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.
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.
Accepted:
Published online:
Apala Majumdar 1; J.M. Robbins 2; Maxim Zyskin 3
@article{CRMATH_2009__347_19-20_1159_0, author = {Apala Majumdar and J.M. Robbins and Maxim Zyskin}, title = {Tangent unit-vector fields: {Nonabelian} homotopy invariants and the {Dirichlet} energy}, journal = {Comptes Rendus. Math\'ematique}, pages = {1159--1164}, publisher = {Elsevier}, volume = {347}, number = {19-20}, year = {2009}, doi = {10.1016/j.crma.2009.09.002}, language = {en}, }
TY - JOUR AU - Apala Majumdar AU - J.M. Robbins AU - Maxim Zyskin TI - Tangent unit-vector fields: Nonabelian homotopy invariants and the Dirichlet energy JO - Comptes Rendus. Mathématique PY - 2009 SP - 1159 EP - 1164 VL - 347 IS - 19-20 PB - Elsevier DO - 10.1016/j.crma.2009.09.002 LA - en ID - CRMATH_2009__347_19-20_1159_0 ER -
%0 Journal Article %A Apala Majumdar %A J.M. Robbins %A Maxim Zyskin %T Tangent unit-vector fields: Nonabelian homotopy invariants and the Dirichlet energy %J Comptes Rendus. Mathématique %D 2009 %P 1159-1164 %V 347 %N 19-20 %I Elsevier %R 10.1016/j.crma.2009.09.002 %G en %F CRMATH_2009__347_19-20_1159_0
Apala Majumdar; J.M. Robbins; Maxim Zyskin. Tangent unit-vector fields: Nonabelian homotopy invariants and the Dirichlet energy. Comptes Rendus. Mathématique, Volume 347 (2009) no. 19-20, pp. 1159-1164. doi : 10.1016/j.crma.2009.09.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.09.002/
[1] Harmonic maps with defects, Comm. Math. Phys., Volume 107 (1986), pp. 649-705
[2] The Physics of Liquid Crystals, Clarendon Press, Oxford, 1974
[3] Harmonic mappings of Riemannian manifolds, Amer. J. Math., Volume 86 (1964), pp. 109-160
[4] Controllable alignment of nematic liquid crystals around microscopic posts: Stabilization of multiple states, Appl. Phys. Lett., Volume 80 (2002), pp. 3635-3637
[5] Combinatorial Group Theory, Dover, 1976
[6] Lower bounds for energies of harmonic tangent unit-vector fields on convex polyhedra, Lett. Math. Phys., Volume 70 (2004), pp. 169-183
[7] Elastic energy of liquid crystals in convex polyhedra, J. Phys. A: Math. Gen., Volume 37 (2004), p. L573-L580
[8] Elastic energy for reflection-symmetric topologies, J. Phys. A: Math. Gen., Volume 39 (2006), pp. 2673-2687
[9] Tangent unit-vector fields: Nonabelian homotopy invariants and the Dirichlet energy (2009, in preparation; preprint) | arXiv
[10] Classification of unit-vector fields in convex polyhedra with tangent boundary conditions, J. Phys. A: Math. Gen., Volume 37 (2004), pp. 10609-10623
[11] A Comprehensive Introduction to Differential Geometry, vol. 2, Publish or Perish Press, Berkeley, CA, 1990
Cited by Sources:
Comments - Policy