**Memoires de la Societe Mathematique de France**

Volume: 133;
2013;
116 pp;
Softcover

MSC: Primary 53; 34; 32;
**Print ISBN: 978-2-85629-766-7
Product Code: SMFMEM/133**

List Price: $48.00

AMS Member Price: $38.40

# Problème de Plateau, Équations Fuchsiennes et Problème de Riemann-Hilbert

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*Laura Desideri*

A publication of the Société Mathématique de France

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.

#### Readership

Graduate students and research mathematicians.