On the minimal surfaces of Riemann
Let \(\gamma_ 1\), \(\gamma_ 2\) be plane Jordan curves in horizontal planes \(P_ 1\), \(P_ 2\) of \(\mathbb{R}^ 3\). It is known, under simple conditions on \(\gamma_ 1\) and \(\gamma_ 2\), that they bound a least area minimal annulus between \(P_ 1\) and \(P_ 2\). \textit{W. H. Meeks III} and \textit{R. White} [Bull. Am. Math. Soc., New Ser. 24, No. 1, 179-184 (1991; Zbl 0752.53007)] proved that if the curves are convex there are at most two minimal annuli bounded by \(\gamma_ 1 \cup \gamma_ 2\). Assuming that \(\gamma_ 1\) and \(\gamma_ 2\) are convex, \textit{M. Shiffman} [Ann. Math., II. Ser. 63, 77-90 (1956; Zbl 0070.168)] proved that if \(M\) is a minimal annulus bounded by \(\gamma_ 1 \cup \gamma_ 2\), then for each horizontal plane \(P\) between \(P_ 1\) and \(P_ 2\), the intersection \(P\cap M\) is again a convex Jordan curve, furthermore if \(\gamma_ 1\) and \(\gamma_ 2\) are circles, then \(P\cap M\) is also a circle. In view of the results of Shiffman, the author deals with the natural question of what happens if the two curves are replaced by straight lines \(D_ 1\), \(D_ 2\). Let \(\theta\) denote the angle between \(D_ 1\) and \(D_ 2\). If \(\theta = 0\), i.e., \(D_ 1\) is parallel to \(D_ 2\), Riemann already constructed a minimal embedded annulus \(S\) between \(P_ 1\) and \(P_ 2\) bounded by \(D_ 1 \cup D_ 2\). Furthermore, its intersection with any horizontal plane between \(P_ 1\) and \(P_ 2\) is a circle. (A detailed description of Riemann's example is given in [\textit{D. Hoffman, H. Karcher} and \textit{H. Rosenberg}, A characterization of Riemann examples (preprint)], where it is proved that if \(\theta = 0\), then the only minimal properly embedded annulus bounded by \(D_ 1 \cup D_ 2\) is precisely this example). The author shows, in the paper under review, that the case \(\theta \neq 0\) does not occur. He also generalizes the example of Riemann to yield a family of minimal surfaces \(S_ k\) with the following properties: 1) \(S_ 0 = S\) (Riemann's example). 2) For every integer \(k \geq 0\), \(S_ k\) is a minimal immersed annulus between \(P_ 1\) and \(P_ 2\), bounded by \(D_ 1\cup D_ 2\) such that after reflection about the lines \(D_ 1,D_ 2,\dots\) we get a complete minimal surface in \(\mathbb{R}^ 3\), which is called again \(S_ k\). Furthermore, \(S_ k\) is invariant under the translation \(X\mapsto X + 2\pi u\), where \(u\) is the vector orthogonal to \(D_ 1\) translating \(D_ 1\) to \(D_ 2\). Also each end of \(S_ k\) in \(\mathbb{R}^ 3\) is a flat horizontal end, i.e. an end asymptotic to a horizontal plane, and the projection of every end over any horizontal plane is a \((4k+1)\) to 1 map. The author also gives the Weierstrass representation of this family. He uses in the proofs the Weierstrass representation of minimal surfaces and the reflection principle.
scientific article; zbMATH DE number 167708