Monopoles on AC 3-manifolds

Oliveira, Gonçalo

doi:10.1112/jlms/jdw017

Cited by 8 publications

(27 citation statements)

References 11 publications

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“…The second example uses Taubes' construction of multi-monopoles on R 3 to produce sequences of charge k 1 monopoles with unbounded masses, such that the corresponding zero set Z is any a priori prescribed set of l pairwise distinct points in X, for any given 1 l k. Next, we include a simple general way to produce, from a given charge k > 1 monopole, examples of sequences of charge k monopoles in R 3 with unbounded masses and for which the zero set Z = {0} and the charge k 0 of the bubble at the origin equals k > 1. Finally, using the multi-monopole construction of [17] in the more general setting of an AC 3-manifold (X 3 , g) with b 2 (X) = 0, we construct sequences of charge k monopoles with unbounded masses whose zero set is any a priori prescribed set of k pairwise distinct points in X.…”

Section: Motivating Examplesmentioning

confidence: 99%

“…The proof of assertion (3.10) In this section, we shall prove assertion (3.10), which says that the zeros of the monopoles constructed via Theorem 3.2 are contained in balls of radius 10m −1/2 around the k-points in X used in the construction. This requires a number of technical ingredients from [17], and so we decided to include this section as an appendix. It follows from [17,Proposition 6] that the monopole (A i , Φ i ) can be written as…”

Section: Convergence As Measuresmentioning

confidence: 99%

“…This requires a number of technical ingredients from [17], and so we decided to include this section as an appendix. It follows from [17,Proposition 6] that the monopole (A i , Φ i ) can be written as…”

Section: Convergence As Measuresmentioning

confidence: 99%

“…is an approximate monopole constructed in [17,Proposition 4]. Moreover, by its own construction, we have that the restriction…”

Section: Convergence As Measuresmentioning

confidence: 99%

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The limit of large mass monopoles

Fadel

Oliveira

2019

Proceedings of London Math Soc

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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. Contents

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Section: Motivating Examplesmentioning

confidence: 99%

Section: Convergence As Measuresmentioning

confidence: 99%

Section: Convergence As Measuresmentioning

confidence: 99%

“…is an approximate monopole constructed in [17,Proposition 4]. Moreover, by its own construction, we have that the restriction…”

Section: Convergence As Measuresmentioning

confidence: 99%

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The limit of large mass monopoles

Fadel

Oliveira

2019

Proceedings of London Math Soc

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“…(3) Recently, Oliveira [26] proved the existence of unframed monopoles of sufficiently large mass on an asymptotically conic 3-manifold X assuming b 2 (X) = 0 (which implies b 0 (∂X) = 1). He produces a 4k + b 1 (X) − 1-dimensional open set of such monopoles, which is consistent with our result.…”

Section: Several Remarks Are In Ordermentioning

confidence: 99%

Dimension of monopoles on asymptotically conic 3-manifolds: Figure 1.

Kottke

2015

Bull. London Math. Soc.

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The Haydys monopole equation

Nagy

Oliveira

2020

Sel. Math. New Ser.

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We study complexified Bogomolny monopoles using the complex linear extension of the Hodge star operator, these monopoles can be interpreted as solutions to the Bogomolny equation with a complex gauge group. Alternatively, these equations can be obtained from dimensional reduction of the Haydys instanton equations to three dimensions, thus we call them Haydys monopoles. We find that (under mild hypotheses) the smooth locus of the moduli space of finite energy Haydys monopoles on R 3 is a Kähler manifold containing the ordinary Bogomolny moduli space as a minimal Lagrangian submanifold-an A-brane. Moreover, using a gluing construction we construct an open neighborhood of this submanifold modeled on a neighborhood of the zero section in the tangent bundle to the Bogomolny moduli space. This is analogous to the case of Higgs bundles over a Riemann surface, where the (co)tangent bundle of holomorphic bundles canonically embeds into the Hitchin moduli space. These results contrast immensely with the case of finite energy Kapustin-Witten monopoles for which we have showed a vanishing theorem in Nagy and Oliveira (Kapustin-Witten equations on ALE and ALF Gravitational Instantons, 2019).

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Monopoles on AC 3-manifolds

Cited by 8 publications

References 11 publications

The limit of large mass monopoles

The limit of large mass monopoles

Dimension of monopoles on asymptotically conic 3-manifolds: Figure 1.

The Haydys monopole equation

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