  We study uniaxial energy-minimizers within the Landau-de Gennes theory for
nematic liquid crystals on a three-dimensional spherical droplet subject to
homeotropic boundary conditions. We work in the low-temperature regime and show
that uniaxial energy-minimizers necessarily have the structure of the
well-studied radial-hedgehog solution in the low-temperature limit. An
immediate consequence of this result is that Landau-de Gennes energy minimizers
cannot be purely uniaxial for sufficiently low temperatures.
