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Small deformations of polygons and polyhedra


Author: Jean-Marc Schlenker
Journal: Trans. Amer. Math. Soc. 359 (2007), 2155-2189
MSC (2000): Primary 53C45, 53C50, 51M16
Published electronically: December 20, 2006
MathSciNet review: 2276616
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Abstract | References | Similar Articles | Additional Information

Abstract: We describe the first-order variations of the angles of Euclidean, spherical or hyperbolic polygons under infinitesimal deformations such that the lengths of the edges do not change. Using this description, we introduce a vector-valued quadratic invariant $ b$ on the space of those isometric deformations which, for convex polygons, has a remarkable positivity property.

We give two geometric applications. The first is an isoperimetric statement for hyperbolic polygons: Among the convex hyperbolic polygons with given edge lengths, there is a unique polygon with vertices on a circle, a horocycle, or on one connected component of the space of points at constant distance from a geodesic, and it has maximal area. The second application is a rigidity result for equivariant polyhedral surfaces in the Minkowski space.

ESUMÉ. On décrit les déformations infinitésimales des angles d'un polygone euclidien, sphérique ou hyperbolique sous les déformations infinitésimales qui préservent les longueurs des arêtes. On en déduit la définition d'un invariant quadratique à valeurs vectorielles $ b$ sur l'espace de ces déformations isométriques qui, pour les polygones convexes, a une propriété remarquable de positivité.

On donne deux applications géométriques. La première est un énoncé isoperimétrique pour les polygones hyperboliques: Parmi les polygones hyperboliques convexes dont les longueurs des arêtes sont données, il existe un unique élément dont les sommets sont sur un cercle, un horocycle, ou dans une composante connexe de l'ensemble des points à distance constante d'une géodésique, et son aire est maximale. La seconde application est un résultat de rigidité pour les surfaces polyèdrales équivariantes dans l'espace de Minkowski.


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Additional Information

Jean-Marc Schlenker
Affiliation: Laboratoire Emile Picard, UMR CNRS 5580, UFR MIG, Université Paul Sabatier, 31062 Toulouse Cedex 9, France
Email: schlenker@math.ups-tlse.fr

DOI: https://doi.org/10.1090/S0002-9947-06-04172-9
Received by editor(s): March 8, 2005
Published electronically: December 20, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.