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

Abstract

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