A connection between minimal surfaces and the two-dimensional analogues of a problem of Euler

If $α\in\r$, an $α$-stationary surface in Euclidean space is a surface $Σ$ whose mean curvature $H$ satisfies $H(p)=α|p|^{-2} \langleν,p\rangle$, $p\inΣ$. These surfaces generalize in dimension two a classical family of curves studied by Euler which are critical points of the moment of inertia of planar curves. In this paper we establish, via inversions, a one-to-one correspondence between $α$-stationary surfaces and $-(α+4)$-stationary surfaces. In particular, there is a correspondence between $-4$-stationary surfaces and minimal surfaces. Using this duality we give some results of uniqueness of $-4$-stationary surfaces and we solve the Börling problem.