Singularity formation of the Yang–Mills Flow

Abstract

We study singularity structure of Yang-Mills flow in dimensions n ≥ 4. First we obtain a description of the singular set in terms of concentration for a localized entropy quantity, which leads to an estimate of its Hausdorff dimension. We develop a theory of tangent measures for the flow, which leads to a stratification of the singular set. By a refined blowup analysis we obtain Yang-Mills connections or solitons as blowup limits at any point in the singular set.Date: February 1, 2016. 1 A corollary of these theorems is the existence of a either Yang-Mills connection or Yang-Mills soliton as a blowup limit of arbitrary finite time singularities. For type I singularities the existence of soliton blowup limits was established in [24], following from the entropy monotonicity for Yang-Mills flow demonstrated in [9]. The existence of soliton blowup limits for arbitrary singularities of mean curvature flow was established in [11], relying on the structure theory associated with Brakke's weak solutions. A preliminary investigation into the entropy-stability of Yang-Mills solitons was undertaken in [3] and [12]. Those results now apply to studying arbitrary finite-time singularities of Yang-Mills flow, as all admit singularity models which are either Yang-Mills connections or Yang-Mills solitons. Corollary 1.4. Fix n ≥ 4 and let E → (M n , g) be a vector bundle over a closed Riemannian manifold. Let ∇ t a smooth solution to Yang-Mills flow on [0, T ) such that lim sup t→T |F ∇tsuch that the corresponding blowup sequence converges modulo gauge transformations to either (1) A Yang-Mills connection on S 4 .(2) A Yang-Mills soliton.

Uhlenbeck compactness for Yang-Mills flow in higher dimensions

“…The next theorem follows from the refined blowup arguments due to Lin [19], Lin and Wang [20], and Tian [28]. The recent paper by Kelleher and Streets [18] is discussed in Remark 4.2 below. Theorem 4.1 (Cf.…”

“…Theorem 4.1 (Cf. Kelleher-Streets [18], §6, Lin-Wang [21], §8.5, Tian [28], §3-4). Let A i be as in Theorem 1.1, satisfying (1.1) and (1.3).…”

“…Remark 4.2. Kelleher and Streets [18] have recently attempted to analyze sequences of smooth solutions of (YM) which are "weakly H 1 -convergent." However, there is a key difficulty in Yang-Mills theory which does not occur with harmonic maps, namely, that the energy does not control the H 1 norm of the connection modulo gauge (unless n < 4 or n = 4 and the energy is small).…”

Nonlinear stability of homothetically shrinking Yang-Mills solitons in the equivariant case

“…The nature of singularities for the Yang-Mills heat flow over compact d-dimensional manifolds Irfan Glogić acknowledges the support of the Austrian Science Fund FWF, Project P 30076. Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -Project-ID 258734477 -SFB 1173. has been investigated by Weinkove in [35] and recently by Kelleher and Streets in [23], [24], showing that homothetically shrinking solitons appear as blowup limits at singular points. Such objects correspond to self-similar solutions of the Yang-Mills heat flow on the trivial bundle over R d , which is the main motivation to study the problem in this geometric setting.…”

“…has been investigated by Weinkove in [35] and recently by Kelleher and Streets in [23], [24], showing that homothetically shrinking solitons appear as blowup limits at singular points. Such objects correspond to self-similar solutions of the Yang-Mills heat flow on the trivial bundle over R d , which is the main motivation to study the problem in this geometric setting.…”