Singularity formation in the Yang-Mills Flow

Weinkove, Ben

doi:10.1007/s00526-003-0217-x

Cited by 30 publications

(51 citation statements)

References 12 publications

(14 reference statements)

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“…We shall introduce the blowup analysis for Type-I singularities, which was carried out by Weinkove [22]. We shall introduce the blowup analysis for Type-I singularities, which was carried out by Weinkove [22].…”

Section: Preliminariesmentioning

confidence: 99%

“…We now follow [22] introducing the blowup procedure around (x 0 , T ). We now follow [22] introducing the blowup procedure around (x 0 , T ).…”

Section: Preliminariesmentioning

confidence: 99%

“…Weinkove [22] proved that Type-I singularities of the Yang-Mills flow are modelled by homothetically shrinking solitons whose curvatures do not vanish identically. On the other hand, homothetically shrinking solitons are closely related to Type-I singularities of the Yang-Mills flow.…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, homothetically shrinking solitons are closely related to Type-I singularities of the Yang-Mills flow. In fact, Weinkove showed in [22] that Type-I singularities of the Yang-Mills flow can be modelled by such solitons. Examples of homothetically shrinking solitons have been found in [10,22].…”

Section: Introductionmentioning

confidence: 99%

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Stabilities of homothetically shrinking Yang-Mills solitons

Chen

Zhang

2015

Trans. Amer. Math. Soc.

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In this paper we introduce entropy-stability and F-stability for homothetically shrinking Yang-Mills solitons, employing entropy and the second variation of the F -functional respectively. For a homothetically shrinking soliton which does not descend, we prove that entropy-stability implies F-stability. These stabilities have connections with the study of Type-I singularities of the Yang-Mills flow. Two byproducts are also included: We show that the Yang-Mills flow in dimension four cannot develop a Type-I singularity, and we obtain a gap theorem for homothetically shrinking solitons.

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Section: Preliminariesmentioning

confidence: 99%

“…We now follow [22] introducing the blowup procedure around (x 0 , T ). We now follow [22] introducing the blowup procedure around (x 0 , T ).…”

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Stabilities of homothetically shrinking Yang-Mills solitons

Chen

Zhang

2015

Trans. Amer. Math. Soc.

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“…A proof for the similar case of Yang-Mills flow is given by [12] and the harmonic map heat flow case is well known. We shall show that, like the case of mean curvature flow in R n , a rescaled solution converges to a self-shrinker in R n .…”

Section: Self-shrinkers and Type I Singularitiesmentioning

confidence: 96%

Convexity and cylindrical estimates for mean curvature flow in the sphere

Nguyen

2015

Trans. Amer. Math. Soc.

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We study mean curvature flow in the sphere with the quadratic curvature condition |A| 2 ≤ 1 n−2 H 2 + 4K which is related to but different from two-convexity for submanifolds of the sphere. We classify type I singularities with no further hypotheses. If H > 0, then we apply the Huisken-Sinestrari convexity estimates to this situation and show that we can classify type II singularities. This shows that at a singularity the surface is asymptotically convex. We then prove cylindrical estimates for the mean curvature flow and a pointwise gradient estimate which shows that near a singularity the surface is quantitatively convex.

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Energy identities and monotonicity for evolving $$\varvec{k}$$ k -forms on moving Riemannian spaces

Afuni

2017

J. Evol. Equ.

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Singularity formation in the Yang-Mills Flow

Cited by 30 publications

References 12 publications

Stabilities of homothetically shrinking Yang-Mills solitons

Stabilities of homothetically shrinking Yang-Mills solitons

Convexity and cylindrical estimates for mean curvature flow in the sphere

Energy identities and monotonicity for evolving $$\varvec{k}$$ k -forms on moving Riemannian spaces

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