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

TL;DR

This paper proves that in three-dimensional space, local minimizers of the Ginzburg-Landau energy with sufficiently small energy growth are necessarily constant, using a novel sharp eta-ellipticity result.

Contribution

It introduces a new sharp eta-ellipticity theorem for Ginzburg-Landau minimizers in three dimensions, leading to a classification of low-energy minimizers.

Findings

Minimizers with energy growth below a specific threshold are constant.

Established a new sharp eta-ellipticity result for 3D Ginzburg-Landau minimizers.

Provided a criterion linking energy growth to the triviality of minimizers.

Abstract

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