Bubble tree convergence for harmonic maps

Parker, Thomas H.

doi:10.4310/jdg/1214459224

Cited by 155 publications

(175 citation statements)

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“…In the case s ¼ 2 we mention in particular the papers [13], [10] and [15]. Parker's paper [13] was the first to present a ''bubble tree analysis'', in this case for an equibounded sequence of harmonic maps from a 2-dimensional manifold; his results basically show that such a sequence can be associated to a limiting configuation, a ''bubble tree'', consisting of a harmonic map f 0 : M ! N and a tree of bubbles f k : S 2 !…”

Section: Introductionmentioning

confidence: 99%

Minimizing conformal energies in homotopy classes

Grotowski¹,

Kronz²

2004

Forum Mathematicum

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Abstract. We consider a minimizing sequence for the conformal energy in a given homotopy class of maps between two compact Riemannian manifolds M and N. In general this sequence will fail to be (strongly) convergent in the natural Sobolev class, but will have a weak limit which is not a priori in the original homotopy class. We prove a topological decomposition theorem: the homotopy class of the original map is given as the composition (in an appropriate sense) of the homotopy class of the weak limit with a finite number of free homotopy classes of maps from the sphere (with dimension that of the manifold M) into N. The method of proof shows that the weak limit is a minimizer in its homotopy class, and also shows that the homotopy classes of maps from the sphere occurring in the decomposition can be represented by minimizers in their respective classes.

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Section: Introductionmentioning

confidence: 99%

Minimizing conformal energies in homotopy classes

Grotowski¹,

Kronz²

2004

Forum Mathematicum

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“…When the domain is two-dimensional, particularly interesting features arise. The conformal invariance of the energy functional leads to non-compactness of the set of harmonic maps in dimension two, and the blow-up behavior has been studied extensively in [5,13,20,23,24,27] for the interior case and [10,15,16] for the boundary case. Roughly speaking, the energy identities for harmonic maps tell us that, during the weak convergence of a sequence of harmonic maps, the loss of energy is concentrated at finitely many points and can be quantized by a sum of energies of harmonic spheres and harmonic disks.…”

Section: Introductionmentioning

confidence: 99%

Bubbling analysis for approximate Lorentzian harmonic maps from Riemann surfaces

et al. 2017

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For a sequence of approximate harmonic maps (u n , v n ) (meaning that they satisfy the harmonic system up to controlled error terms) from a compact Riemann surface with smooth boundary to a standard static Lorentzian manifold with bounded energy, we prove that identities for the Lorentzian energy hold during the blow-up process. In particular, in the special case where the Lorentzian target metric is of the form g N −βdt 2 for some Riemannian metric g N and some positive function β on N , we prove that such identities also hold for the positive energy (obtained by changing the sign of the negative part of the Lorentzian energy) and there is no neck between the limit map and the bubbles. As an application, we complete the blow-up picture of singularities for a harmonic map flow into a standard static Lorentzian manifold. We prove that the energy identities of the flow hold at both finite and infinite singular times. Moreover, the no neck property of the flow at infinite singular time is true.

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“…This means that the loss of energy under the limiting process can be recovered by the energy of a finite number of bubbles. Readers can refer to the pioneering work on the energy identity by Jost [18], Parker [22] regarding the harmonic maps from surfaces and by Ding-Tian [6] for the harmonic map flow. For n-harmonic maps with n ≥ 3, the isolated singularities are removable due to Duzaar-Fuchs [7] and the energy identity was provided by Wang-Wei [27] for a sequence of approximate n-harmonic maps.…”

Section: Introductionmentioning

confidence: 99%