In this paper, we consider SU(2) monopoles on an asymptotically conical, oriented, Riemannian 3-manifold with one end. The connected components of the moduli space of monopoles in this setting are labeled by an integer called the charge. We analyze the limiting behavior of sequences of monopoles with fixed charge, and whose sequence of Yang-Mills-Higgs energies is unbounded. We prove that the limiting behavior of such monopoles is characterized by energy concentration along a certain set, which we call the blow-up set. Our work shows that this set is finite, and using a bubbling analysis obtains effective bounds on its cardinality, with such bounds depending solely on the charge of the monopole. Moreover, for such sequences of monopoles there is another naturally associated set, the zero set, which consists of the set at which the zeros of the Higgs fields accumulate. Regarding this, our results show that for such sequences of monopoles, the zero set and the blow-up set coincide. In particular, proving that in this 'large mass' limit, the zero set is a finite set of points.Some of our work extends for sequences of finite mass critical points of the Yang-Mills-Higgs functional for which the Yang-Mills-Higgs energies are O(mi) as i → ∞, where mi are the masses of the configurations.
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This article investigates the asymptotics of G 2 -monopoles. First, we find that when the underlying G 2 -manifold has polynomial volume growth strictly greater than r 7/2 , finite intermediate energy monopoles with bounded curvature have finite mass.The second main result restricts to the case when the underlying G 2 -manifold is asymptotically conical. In this situation, we deduce sharp decay estimates and that the connection converges, along the end, to a pseudo-Hermitian-Yang-Mills over the asymptotic cone. Contents 1. Introduction 1 2. Preliminaries 6 3. Consequences of Moser iteration and ε-regularity 10 4. Finite mass from finite intermediate energy 13 5. Bochner/Weitzenböck formulas along the end 19 6. Refined asymptotics in the AC case 25 7. Boundary data 36 8. Decay of linearized solutions 39 References 42
Ao meu orientador, Henrique N. Sá Earp, por me introduzir a uma área de pesquisa tão rica e cativante, pela paciência, inúmeros conselhos, e toda ajuda como acadêmico e amigo.Aos professores da banca examinadora, Andrew Clarke e Marcos Jardim, pelo tempo prestado na avaliação deste trabalho e as decorrentes sugestões e correções que ajudaram o mesmo a se tornar mais apresentável.Aos meus familiares -particularmente, aos meus pais -por todo apoio fundacional, sem o qual nada disto seria possível. À minha namorada, Bianca Fujita Castilho, por todo carinho, companheirismo, paciência e incentivo diário.
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