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Stable solutions in the abelian Higgs model

  • Marco Badran
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Abstract

We classify entire stable solutions with quadratic energy growth in the $4$-dimensional abelian Higgs model. More precisely, we prove that such solutions are holomorphic with respect to some complex structure, namely they satisfy the first order equations introduced by Bradlow, and consequently their nodal sets are holomorphic curves. Conversely, we construct solutions whose nodal set is any prescribed holomorphic curve with quadratic area growth. As a corollary, we show that all stable solutions with linear energy growth in ℝ3 are two-dimensional Bogomolnyi vortices.

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Cite

BibTeXRIS
BibTeX
@misc{badran2026stable,
  title = {{Stable solutions in the abelian Higgs model}},
  author = {Marco Badran},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.11647},
  eprint = {2609.11647},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.11647},
}
RIS
TY  - UNPB
TI  - Stable solutions in the abelian Higgs model
AU  - Marco Badran
PY  - 2026
DA  - 2026/09/10
PB  - arXiv
DO  - 10.48550/arxiv.2609.11647
UR  - https://arxiv.org/abs/2609.11647
AB  - We classify entire stable solutions with quadratic energy growth in the $4$-dimensional abelian Higgs model. More precisely, we prove that such solutions are holomorphic with respect to some complex structure, namely they satisfy the first order equations introduced by Bradlow, and consequently their nodal sets are holomorphic curves. Conversely, we construct solutions whose nodal set is any prescribed holomorphic curve with quadratic area growth. As a corollary, we show that all stable solutions with linear energy growth in ℝ^3 are two-dimensional Bogomolnyi vortices.
ER  -