Manchester Geometry Seminar 2003/2004

Homotopy classes of unit-vector fields in convex polyhedra, subject to tangent boundary conditions, are classified by a complete set of invariants - edge orientations, kink numbers and wrapping numbers, subject to certain sum rules. We obtain a lower bound for the minimum energy of harmonic maps in a fixed homotopy class, in terms of the corresponding invariants. These results are applied to the unit cube, where we also obtain an upper bound for the minimum energy in a family of topological types. We also discuss results of numerical computations of minimal energies, with representative maps used as initial configurations. Our study is motivated in part by problems in liquid crystal display design. This is joint work with C. P. Newton, J. M. Robbins and M. Zyskin.