Asymptotics of Finite Energy Monopoles on AC 3-Manifolds

Autor: Daniel Fadel
Rok vydání: 2022
Předmět:
Zdroj: Daniel Fadel
ISSN: 1559-002X
1050-6926
DOI: 10.1007/s12220-022-01095-8
Popis: We study the asymptotic behavior of finite energy $\rm{SU}(2)$ monopoles, and general critical points of the $\rm{SU}(2)$ Yang--Mills--Higgs energy, on asymptotically conical $3$-manifolds with only one end. Our main results generalize classical results due to Groisser and Taubes in the particular case of the flat $3$-dimensional Euclidean space $\mathbb{R}^3$. Indeed, we prove the integrality of the monopole number, or charge, of finite energy configurations, and derive the classical energy formula establishing monopoles as absolute minima. Moreover, we prove that the covariant derivative of the Higgs field of a critical point of the energy decays quadratically along the end, and that its transverse component with respect to the Higgs field, as well as the corresponding component of the curvature of the underlying connection, actually decay exponentially. Additionally, under the assumption of positive Gaussian curvature on the asymptotic link, we prove that the curvature of any critical point connection decays quadratically. Furthermore, we deduce that any irreducible critical point converges uniformly along the conical end to a limiting configuration at infinity consisting of a reducible Yang--Mills connection and a parallel Higgs field.
v4: 71 pages; Final version to appear in The Journal of Geometric Analysis; corrected minor mistakes and typos, and some proofs were expanded and further clarified
Databáze: OpenAIRE
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