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 nonlinear 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.
KeywordsLiquid Crystal Singular Point Boundary Energy Finite Energy Cayley Tree
Unable to display preview. Download preview PDF.
- [ABL]F. Almgren, W. Browder and E. Lieb, Co-area, liquid crystals and minimal surfaces, in Partial Differential Equations, ed. S.S. Chern, Springer Lecture Notes in Math. 1306, 1–12 (1988).Google Scholar
- [AL]F. Almgren and E. Lieb, Singularities of energy minimizing maps from the ball to the sphere: examples counterexamples and bounds, Ann. of Math., Nov. 1988. See also Singularities of energy minimizing maps from the ball to the sphere, Bull. Amer. Math. Soc. 17, 304–306 (1987).MathSciNetzbMATHGoogle Scholar
- [HL2]R. Hardt and F. H. Lin, Stability of singularities of minimizing harmonic maps,J. Diff. Geom., to appear.Google Scholar
- [K]M. KlémanPoints, lignes, parois dans les fluides anisotropes et les solides cristalline Les Éditions de Physique (Orsay), I, 36–37.Google Scholar