A minimal surface of \(\mathbb{R}^3\) with planar ends (Q2772940)

\textit{W. H. Meeks}, \textit{J. Pérez} and \textit{A. Ros} [Invent. Math. 133, 107-132 (1998; Zbl 0916.53004)] showed that a genus zero, periodic, properly embedded minimal surface in \(\mathbb{R}^3\) with two limit ends must be one of Riemann's examples. The present author proves a related uniqueness result, weakening the periodicity assumption. His precise assumptions are: let \(M\) be a genus zero properly embedded minimal surface which has a plane of symmetry and planar ends \(E_n\) at the height \(n\in Z\) for all such \(n\). Moreover, the torque vectors of the ends should fulfill the relation torque\((E_{n+2k})= \text{torque}(E_n)\) for some \(k\) and all \(n\in\mathbb{Z}\). (The torque of an annular end \(E\) is defined as the integral \(\int_\gamma X\wedge\nu\), where \(\gamma\) is a closed curve in \(E\) generating the one-dimensional homology group of \(E\), \(X\) the position vector, and \(\nu\) the conormal of \(\gamma\).) Under these assumptions it is shown that \(M\) is one of Riemann's examples. The proof proceeds by analyzing the Weierstrass data of \(M\) and showing that \(M\) must actually be invariant under a translation, reducing in this way the statement to the theorem of Meeks-Pérez-Ros.