)3 varies radially or K1(m)4 + . + Kn-1 (m)4 varies radially, then M must be either a hyperplane or a Clifford minimal cone.