Counting Singularities in Liquid Crystals

Energy minimizing harmonic maps from the ball to the sphere arise in the study of liquid crystal geometries and in the classical nonhnear sigma model. We linearly dominate the number of points of discontinuity of such a map by the energy of its boundary value function. Our bound is optimal (modulo the best constant) and is the first bound of its kind. We also show that the locations and numbers of singular points of minimizing maps is often counterintuitive; in particular, boundary symmetries need not be respected.