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Mémoires de la Société Mathématique de France
2013; 116 pp; softcover
List Price: US$48
Member Price: US$38.40
Order Code: SMFMEM/133
A note to readers: This book is in French.
This dissertation is devoted to the resolution of the Plateau problem in the case of a polygonal boundary in the three-dimensional euclidean space. It relies on a method developed by René Garnier and published in 1928 in a paper which seems today to be totally forgotten. Even if Garnier's method is more geometrical and constructive than the variational one, it is sometimes really complicated, and even obscure or incomplete. The authors rewrite his proof with a modern formalism, fill some gaps, and propose some alternative easier proofs.
This work mainly relies on a systematic use of Fuchsian systems and on the relation that we establish between the reality of such systems and their monodromy. Garnier's method is based on the following result: using the spinorial Weierstrass representation for minimal surfaces, the authors can associate to each minimal disk with a polygonal boundary a real Fuchsian second order equation defined on the Riemann sphere. The monodromy of the equation is encoded by the oriented directions of the edges of the boundary.
To solve the Plateau problem, the authors are thus led to solve a Riemann-Hilbert problem. Then, they proceed in two steps: first, by means of isomonodromic deformations, they construct the family of all minimal disks with a polygonal boundary with given oriented directions. Then, by studying the edges's lengths of these polygonal boundaries, they show that every polygon is the boundary of a minimal disk.
A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.
Graduate students and research mathematicians.
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