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

TL;DR

This paper explores a duality between certain stationary surfaces in Euclidean space and classical minimal surfaces, establishing a correspondence that leads to uniqueness results and solutions to the B"{o}rling problem.

Contribution

It introduces a novel inversion-based duality between $ ext{α}$-stationary surfaces and $-( ext{α}+4)$-stationary surfaces, connecting them to minimal surfaces and solving related problems.

Findings

Established a one-to-one correspondence between $ ext{α}$-stationary and $-( ext{α}+4)$-stationary surfaces.

Provided uniqueness results for $-4$-stationary surfaces.

Solved the B"{o}rling problem for these surfaces.

Abstract

If $\alpha\in\r$, an $\alpha$-stationary surface in Euclidean space is a surface $\Sigma$ whose mean curvature $H$ satisfies $H(p)=\alpha |p|^{-2} \langle\nu,p\rangle$, $p\in\Sigma$. 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 $\alpha$-stationary surfaces and $-(\alpha+4)$-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 B\"{o}rling problem.