标题： 最小曲面与欧拉问题二维类比之间的联系 显示英文标题

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

摘要： 如果$\alpha\in\r$是欧几里得空间中的一个$\alpha$-平稳曲面，那么它是一个$\Sigma$的曲面，其平均曲率$H$满足$H(p)=\alpha |p|^{-2} \langle\nu,p\rangle$，$p\in\Sigma$。这些曲面在二维情况下推广了欧拉研究的一类经典曲线，这些曲线是平面曲线的惯性矩的临界点。 在本文中，我们通过反演建立了一个$\alpha$-平稳曲面与$-(\alpha+4)$-平稳曲面之间的一一对应关系。特别地，存在一个$-4$-平稳曲面与极小曲面之间的对应关系。利用这种对偶性，我们给出了一些$-4$-平稳曲面的唯一性结果，并解决了 Börling 问题。 显示英文摘要

摘要： 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.