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 + ... + K_n-1 (m)² varies radially. Here the K_i are the principal curvatures at m ∈ M. Under the same hypothesis, for M ⊏ ℝ¹⁰  we prove that if not only K₁(m)² + ... + K_n-1 (m)²   varies radially but either K₁(m)³ + ... + K_n-1 (m)³ varies radially or K₁(m)⁴ + ... + K_n-1 (m)⁴ varies radially, then M must be either a hyperplane or a Clifford minimal cone.