Morse theory for the Yang-Mills functional via equivariant homotopy theory

Gritsch, Ursula

doi:10.1090/s0002-9947-00-02562-9

Cited by 4 publications

(2 citation statements)

References 17 publications

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“…More recently Gritsch [40] (in 2000) has proved existence of non-minimal Yang-Mills connections over a broader family of four-dimensional manifolds. We recall her construction of those manifolds.…”

Section: Discreteness Of Critical Values Of the Yang-mills Energy Fun...mentioning

confidence: 99%

Discreteness for energies of Yang-Mills connections over four-dimensional manifolds

Feehan

2015

Preprint

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We generalize our previous results [29, Theorem 1 and Corollary 2], [30, Theorem 1] on the existence of an L 2 -energy gap for Yang-Mills connections over closed four-dimensional manifolds and energies near the ground state (occupied by flat, anti-self-dual, or self-dual connections) to the case of Yang-Mills connections with arbitrary energies. We prove that for any principal bundle with compact Lie structure group over a closed, four-dimensional, Riemannian manifold, the L 2 energies of Yang-Mills connections on that principal bundle form a discrete sequence without accumulation points. Our proof employs a version of our Lojasiewicz-Simon gradient inequality for the Yang-Mills L 2 -energy functional from our monograph [31] and extensions of our previous results on the bubble-tree compactification for the moduli space of anti-self-dual connections [32] to the moduli space of Yang-Mills connections with a uniform L 2 bound on their energies.

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Section: Discreteness Of Critical Values Of the Yang-mills Energy Fun...mentioning

confidence: 99%

Discreteness for energies of Yang-Mills connections over four-dimensional manifolds

Feehan

2015

Preprint

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“…By a result due to Sibner, Sibner, and Uhlenbeck [40], there exist non-minimal critical points of the Yang-Mills energy functional on P = S 4 × SU(2) and, more generally, principal SU(2)bundles, P , over S 4 for any c 2 (P ) ≥ 2 by work of Bor and Montgomery [6,7], Sadun and Segert [33,34,35,36,37], and other four-dimensional manifolds by work of Gritsch [22] and Parker [31].…”

Section: Introductionmentioning

confidence: 99%

Energy gap for Yang–Mills connections, I: Four-dimensional closed Riemannian manifolds

Feehan

2016

Advances in Mathematics

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We extend an L 2 energy gap result due to Theorem 2] and Parker [30, Proposition 2.2] for Yang-Mills connections on principal G-bundles, P , over closed, connected, four-dimensional, oriented, smooth manifolds, X, from the case of positive Riemannian metrics to the more general case of good Riemannian metrics, including metrics that are generic and where the topologies of P and X obey certain mild conditions and the compact Lie group, G, is SU (2) or SO(3). A 13 4.2. Completion of the proofs of Theorem 1 and Corollary 2 14 Appendix A. Uhlenbeck continuity of the least eigenvalue of d + A d +, * A with respect to the connection 15 References 22

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The energy identity for a sequence of Yang–Mills $$\alpha $$-connections

Hong

Schabrun

2019

Calc. Var.

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We prove that the Yang-Mills α-functional satisfies the Palais-Smale condition, implying the existence of critical points, which are called Yang-Mills α-connections. It was shown in [10] that as α → 1, a sequence of Yang-Mills α-connections converges to a Yang-Mills connection away from finitely many points. We prove an energy identity for such a sequence of Yang-Mills α-connections. As an application, we also prove an energy identity for the Yang-Mills flow at the maximal existence time.

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Morse theory for the Yang-Mills functional via equivariant homotopy theory

Cited by 4 publications

References 17 publications

Discreteness for energies of Yang-Mills connections over four-dimensional manifolds

Discreteness for energies of Yang-Mills connections over four-dimensional manifolds

Energy gap for Yang–Mills connections, I: Four-dimensional closed Riemannian manifolds

The energy identity for a sequence of Yang–Mills $$\alpha $$-connections

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