  In the present work, we study minimizers of the Landau-de Gennes free energy
in a bounded domain $\Omega\subset \mathbb{R}^3$. We prove that at low
temperature minimizers do not vanish, even for topologically non-trivial
boundary conditions. This is in contrast with a simplified Ginzburg-Landau
model for superconductivity studied by Bethuel, Brezis and H\'elein. Merging
this with an observation of Canevari we obtain, as a corollary, the occurence
of biaxial escape: the tensorial order parameter must become strongly biaxial
at some point in $\Omega$. In particular, while it is known that minimizers
cannot be purely uniaxial, we prove the much stronger and physically relevant
fact that they lie in a different homotopy class.
