A Conformally Invariant Gap Theorem in Yang–Mills Theory

Gursky, Matthew J.; Kelleher, Casey Lynn; Streets, Jeffrey

doi:10.1007/s00220-017-3070-z

Cited by 7 publications

(11 citation statements)

References 31 publications

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“…It is easy to see that the expression γ 1 (E)|F | is independent of the choice of norms. Therefore, despite the difference of conventions pointed out in Remark 3.5, the definition of Φ g in (3.9) agrees with the corresponding formula (3.5) in [GKS18].…”

Section: A Canonical Conformal Representativesupporting

confidence: 75%

“…Proof. Using the Bochner formula for Yang-Mills connections, in [GKS18] we showed that either F + ≡ 0, or else λ 1 (L g ) = λ 1 (L 1 g ) ≤ 0. Since we are ruling out the former by assumption, the latter condition must hold.…”

Section: A Canonical Conformal Representativementioning

confidence: 90%

“…The definition of the inner product on Λ 2 + (g E ) given in [GKS18] differs from the definition of this paper. In particular, the estimate for γ 1 (E) in Section 2 of [GKS18] needs to be adjusted. With respect to our current conventions, we have the estimate…”

Section: A Canonical Conformal Representativementioning

confidence: 97%

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Index-Energy Estimates for Yang–Mills Connections and Einstein Metrics

Gursky

Kelleher

Streets

2019

Commun. Math. Phys.

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We prove a conformally invariant estimate for the index of Schrödinger operators acting on vector bundles over four-manifolds, related to the classical Cwikel-Lieb-Rozenblum estimate. Applied to Yang-Mills connections we obtain a bound for the index in terms of its energy which is conformally invariant, and captures the sharp growth rate. Furthermore we derive an index estimate for Einstein metrics in terms of the topology and the Einstein-Hilbert energy. Lastly we derive conformally invariant estimates for the Betti numbers of an oriented four-manifold with positive scalar curvature.

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Section: A Canonical Conformal Representativesupporting

confidence: 75%

Section: A Canonical Conformal Representativementioning

confidence: 90%

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Index-Energy Estimates for Yang–Mills Connections and Einstein Metrics

Gursky

Kelleher

Streets

2019

Commun. Math. Phys.

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“…This result is a combination of classical results [2,7] concerning energy minimizing solutions within a topological class (called instantons), as well as a recent energy lower bound for the non-minimizing solutions proved by Gursky-Kelleher-Streets [16]. For a derivation, see [39,Section 6].…”

Section: Such a Function O Induces The Gauge Transformationmentioning

confidence: 53%

The threshold conjecture for the energy critical hyperbolic Yang–Mills equation

Tataru

2021

Ann. of Math. (2)

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This article represents the fourth and final part of a four-paper sequence whose aim is to prove the Threshold Conjecture as well as the more general Dichotomy Theorem for the energy critical 4 + 1 dimensional hyperbolic Yang-Mills equation. The Threshold Theorem asserts that topologically trivial solutions with energy below twice the ground state energy are global and scatter. The Dichotomy Theorem applies to solutions in arbitrary topological class with large energy, and provides two exclusive alternatives: Either the solution is global and scatters, or it bubbles off a soliton in either finite time or infinite time.Using the caloric gauge developed in the first paper [37], the continuation/scattering criteria established in the second paper [38], and the large data analysis in an arbitrary topological class at optimal regularity in the third paper [39], here we perform a blowup analysis which shows that the failure of global well-posedness and scattering implies either the existence of a soliton with at most the same energy bubbling off, or the existence existence of a nontrivial self-similar solution. The proof is completed by showing that the latter solutions do not exist.

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“…On the other hand, the question of existence of finite energy harmonic Yang-Mills connections is best phrased in terms of the topological classes described above: Parts (1) and (2) are classical. We remark that part (3), which follows from a recent result of [8], is nontrivial due to existence of nonminimizing harmonic Yang-Mills connections [39]. We also note that harmonic Yang-Mills connections which are not energy minimizers no longer have to be self-dual or anti-self-dual.…”

Section: Topological Classes Of Finite Energy Connections On Rmentioning

confidence: 71%

Untitled

2019

Bull. Amer. Math. Soc.

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This article is devoted to the energy critical hyperbolic Yang-Mills equation in the (4 + 1)-dimensional Minkowski space, which is considered by the authors in a sequence of four papers. The final outcome of these papers is twofold: (i) the Threshold Theorem, which asserts that global well-posedness and scattering hold for all topologically trivial initial data with energy below twice the ground state energy; and (ii) the Dichotomy Theorem, which for larger data in arbitrary topological classes provides a choice of two outcomes, either a global scattering solution or a soliton bubbling off. In the last case, the bubbling-off phenomena can happen in one of two ways: (a) in finite time, triggering a finite time blowup; or (b) in infinite time. Our goal here is to first describe the equation and the results, and then to provide an overview of the flow of ideas within their proofs in the above-mentioned four papers. Contents 1. Introduction 171 2. The caloric gauge 183 3. Energy dispersed caloric Yang-Mills waves 192 4. Large data, causality, and the temporal gauge 197 5. To bubble or not to bubble 201 About the authors 208 References 208

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A Conformally Invariant Gap Theorem in Yang–Mills Theory

Abstract: We show a sharp conformally invariant gap theorem for Yang-Mills connections in dimension 4 by exploiting an associated Yamabe-type problem.

Cited by 7 publications

References 31 publications

Index-Energy Estimates for Yang–Mills Connections and Einstein Metrics

Index-Energy Estimates for Yang–Mills Connections and Einstein Metrics

The threshold conjecture for the energy critical hyperbolic Yang–Mills equation

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