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On the minimal surfaces of Riemann. (English) Zbl 0787.53005
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 \(P_ 2\). W. H. Meeks III and 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, 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 [D. Hoffman, H. Karcher and 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.

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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