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

Let O be a closed geodesic polygon in S 2 . 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 S 2 , we compute the infimum Dirichlet energy, E(H), for continuous maps satisfying tangent boundary conditions 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 }, ∗). The lower bound for E(H) is obtained from combinatorial group theory arguments, while the upper bound is obtained by constructing explicit representatives which, on all but an arbitrarily small subset of O, are alternatively locally conformal or anticonformal. For conformal and anticonformal classes (classes containing wholly conformal and anticonformal representatives respectively), the expression for E(H) reduces to a previous result involving the degrees of a set of regular values s1 , . . . , sn in the target S 2 space. These degrees may be viewed as invariants associated with the abelianization of π1 (S 2 − {s1 , . . . , sn }, ∗). For nonconformal classes, however, E(H) may be strictly greater than the abelian bound. This stems from the fact that, for nonconformal maps, the number of preimages of certain regular values may necessarily be strictly greater than the absolute value of their degrees.\ud \ud This work is motivated by the theoretical modelling of nematic liquid crystals in confined polyhedral geometries. The results imply new lower and upper bounds for the Dirichlet energy (one-constant Oseen-Frank energy) of reflection-symmetric tangent unit-vector fields in a rectangular prism

Lower bound for energies of harmonic tangent unit-vector fields on convex polyhedra

We derive a lower bound for energies of harmonic maps of convex polyhedra in $\R^3$ to the unit sphere $S^2,$ with tangent boundary conditions on the faces. We also establish that $C^\infty$ maps, satisfying tangent boundary conditions, are dense with respect to the Sobolev norm, in the space of continuous tangent maps of finite energy.Comment: Acknowledgment added, typos removed, minor correction

Geometric analysis of optical frequency conversion and its control in quadratic nonlinear media

We analyze frequency conversion and its control among three light waves using a geometric approach that enables the dynamics of the waves to be visualized on a closed surface in three dimensions. It extends the analysis based on the undepleted-pump linearization and provides a simple way to understand the fully nonlinear dynamics. The Poincaré sphere has been used in the same way to visualize polarization dynamics. A geometric understanding of control strategies that enhance energy transfer among interacting waves is introduced, and the quasi-phase-matching strategy that uses microstructured quadratic materials is illustrated in this setting for both type I and II second-harmonic generation and for parametric three-wave interactions

Domain wall motion in ferromagnetic nanowires driven by arbitrary time-dependent fields: An exact result

We address the dynamics of magnetic domain walls in ferromagnetic nanowires under the influence of external time-dependent magnetic fields. We report a new exact spatiotemporal solution of the Landau-Lifshitz-Gilbert equation for the case of soft ferromagnetic wires and nanostructures with uniaxial anisotropy. The solution holds for applied fields with arbitrary strength and time dependence. We further extend this solution to applied fields slowly varying in space and to multiple domain walls.Comment: 3 pages, 1 figur

Geometric phases and anholonomy for a class of chaotic classical systems

Berry's phase may be viewed as arising from the parallel transport of a quantal state around a loop in parameter space. In this Letter, the classical limit of this transport is obtained for a particular class of chaotic systems. It is shown that this ``classical parallel transport'' is anholonomic --- transport around a closed curve in parameter space does not bring a point in phase space back to itself --- and is intimately related to the Robbins-Berry classical two-form.Comment: Revtex, 11 pages, no figures