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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.
Reviewer: C.Olmos (Córdoba)

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