  We study the radial-hedgehog solution in a three-dimensional spherical
droplet, with homeotropic boundary conditions, within the Landau-de Gennes
theory for nematic liquid crystals. The radial-hedgehog solution is a candidate
for a globally stable configuration in this model framework and is also a
prototype configuration for studying isolated point defects in condensed matter
physics. The static properties of the radial-hedgehog solution are governed by
a nonlinear singular ordinary differential equation. We consider two different
limits separately - the \emph{vanishing core limit} and \emph{low-temperature
limit} respectively. We use a combination of Ginzburg-Landau techniques,
perturbation methods and stability analysis to study the qualitative properties
of the radial-hedgehog solution, both in the vicinity of and away from the
defect core. We establish the instability of the radial-hedgehog solution with
respect to biaxial perturbations in certain parameter regimes and demonstrate
the stability of the radial-hedgehog solution in other parameter regimes. Our
results complement previous work in the field, are rigorous in nature and give
information about the role of geometry and temperature on the properties of the
radial-hedgehog solution and the associated biaxial instabilities.
