  We study the stability of 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 show that the radial-hedgehog
solution has no zeroes for a sufficiently narrow shell, for all temperatures
below the nematic supercooling temperature. We prove that the radial-hedgehog
solution is the unique global Landau-de Gennes energy minimizer for a
sufficiently narrow 3D spherical shell, for all temperatures below the nematic
supercooling temperature. We provide explicit geometry-dependent criteria for
the global minimality of the radial-hedgehog solution in this temperature
regime. In the low temperature limit, we prove the global minimality of the
radial-hedgehog solution on a 3D spherical shell, for all values of the inner
and outer radii.
