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

Jürgen Jost,Lei Liu,Miaomiao Zhu

doi:10.1007/s00526-017-1202-0

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.

Highlights

Dirac-harmonic maps were introduced and studied in [2,3]. They were motivated by the supersymmetric nonlinear sigma model from quantum field theory [6,10], and they combine and generalize the theories of harmonic maps and harmonic spinors
Choosing a local orthonormal basis eα, α = 1, 2 on M, the usual Dirac operator is defined as ∂/ := eα · ∇eα, where ∇ is the spin connection on M
As an application of Theorem 1.2, we study the asymptotic behavior at infinite time for the Dirac-harmonic map flow in dimension 2

Summary

Introduction

Dirac-harmonic maps were introduced and studied in [2,3]. They were motivated by the supersymmetric nonlinear sigma model from quantum field theory [6,10], and they combine and generalize the theories of harmonic maps and harmonic spinors. We will extend the results from [2,18,32] to the approximate Dirac-harmonic maps from a closed Riemann surface M to a compact Riemannian manifold N. Our first main result is Theorem 1.2 For a sequence of smooth approximate Dirac-harmonic maps {(φn, ψn)} from a closed Riemann surface M to a compact Riemannian manifold N with uniformly bounded energy. There exist tn ↑ ∞, a Dirac-harmonic map (φ∞, ψ∞) ∈ C2+α(M, N ) ×C1+α(M, M ⊗ φ∞∗ T N ) with boundary data φ∞|∂ M = φ and Bψ∞|∂ M = Bχ , and a nonnegative integer I and finitely many points { p1, ..., pI } ∈ M such that (1) (φn, ψn) := (φ(·, tn), ψ(·, tn)).

Some basic lemmas

Three circle theorem for approximate Dirac-harmonic maps

Energy identity and no neck result