A rigidity theorem for Riemann's minimal surfaces
Annales de l'Institut Fourier, Volume 43 (1993) no. 2, p. 485-502

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.

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.

@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},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {43},
     number = {2},
     year = {1993},
     pages = {485-502},
     doi = {10.5802/aif.1342},
     zbl = {0780.53011},
     mrnumber = {94c:53010},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1993__43_2_485_0}
}
Romon, Pascal. A rigidity theorem for Riemann's minimal surfaces. Annales de l'Institut Fourier, Volume 43 (1993) no. 2, pp. 485-502. doi : 10.5802/aif.1342. http://www.numdam.org/item/AIF_1993__43_2_485_0/

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