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

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.

Energy identity for a class of approximate Dirac-harmonic maps from surfaces with boundary

“…In Theorem 1.1, for those Dirac-harmonic spheres splitting off at the interior blow-up points, i.e. (σ l i , ξ l i ) : S 2 → N, i = 1, ..., q; l = 1, ..., L i , we know that the image of the map parts σ l i , i = 1, ..., q; l = 1, ..., L i , are connected to the map part φ of the base field (φ, ψ) in the target manifold; this is proved in [17], the refined bubble tree can be constructed by applying similar arguments as in the harmonic map case given by [2,Section 3] and [20,Appendix]. However, for those Dirac-harmonic spheres splitting off at the boundary blow-up points, i.e.…”

“…Before we state our main results, let us recall a definition of approximate Dirac-harmonic map in [17]. Denote…”

“…In order to study the blow-up behavior of the Dirac-harmonic map flow from surfaces with boundary considered in [5,14], we introduced the notion of approximate Dirac-harmonic maps in [17] and proved the energy identity and no neck result in the interior blow-up case for a sequence of such maps. In general, this sequence might blow up at a boundary point.…”

“…Based on the interior blow-up results for approximate Dirac-harmonic maps studied in [17], we state our first main result in this paper concerning the boundary blow-up case:…”

“…To prove the energy quantization result near the boundary in Theorem 1.1, we shall follow the general blow-up scheme developed for harmonic map type problems, however, the proofs in this case are subtle and there are new difficulties arising when carrying out the neck analysis. Firstly, the method of the three circle type theorem used in the interior case in [17] can not be applied to the boundary case and we need to apply certain integration argument to show the no neck energy property. Secondly, we need to establish a new Pohozaev type identity for approximate Dirac-harmonic maps from surfaces with boundary (see Lemma 2.2) which requires some algebraic property for the spinors, see (2.10).…”