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

Abstract

If α ∈ R, an α-stationary surface in Euclidean space is a surface whose mean curvature H satisfies H ( p) = α| p| −2〈ν, p〉, p ∈ . 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.