We show under natural assumptions that stable solutions to the abelian Yang–Mills–Higgs equations on Hermitian line bundles over the round 2-sphere actually satisfy the vortex equations, which are a first-order reduction of the (second-order) abelian Yang–Mills–Higgs equations. We also obtain a similar result for stable solutions on a flat 2-torus. Our method of proof comes from the work of Bourguignon–Lawson (Commun Math Phys 79(2):189–230, 1981) concerning stable SU(2) Yang–Mills connections on compact homogeneous 4-manifolds.

中文翻译：

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$$ S ^ 2 $$ S 2和$$ T ^ 2 $$ T 2上的Abelian Yang-Mills-Higgs方程的稳定解

在自然假设下，我们证明了圆形2球面上Hermitian线束上的Abelian Yang-Mills-Higgs方程的稳定解实际上满足涡旋方程，这是（二阶）Abelian Yang-的一阶约简。 Mills–Higgs方程。我们还获得了在平直的2阶上稳定解决方案的相似结果。我们的证明方法来自于Bourguignon-Lawson（Commun Math Phys 79（2）：189-230，1981）关于稳定SU（2）Yang-Mills在紧凑的同构4流形上的连接的工作。