The n-dimensional analogue of a variational problem of Euler

For the f-weighted area-functional epsilon( f )(M) = integral(M) (x)dH(n)(x)we prove non-existence of compact stationary surfaces when f = |x|(alpha) and alpha > -n, while for alpha = -n all spheres S-R(0) are shown to be stable and even minimizers for epsilon( f). Moreover any compact and f-stable surface is a sphere S-R(0) in case alpha = -n. Furthermore we prove stability of the minimal cones over products of spheres under suitable conditions on alpha and show non-existence of nontrivial cones in case these conditions do not hold. Finally we show that the cones over products of spheres Sk-1 x Sk-1 c R-k x R-k, k > 2, are in fact minimizers for epsilon( f ), if 1 <= alpha <= 2(k - 1). In particular the cone over the Clifford torus minimizes epsilon( f ) , f = |x|(alpha) and 1 < alpha < 2.