  We study global minimizers of the Landau-de Gennes (LdG) energy functional
for nematic liquid crystals, on arbitrary three-dimensional simply connected
geometries with topologically non-trivial and physically relevant Dirichlet
boundary conditions. Our results are specific to the low-temperature limit. We
prove (i) that (re-scaled) global LdG minimizers converge uniformly to a
(minimizing) limiting harmonic map, away from the singular set of the limiting
map; (ii) there exist both a point of maximal biaxiality and a nonempty
Lebesgue-null set of uniaxial points near each singular point of the limiting
harmonic map (this improves the recent results of \cite{contreraslamy});
  (iii) estimates for the size of "strongly biaxial" regions in terms of the
reduced temperature $t$. We further show that global LdG minimizers in the
restricted class of uniaxial $\Qvec$-tensors cannot be stable critical points
of the LdG energy for low temperatures.
