Majumdar, A. and Robbins, J M and Zyskin, M (2009) Tangent unitvector fields: nonabelian homotopy invariants and the Dirichlet energy. Not specified . (Submitted)

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Abstract
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 compute the infimum Dirichlet energy, E(H), for continuous maps satisfying tangent boundary conditions 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 ntimes punctured twosphere, π1 (S 2 − , ∗). The lower bound for E(H) is obtained from combinatorial group theory arguments, while the upper bound is obtained by constructing explicit representatives which, on all but an arbitrarily small subset of O, are alternatively locally conformal or anticonformal. For conformal and anticonformal classes (classes containing wholly conformal and anticonformal representatives respectively), the expression for E(H) reduces to a previous result involving the degrees of a set of regular values s1 , . . . , sn in the target S 2 space. These degrees may be viewed as invariants associated with the abelianization of π1 (S 2 − , ∗). For nonconformal classes, however, E(H) may be strictly greater than the abelian bound. This stems from the fact that, for nonconformal maps, the number of preimages of certain regular values may necessarily be strictly greater than the absolute value of their degrees.
This work is motivated by the theoretical modelling of nematic liquid crystals in confined polyhedral geometries. The results imply new lower and upper bounds for the Dirichlet energy (oneconstant OseenFrank energy) of reflectionsymmetric tangent unitvector fields in a rectangular prism.
Item Type:  Article 

Subjects:  D  G > General 
Research Groups:  Oxford Centre for Collaborative Applied Mathematics 
ID Code:  844 
Deposited By:  Dr M. Stoll 
Deposited On:  16 Oct 2009 06:36 
Last Modified:  29 May 2015 18:32 
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