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

Jürgen Jost, Lei Liu, and Miaomiao Zhu

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Submission date: 29. Sep. 2016
Pages: 28
published in: Calculus of variations and partial differential equations, 56 (2017) 4, art-no. 108 
DOI number (of the published article): 10.1007/s00526-017-1202-0
Bibtex
Keywords and phrases: approximate Dirac-harmonic maps, Dirac-harmonic map flow, energy identity, no neck
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Abstract:
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.