Higher order Seiberg–Witten functionals and their associated gradient flows

Saratchandran, Hemanth

doi:10.1007/s00229-018-1092-2

Cited by 4 publications

(23 citation statements)

References 23 publications

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“…Following [7,10], we can derive estimates of Bernstein-Bando-Shi type, which is similar to [17, Proposition 4.10]. Then for t ∈ [0, T ) ⊂ I with T < 1 (QK) 4 , there exists a positive constant C q := C q (dim(M ), rk(E), G, q, k, s, g, γ) ∈ R >0 such that…”

Section: Long Time Existence Obstructionmentioning

confidence: 90%

“…In this section we introduce the basic setup and notation that will be used throughout the paper. Our approach follows the notation of [7], [10], [17].…”

Section: Preliminariesmentioning

confidence: 99%

“…We can use De Turck's trick to establish the local existence of the Yang-Mills-Higgs k-flow. We refer to [7,10,17] for more details. As the proof is standard, we will omit the details.…”

Section: Long Time Existence Obstructionmentioning

confidence: 99%

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A new higher order Yang--Mills--Higgs flow in Riemannian $4$-manifold

Saratchandran¹,

Zhang²

2022

Preprint

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Let (M, g) be a closed Riemannian 4-manifold and let E be a vector bundle over M with structure group G, where G is a compact Lie group. In this paper, we consider a new higher order Yang-Mills-Higgs functional, in which the Higgs field is a section of Ω 0 (adE). We show that, under suitable conditions, solutions to the gradient flow do not hit any finite time singularities. In the case that E is a line bundle, we are able to use a different blow up procedure and obtain an improvement of the long time result in [17].

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Section: Long Time Existence Obstructionmentioning

confidence: 90%

“…In this section we introduce the basic setup and notation that will be used throughout the paper. Our approach follows the notation of [7], [10], [17].…”

Section: Preliminariesmentioning

confidence: 99%

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A new higher order Yang--Mills--Higgs flow in Riemannian $4$-manifold

Saratchandran¹,

Zhang²

2022

Preprint

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“…To meet the requirements in the next sections, here, in this short section the setup and notation are briefly presented. We will use some of Kelleher's notations in [25] and Saratchandran's in [40]. For a more concentrate elements about Yang-Mills theory, we refer to Donaldson-Kronheimer [11] and Jost's [24] books and references therein.…”

Section: Preliminarymentioning

confidence: 99%

“…It is not surprising to consider such higher order flow. Just recently, in [25], Kelleher studied higher order Yang-Mills flow (with vanishing Higgs field) and in [40], Saratchandran studied higher order Seiberg-Witten flow (flow of connections coupled with spinor fields). The study of higher order flow also has a long history.…”

Section: Introductionmentioning

confidence: 99%

Gradient Flows of Higher Order Yang-Mills-Higgs Functionals

Zhang¹

2020

Preprint

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In this paper, we define a family of functionals generalizing the Yang-Mills-Higgs functional on a closed Riemannian manifold. Then we prove the short time existence of the corresponding gradient flow by a gauge fixing technique. The lack of maximal principle for the higher order operator brings us a lot of inconvenience during the estimates for the Higgs field. We observe that the L 2 -bound of the Higgs field is enough for energy estimates in 4 dimension, and we show that, provided the order of derivatives, appearing in the higher order Yang-Mills-Higgs functionals, is strictly greater than 1, solutions to the gradient flow do not hit any finite time singularities. As for the Yang-Mills-Higgs k-functional with Higgs self-interaction, we show that, provided dim(M ) < 2(k + 1), the associated gradient flow admits long time existence with smooth initial data. The proof depends on local L 2 -derivative estimates, energy estimates and blow-up analysis.

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A New Higher Order Yang–mills–higgs Flow on Riemannian -Manifolds

Saratchandran¹,

ZHANG²,

ZHANG

2022

Bull. Aust. Math. Soc.

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Let $(M,g)$ be a closed Riemannian $4$ -manifold and let E be a vector bundle over M with structure group G, where G is a compact Lie group. We consider a new higher order Yang–Mills–Higgs functional, in which the Higgs field is a section of $\Omega ^0(\text {ad}E)$ . We show that, under suitable conditions, solutions to the gradient flow do not hit any finite time singularities. In the case that E is a line bundle, we are able to use a different blow-up procedure and obtain an improvement of the long-time result of Zhang [‘Gradient flows of higher order Yang–Mills–Higgs functionals’, J. Aust. Math. Soc.113 (2022), 257–287]. The proof relies on properties of the Green function, which is very different from the previous techniques.

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Higher order Seiberg–Witten functionals and their associated gradient flows

Cited by 4 publications

References 23 publications

A new higher order Yang--Mills--Higgs flow in Riemannian $4$-manifold

A new higher order Yang--Mills--Higgs flow in Riemannian $4$-manifold

Gradient Flows of Higher Order Yang-Mills-Higgs Functionals

A New Higher Order Yang–mills–higgs Flow on Riemannian -Manifolds

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