We consider nematic liquid crystals in a bounded, convex polyhedron described by a director field n(r) subject to tangent boundary conditions. We derive lower bounds for the one-constant elastic energy in terms of topological invariants. For a right rectangular prism and a large class of topologies, we derive upper bounds by introducing, test configurations constructed from local conformal solutions of the Euler-Lagrange equation. The ratio of the upper and lower bounds depends only on the aspect ratios of the prism. As the aspect ratios are varied, the minimum-energy conformal state undergoes a sharp transition from being smooth to having singularities on the edges.
Majumdar, A., Robbins, J. M., & Zyskin, M. (2004). Elastic energy of liquid crystals in convex polyhedra. Journal of Physics A: Mathematical and General, 37(44), [L573]. https://doi.org/10.1088/0305-4470/37/44/L05