Nonstationary weak limit of a stationary harmonic map sequence

The aim of this note is to extend the results in [NV] to the case of approximate harmonic maps. More precisely, we will proved that the singular strata S k (u) of an approximate harmonic map are k-rectifiable, and we will show effect bounds on the quantitative strata. In the process we will simplify many of the arguments from [NV], and in particular we produce a new main covering lemmas which vastly simplifies the older argument.

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“…if f i = 0 for all i), their limit might not be stationary. An interesting example of this is given in [DLL03].…”

Section: Weak Convergencementioning

confidence: 99%

Stratification for the singular set of approximate harmonic maps

Naber

Valtorta

2018

Math. Z.

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“…This is not true for stationary maps, as bubbling can occur (see example 3.10). Moreover, it is worth mentioning that weak W 1,2 limits of stationary maps are in general not stationary, see [DLL03].…”

Section: Preliminariesmentioning

confidence: 99%

“…where the first convergence is weak in H 1 loc while the second is in the sense of measures. Note that the limit u will be weakly harmonic, but need not be stationary harmonic (see [DLL03]). We have by [Lin99] that the defect measure ν is m − 2 rectifiable and so can be written…”

Section: Introductionmentioning

confidence: 99%

Energy identity for stationary Yang Mills

Naber

Valtorta

2019

Invent. math.

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Given a principal bundle P → M over a Riemannian manifold with compact structure group G, let us consider a stationary Yang-Mills connection A with energy M |F A | 2 ≤ Λ. If we consider a sequence of such connections A i , then it is understood by [Tia00] that up to subsequence we can converge A i → A to a singular limit connection such that the energy measures converge |F A i | 2 dv g → |F A | 2 dv g + ν, where ν = e(x)dλ n−4 is the n − 4 rectifiable defect measure. Our main result is to show, without additional assumptions, that for n − 4 a.e. point the energy density e(x) may be computed explicitly as the sum of the bubble energies arising from blow ups at x. Each of these bubbles may be realized as a Yang Mills connection over S 4 itself.This energy quantization was proved in [Riv02] assuming a uniform L 1 hessian bound on the curvatures in the sequence. In fact, our second main theorem is to show this hessian bound holds automatically. Precisely, given a connection A as above we have the apriori estimate M |∇ 2 F A | < C(Λ, dim G, M) for the curvature. It is important to note this result is proved in tandem with the energy quantization, and not before it. Indeed, we will in fact prove an effective version of the energy identity, and it is this effective version which will lead to both the L 1 hessian bound and the classical energy quantization results. In the course of the proof we will provide a quantitative version of the bubble tree decomposition which hold in all dimensions with effective estimates for a fixed stationary connections. To produce to strongest estimates in the paper we introduce an ǫ-gauge condition, which generalizes the usual Coulomb gauge and which will exist, with effective control, even over singular regions. On these ǫ-gauges we will provide a new superconvexity estimate which will be a key tool in analyzing higher dimensional annular regions.

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“…Willmore [37] for a definition and basic properties of the Willmore functional). In general, however, the limit map u 0 may not be in H 2 loc (Ω, N), and (1.17) might be false (at least this is suggested by the related results of Ding, J. Li, and W. Li [5]). We only know that τ 1 (u 0 ) ∈ L 2 (Ω, R n ) by (1.16).…”

Section: Theorem 12 There Exists Anmentioning

confidence: 99%

Untitled

Moser¹

2005

Internat. Math. Res. Papers

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Under reasonable conditions, an initial/boundary value problem belonging to (1.9) with smooth data can be solved by standard methods at least for short times. For the corresponding problem on a four-or lower-dimensional manifold without boundary, Lamm [19] even proved long-time existence of solutions under the assumption that N has nonpositive sectional curvature. In general, however, it must be expected that a solution may develop singularities in finite time, although no such example is known. The aim of this paper is to examine the behavior of the flow as the first singular time is approached.Our results can be seen as a first step towards classifying the singularities of the biharmonic map heat flow, also with a view to ruling out certain types of singularities under special assumptions.Before we state the results, we briefly discuss some known facts about the harmonic map heat flow. The dimension two is critical for the harmonic map heat flow in the sense that E 1 is invariant under rescalings of the domain. Similarly, the dimension four is critical for the biharmonic map heat flow, and we may therefore expect some analogies between these two problems. For the initial/boundary value problem for the harmonic map heat flow on a two-dimensional domain, it was shown by Struwe [32] and Chang [4] that a weak solution exists under reasonable conditions. Moreover, this solution is smooth away from a finite number of points in time space. The singular points are characterized by a concentration of the energy, and at each of them, at least one nonconstant harmonic map from R 2 into N can be separated by a rescaling (or "blowup") procedure.Such a harmonic map gives rise to a harmonic 2-sphere (i.e., a harmonic map from the standard 2-sphere S 2 into N) via the stereographic projection. The harmonic spheres obtained thus are often called "bubbles," and the whole process is referred to as "bubblingoff." The mentioned results were refined by Ding and Tian [6], Qing and Tian [28], Wang [35], Topping [34], and others.For the biharmonic map heat flow, we need to study the concentration points for not one, but two energies, E 1 and E 2 . Consequently, we identify two distinct sets where singularities may occur. The dimension four is supercritical for E 1 , therefore the singular set belonging to concentrations of the first energy must be expected to be quite large in general. We can say, however, that its Hausdorff dimension is at most 2 (and we are able to derive a few other facts about its structure). At each point of this set, we find a harmonic map bubbling off. It turns out that these bubbles give rise either to harmonic 2-spheres or weakly harmonic 3-spheres in N. Concentration points of E 2 , on the other hand, admit a blowup that converges to either a (weakly) harmonic sphere of the same type or a biharmonic map. In analogy to the harmonic map heat flow, it might be suspected that this part of the singular set is finite. Unfortunately, we have no such result, owing to an insufficient control of the local evolution of...

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Nonstationary weak limit of a stationary harmonic map sequence

Cited by 7 publications

References 14 publications

Stratification for the singular set of approximate harmonic maps

Stratification for the singular set of approximate harmonic maps

Energy identity for stationary Yang Mills

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