Energies of $S^2$-valued harmonic maps on polyhedra with tangent boundary conditions

Abstract

A unit-vector field $\mathbf{n}\text{:}P\to S^2$ on a convex polyhedron $P\subset {\mathbb{R}}^3$ satisfies tangent boundary conditions if, on each face of P, n takes values tangent to that face. Tangent unit-vector fields are necessarily discontinuous at the vertices of P. We consider fields which are continuous elsewhere. We derive a lower bound $E_P^{-}(h)$ for the infimum Dirichlet energy $E_P^{\inf }(h)$ for such tangent unit-vector fields of arbitrary homotopy type h. $E_P^{-}(h)$ is expressed as a weighted sum of minimal connections, one for each sector of a natural partition of $S^2$ induced by P. For P a rectangular prism, we derive an upper bound for $E_P^{\inf }(h)$ whose ratio to the lower bound may be bounded independently of h. The problem is motivated by models of nematic liquid crystals in polyhedral geometries. Our results improve and extend several previous results.