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Comptes Rendus Mathématique
Volume 347, n° 19-20
pages 1159-1164 (octobre 2009)
Doi : 10.1016/j.crma.2009.09.002
Received : 3 April 2009 ;  accepted : 12 July 2009
Tangent unit-vector fields: Nonabelian homotopy invariants and the Dirichlet energy
Champs de vecteurs unités tangents : Invariants homotopiques non abelien et énergie de Dirichlet
 

Apala Majumdar a , J.M. Robbins b , Maxim Zyskin c
a Mathematical Institute, University of Oxford, 24 - 29 St. Giles, Oxford OX1 3LB, UK 
b School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK 
c Department of Mathematics, SETB 2.454 - 80 Fort Brown, Brownsville, TX 78520, USA 

Abstract

Let O be a closed geodesic polygon in  . Maps from O into   are said to satisfy tangent boundary conditions if the edges of O are mapped into the geodesics which contain them. Taking O to be an octant of  , we evaluate the infimum Dirichlet energy,  , for continuous tangent maps of arbitrary homotopy type H . The expression for   involves a topological invariant – the spelling length – associated with the (nonabelian) fundamental group of the n -times punctured two-sphere,  . These results have applications for the theoretical modelling of nematic liquid crystal devices. To cite this article: A. Majumdar et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).

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Résumé

Soit O un polygone géodésique fermé de  . On dit qu’une application de O dans   vérifie des conditions aux limites tangentes si elle associe à chaque côté de O la géodésique qui le contient. Dans le cas où O est un octant de  , on calcule l’infimum d’énergie de Dirichlet,  , pour des applications tangentes continues d’un type d’homotopie quelconque H . L’expression de   utilise un invariant topologique, la longueur nominale, lié au groupe fondamentel (non abélien) de la sphère   à n trous ponctuels,  . Les réeultats obtenus ont des applications pratiques, notamment dans la modélisation des systèmes contenant de cristaux liquides nématiques. Pour citer cet article : A. Majumdar et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).

The full text of this article is available in PDF format.


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