  We introduce a functional framework taylored to investigate the minimality
and stability properties of the Ginzburg-Landau vortex of degree one on the
whole plane. We prove that a renormalized Ginzburg-Landau energy is
well-defined in that framework and that the vortex is its unique global
minimizer up to the invariances by translation and phase shift. Our main result
is a nonlinear coercivity estimate for the renormalized energy around the
vortex, from which we can deduce its orbital stability as a solution to the
Gross-Pitaevskii equation, the natural Hamiltonian evolution equation
associated to the Ginzburg-Landau energy.
