Small energy Ginzburg-Landau minimizers in ${\mathbb{R}}^3$

We prove that a local minimizer of the Ginzburg-Landau energy in ${\mathbb R}^3$satisfyingthecondition$\liminf_{R\to\infty}E(u;B_R)/RlnR < 2\pi$ must be constant. The main tool is a new sharp eta-ellipticity result for minimizers in dimension three that might be of independent interest.