Stable minimal cones in R and R with constant scalar curvature

In this paper we prove that if M ⊂ R, 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 κ1(m) 2 + · · ·+ κn−1(m) 2 varies radially. Here the κi are the principal curvatures at m ∈ M . Under the same hypothesis, for M ⊂ R 10 we prove that if not only κ1(m) 2 + · · · + κn−1(m) 2 varies radially but either κ1(m) 3 + · · ·+ κn−1(m) 3 varies radially or κ1(m) 4 + · · ·+ κn−1(m) 4 varies radially, then M must be either a hyperplane or a Clifford minimal cone.