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

Journal article

Let O be a closed geodesic polygon in S2 . 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 S2 , 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, π1(S 2 −{s1 , . . . , sn },∗). These results have applications for the theoretical modelling of nematic liquid crystal devices.

Authors: Majumdar, A; Robbins, J; Zyskin, M

• Publication date: 2009-01-01