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 $P$, $\mathbf{n}$ 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.