Nematic liquid crystals and harmonic maps on polyhedra

Abstract:

States of nematic liquid crystal in polyhedral cells
are described by director fields (ie maps to RP^2)
which are local minima of the Frank energy,  subject
to tangent (or periodic, or normal) boundary
conditions on faces. In the so-called one constant
approximation, the states are harmonic maps. Tangent
boundary conditions disallow continuity at the
vertices. Regular states are those which are
continuous away from the vertices. We obtain complete
topological classification of regular states in terms
of homotopy invariants. We obtain lower energy bounds
as a function of homotopy invariants, and for the case
of a rectangular prism upper bounds proportional to
lower bounds (with a topology and geometry independent
factor). An interesting question is the number of
regular states. Analytic and numerical evidence
suggests that in most homotopic classes minimum is not
achieved, and there are just few regular states. Our
work has applications to new types of liquid crystal
displays, bi-stable or multi stable displays.