Blow-up analysis for approximate Dirac-harmonic maps in dimension 2 with applications to the Dirac-harmonic heat flow |
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Authors: | Email author" target="_blank">Jürgen?JostEmail author Lei?Liu Miaomiao?Zhu |
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Institution: | 1.Max Planck Institute for Mathematics in the Sciences,Leipzig,Germany;2.School of Mathematical Sciences,Shanghai Jiao Tong University,Shanghai,China |
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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 \(\{(\phi _n,\psi _n)\}\), 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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