Tangent unit-vector fields: Nonabelian homotopy invariants and the Dirichlet energy

Majumdar, Apala; Robbins, J.M.; Zyskin, Maxim

doi:10.1016/j.crma.2009.09.002

Partial Differential Equations/Topology

Tangent unit-vector fields: Nonabelian homotopy invariants and the Dirichlet energy
[Champs de vecteurs unités tangents : Invariants homotopiques non abelien et énergie de Dirichlet]

Présenté par : Haïm Brezis

Majumdar, Apala ¹ ; Robbins, J.M.² ; Zyskin, Maxim ³

¹ Mathematical Institute, University of Oxford, 24 - 29 St. Giles, Oxford OX1 3LB, UK
² School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK
³ Department of Mathematics, SETB 2.454 - 80 Fort Brown, Brownsville, TX 78520, USA

Comptes Rendus. Mathématique, Tome 347 (2009) no. 19-20, pp. 1159-1164.

Résumé
Abstract

Soit O un polygone géodésique fermé de $S^{2}$ . On dit qu'une application de O dans $S^{2}$ 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 $S^{2}$ , on calcule l'infimum d'énergie de Dirichlet, $E (H)$ , pour des applications tangentes continues d'un type d'homotopie quelconque H. L'expression de $E (H)$ utilise un invariant topologique, la longueur nominale, lié au groupe fondamentel (non abélien) de la sphère $S^{2}$ à n trous ponctuels, $π_{1} (S^{2} - {s_{1}, \dots, s_{n}}, *)$ . Les réeultats obtenus ont des applications pratiques, notamment dans la modélisation des systèmes contenant de cristaux liquides nématiques.

Let O be a closed geodesic polygon in $S^{2}$ . Maps from O into $S^{2}$ 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 $S^{2}$ , we evaluate the infimum Dirichlet energy, $E (H)$ , for continuous tangent maps of arbitrary homotopy type H. The expression for $E (H)$ involves a topological invariant – the spelling length – associated with the (nonabelian) fundamental group of the n-times punctured two-sphere, $π_{1} (S^{2} - {s_{1}, \dots, s_{n}}, *)$ . These results have applications for the theoretical modelling of nematic liquid crystal devices.

Reçu le : 2009-04-03
Accepté le : 2009-07-12
Publié le : 2009-09-25

DOI : 10.1016/j.crma.2009.09.002

Affiliations des auteurs :

Majumdar, Apala ¹ ; Robbins, J.M. ² ; Zyskin, Maxim ³

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     title = {Tangent unit-vector fields: {Nonabelian} homotopy invariants and the {Dirichlet} energy},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1159--1164},
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Majumdar, Apala; Robbins, J.M.; Zyskin, Maxim. Tangent unit-vector fields: Nonabelian homotopy invariants and the Dirichlet energy. Comptes Rendus. Mathématique, Tome 347 (2009) no. 19-20, pp. 1159-1164. doi : 10.1016/j.crma.2009.09.002. http://www.numdam.org/articles/10.1016/j.crma.2009.09.002/

Bibliographie
Cité par

[1] Brezis, H.; Coron, J.M.; Lieb, E.H. Harmonic maps with defects, Comm. Math. Phys., Volume 107 (1986), pp. 649-705

[2] De Gennes, P.G. The Physics of Liquid Crystals, Clarendon Press, Oxford, 1974

[3] Eells, J.; Sampson, J.H. Harmonic mappings of Riemannian manifolds, Amer. J. Math., Volume 86 (1964), pp. 109-160

[4] Kitson, S.; Geisow, A. Controllable alignment of nematic liquid crystals around microscopic posts: Stabilization of multiple states, Appl. Phys. Lett., Volume 80 (2002), pp. 3635-3637

[5] Magnus, W.; Karras, A.; Solitar, D. Combinatorial Group Theory, Dover, 1976

[6] Majumdar, A.; Robbins, J.M.; Zyskin, M. Lower bounds for energies of harmonic tangent unit-vector fields on convex polyhedra, Lett. Math. Phys., Volume 70 (2004), pp. 169-183

[7] Majumdar, A.; Robbins, J.M.; Zyskin, M. Elastic energy of liquid crystals in convex polyhedra, J. Phys. A: Math. Gen., Volume 37 (2004), p. L573-L580

[8] Majumdar, A.; Robbins, J.M.; Zyskin, M. Elastic energy for reflection-symmetric topologies, J. Phys. A: Math. Gen., Volume 39 (2006), pp. 2673-2687

[9] Majumdar, A.; Robbins, J.M.; Zyskin, M. Tangent unit-vector fields: Nonabelian homotopy invariants and the Dirichlet energy (2009, in preparation; preprint) | arXiv

[10] Robbins, J.M.; Zyskin, M. Classification of unit-vector fields in convex polyhedra with tangent boundary conditions, J. Phys. A: Math. Gen., Volume 37 (2004), pp. 10609-10623

[11] Spivak, M. A Comprehensive Introduction to Differential Geometry, vol. 2, Publish or Perish Press, Berkeley, CA, 1990

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