Bubbling of the heat flows for harmonic maps from surfaces

Qing, Jie; Тиан, Ганг

doi:10.1002/(sici)1097-0312(199704)50:4<295::aid-cpa1>3.0.co;2-5

Cited by 122 publications

(138 citation statements)

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“…This has been studied extensively by [2,18,32], and in [11] for a more general case. For the harmonic map case, we refer to [7,[15][16][17]22,29,31]. Roughly speaking, the results of those papers assert that the failure of strong convergence occurs at finitely many concentration points of the energy.…”

Section: φ = A(φ)(dφ Dφ) + Re(p(a(dφ(e α ) E α · ψ); ψ)) (15)mentioning

confidence: 99%

“…The idea is from Qing-Tian's paper [22], which used a special case of the three circle theorem due to Simon [25] to show that the tangential energy of the sequence in the neck region decays exponentially. The second author in cooperation with H.Yin has extended this idea to some fourth order equations, see [19,20].…”

Section: Three Circle Theorem For Approximate Dirac-harmonic Mapsmentioning

confidence: 99%

“…The second author in cooperation with H.Yin has extended this idea to some fourth order equations, see [19,20]. Let us first state the three circle theorem for harmonic functions (see [18,22,25]). …”

Section: Three Circle Theorem For Approximate Dirac-harmonic Mapsmentioning

confidence: 99%

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Blow-up analysis for approximate Dirac-harmonic maps in dimension 2 with applications to the Dirac-harmonic heat flow

2017

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Dirac-harmonic maps couple a second order harmonic map type system with a first nonlinear Dirac equation. We consider approximate Dirac-harmonic maps {(φ n , ψ n )}, that is, maps that satisfy the Dirac-harmonic system up to controlled error terms. We show that such approximate Dirac-harmonic maps defined on a Riemann surface, that is, in dimension 2, continue to satisfy the basic properties of blow-up analysis like the energy identity and the no neck property. The assumptions are such that they hold for solutions of the heat flow of Dirac-harmonic maps. That flow turns the harmonic map type system into a parabolic system, but simply keeps the Dirac equation as a nonlinear first order constraint along the flow. As a corollary of the main result of this paper, when such a flow blows up at infinite time at interior points, we obtain an energy identity and the no neck property.

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Section: φ = A(φ)(dφ Dφ) + Re(p(a(dφ(e α ) E α · ψ); ψ)) (15)mentioning

confidence: 99%

Section: Three Circle Theorem For Approximate Dirac-harmonic Mapsmentioning

confidence: 99%

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Blow-up analysis for approximate Dirac-harmonic maps in dimension 2 with applications to the Dirac-harmonic heat flow

2017

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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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“…On the other hand, it should be pointed out that one has also established some effective methods to prove successfully the energy identity and give the detailed description of the connecting necks for the heat flow of harmonic maps from a Riemann surface, or more generally, a sequence of maps from a Riemann surface with tension fields τ bounded in the sense of L 2 [3,4,13,14].…”

Section: Introductionmentioning

confidence: 99%