The n-dimensional analogue of a variational problem of Euler

Abstract

For the f-weighted area-functional

$$\begin{aligned} \mathcal {E}_f (M) = \int _M f(x) d\mathcal {H}_n (x) \end{aligned}$$

we prove non-existence of compact stationary surfaces when $f = \vert x \vert ^\alpha$ and $\alpha > -n$, while for $\alpha = -n$ all spheres $S_R (0)$ are shown to be stable and even minimizers for $\mathcal {E}_f$. 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 $S^{k-1} \times S^{k-1} \subset \mathbb {R}^k \times \mathbb {R}^k, ~k \ge 2$, are in fact minimizers for $\mathcal {E}_f$, if $1 \le \alpha \le 2 (k-1)$. In particular the cone over the Clifford torus minimizes $\mathcal {E}_f, ~f=\vert x \vert ^\alpha$ and $1 \le \alpha \le 2$.