A rigidity theorem for Riemann's minimal surfaces
Romon, Pascal
Annales de l'Institut Fourier, Tome 43 (1993), p. 485-502 / Harvested from Numdam
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Nous exposons d’abord la structure complexe de la famille de surfaces minimales simplement périodiques découverte par Riemann; elles sont caractérisées comme extensions analytiques des anneaux minimaux bordés par deux droites parallèles dans deux plans parallèles. Nous montrons alors leur unicité en tant que solutions du problème généralisé aux anneaux épointés. Nous présenterons ce faisant les méthodes usuelles de détermination des surfaces minimales simplement périodiques de courbure totale finie, et d’élimination des périodes.

We describe first the analytic structure of Riemann’s examples of singly-periodic minimal surfaces; we also characterize them as extensions of minimal annuli bounded by parallel straight lines between parallel planes. We then prove their uniqueness as solutions of the perturbed problem of a punctured annulus, and we present standard methods for determining finite total curvature periodic minimal surfaces and solving the period problems.

@article{AIF_1993__43_2_485_0,
     author = {Romon, Pascal},
     title = {A rigidity theorem for Riemann's minimal surfaces},
     journal = {Annales de l'Institut Fourier},
     volume = {43},
     year = {1993},
     pages = {485-502},
     doi = {10.5802/aif.1342},
     mrnumber = {94c:53010},
     zbl = {0780.53011},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1993__43_2_485_0}
}

Romon, Pascal. A rigidity theorem for Riemann's minimal surfaces. Annales de l'Institut Fourier, Tome 43 (1993) pp. 485-502. doi : 10.5802/aif.1342. http://gdmltest.u-ga.fr/item/AIF_1993__43_2_485_0/

[1] M. Callahan, D. Hoffman and W. H. Meeks Iii, Embedded minimal surfaces with an infinite number of ends, Invent. Math., 96 (1989), 459-505. | MR 90b:53005 | Zbl 0676.53004

[2] M. Callahan, D. Hoffman and W. H. Meeks Iii, The structure of singly-periodic minimal surfaces, Invent. Math., 99 (1990), 455-581. | MR 92a:53005 | Zbl 0695.53005

[3] D. Hoffman, H. Karcher and H. Rosenberg, Embedded minimal annuli in ℝ3 bounded by a pair of straight lines, Comment. Math. Helvetici, 66 (1991), 599-617. | MR 93d:53012 | Zbl 0765.53004

[4] L. Jorge and W. H. Meeks Iii, The topology of complete minimal surfaces of finite total Gaussian curvature, Topology, Vol 22, no 2 (1983), 203-221. | MR 84d:53006 | Zbl 0517.53008

[5] H. Karcher, Embedded minimal surfaces derived from Scherk's examples, Manuscripta Math., 62 (1988), 83-114. | MR 89i:53009 | Zbl 0658.53006

[6] H. Karcher, Construction of minimal surfaces. Surveys in Geometry, pages 1-96, 1989, University of Tokyo, and Lectures Notes No. 12, SFB256, Bonn, 1989.

[7] H. B. Lawson, Lectures on minimal submanifolds, Vol 1, Math-Lectures Series 9, Publish or Perish. | Zbl 0434.53006

[8] R. Osserman, A survey of minimal surfaces, Van Nostrand Reinhold Math., 25 (1969). | MR 41 #934 | Zbl 0209.52901

[9] J. Pérez and A. Ros, Some uniqueness and nonexistence theorems for embedded minimal surfaces. | Zbl 0789.53004

[10] B. Riemann, Über die Fläche vom kleinsten Inhalt bei gegebener Begrenzung, Abh. Königl., d. Wiss. Göttingen, Mathem. Cl., 13 (1967), 3-52.

[11] E. Toubiana, On the minimal surfaces of Riemann. | Zbl 0787.53005