Decay of excess for the abelian Higgs model

Abstract

We prove that entire critical points $(u,\nabla )$ of the self-dual $U(1)$-Yang–Mills–Higgs functional $E_1$, with energy $E_1(u,\nabla ;B_R)\le (1+\tau (n))\cdot 2\pi {\omega }_{n-2}{R}^{n-2}$ for all $R>0$, have unique blow-down. Moreover, we show that they are two-dimensional in ambient dimension $2\le n\le 4$, or in any dimension $n\ge 2$ assuming that $(u,\nabla )$ is a local minimizer, thus establishing a codimension-two analogue of Savin’s theorem. The main ingredient is an Allard-type improvement of flatness.