Removable singularities in Yang-Mills fields

Uhlenbeck, Karen

doi:10.1090/s0273-0979-1979-14632-9

Cited by 168 publications

(268 citation statements)

References 4 publications

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“…As this case is rather special, it turns out that control on the full Ricci curvature is not necessary and some results are already available when one has only finiteness of the L n/2 -norm of the scalar curvature (see Theorem 2.1 for details). Our result may be compared with a classical result by K. Uhlenbeck: any Hermitian vector bundle on the euclidean ball B n − {0} whose curvature is in L n/2 extends W 1,n on the whole ball [27].…”

Section: R(x) ρ} ρ Dρ < ∞mentioning

confidence: 83%

The Huber theorem for non-compact conformally flat manifolds

Carron¹,

Herzlich²

2002

Commentarii Mathematici Helvetici

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Abstract.It was proved in 1957 by Huber that any complete surface with integrable Gauss curvature is conformally equivalent to a compact surface with a finite number of points removed. Counterexamples show that the curvature assumption must necessarily be strengthened in order to get an analogous conclusion in higher dimensions. We show in this paper that any non compact Riemannian manifold with finite L n/2 -norm of the Ricci curvature satisfies Huber-type conclusions if either it is a conformal domain with volume growth controlled from above in a compact Riemannian manifold or if it is conformally flat of dimension 4 and a natural Sobolev inequality together with a mild scalar curvature decay assumption hold. We also get partial results in other dimensions. Mathematics Subject Classification (2000). 53C21, 58J60.

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Section: R(x) ρ} ρ Dρ < ∞mentioning

confidence: 83%

The Huber theorem for non-compact conformally flat manifolds

Carron¹,

Herzlich²

2002

Commentarii Mathematici Helvetici

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“…The codimension of the singular set is motivated by the fact that the right-hand side u|Rm+· · · | 2 is unitless for n = 4, and is supported by familiar convergence results for Einstein 4-manifolds ( [1,5,6,14,27,28,29] and many others), as well as analogous results for mean curvature flow in the critical case [17] and other flows. Later Professor Tian informed us that he had independently formulated a similar conjecture.…”

Section: Conjecture 16 (Long-term Behavior) Let G(t) Be a Ricci Flomentioning

confidence: 99%

Entropy and reduced distance for Ricci expanders

Feldman

Ilmanen

2005

J Geom Anal

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ABSTRACT. Perelman has discovered two integral quantities, the shrinker entropy A natural conjecture asserts that g(t)/t converges as t → ∞ to a negative Einstein manifold in some weak sense (in particular ignoring collapsing parts). If the limit is known a-priori to be smooth and compact, this statement follows easily from any monotone quantity that is constant on expandersSmall, large and distant parts of a Ricci flow are known to be modeled by various kinds of Ricci solitons: Steady, shrinking, and expanding. Perelman has discovered monotone quantities corresponding to the first two of these. The functional F is monotone on the Ricci flow and constant precisely on steadies [25, Section 1]. The shrinking, or localizing, entropy W is monotone in general and constant precisely on shrinking solitons [25, Section 3]. Closely related is the notion of the (backward) reduced volume [25, Section 7]. The latter two show that developing singularities and ancient histories are modeled on shrinkers, when an appropriate blowup, or blowdown is taken [25,27].In this note, by tweaking some signs, we observe similar monotonicity formulae for the expanding case. In Section 1, analogous to [25, Section 1], we define an expanding, or delocalizing entropy W + , which is monotone in general and constant precisely on expanders. It follows from Math Subject Classifications. Primary: 58G11.

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“…We produce our one-parameter (i.e. λ ∈ (0, ∞]) family of instantons (see (13) for an explicit form) via rescaling the metric by f and then projecting the obtained Levi-Civitá connection onto the negative su(2) − ⊂ so(4) subalgebra, and finally removing the singularity at the NUT by applying Uhlenbeck's theorem [28].…”

Section: Introductionmentioning

confidence: 99%

Geometric construction of new Yang–Mills instantons over Taub-NUT space

Etesi

Hausel

2001

Physics Letters B

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In this paper we exhibit a one-parameter family of new Taub-NUT instantons parameterized by a half-line. The endpoint of the half-line will be the reducible Yang-Mills instanton corresponding to the Eguchi-Hanson-Gibbons L 2 harmonic 2-form, while at an inner point we recover the Pope-Yuille instanton constructed as a projection of the LeviCivitá connection onto the positive su(2) + ⊂ so(4) subalgebra. Our method imitates the Jackiw-Nohl-Rebbi construction originally designed for flat R 4 . That is we find a oneparameter family of harmonic functions on the Taub-NUT space with a point singularity, rescale the metric and project the obtained Levi-Civitá connection onto the other negative su(2) − ⊂ so(4) part. Our solutions will possess the full U (2) symmetry, and thus provide more solutions to the recently proposed U (2) symmetric ansatz of Kim and Yoon.

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Removable singularities in Yang-Mills fields

Cited by 168 publications

References 4 publications

The Huber theorem for non-compact conformally flat manifolds

The Huber theorem for non-compact conformally flat manifolds

Entropy and reduced distance for Ricci expanders

Geometric construction of new Yang–Mills instantons over Taub-NUT space

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