Abstract
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.
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Acknowledgements
I would like to thank André Neves and Guangbo Xu for helpful conversations related to this work. Thanks also go to the referee for many helpful comments.
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Cheng, D.R. Stable Solutions to the Abelian Yang–Mills–Higgs Equations on \(S^2\) and \(T^2\). J Geom Anal 31, 9551–9572 (2021). https://doi.org/10.1007/s12220-021-00619-y
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DOI: https://doi.org/10.1007/s12220-021-00619-y