Radial symmetry on three-dimensional shells in the Landau–de Gennes theory

• The radial-hedgehog solution is a Landau–de Gennes critical point on a 3D shell.
• We prove global minimality of the radial-hedgehog solution for thin shells.
• We prove global minimality of the radial-hedgehog solution for low temperatures.

We study the radial-hedgehog solution on a three-dimensional (3D) spherical shell with radial boundary conditions, within the Landau–de Gennes theory for nematic liquid crystals. We prove that the radial-hedgehog solution is the unique minimizer of the Landau–de Gennes energy in two separate regimes: (i) for thin shells when the temperature is below the critical nematic supercooling temperature and (ii) for a fixed shell width at sufficiently low temperatures. In case (i), we provide explicit geometry-dependent criteria for the global minimality of the radial-hedgehog solution.