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

Dirac-harmonic maps couple a second order harmonic map type system with a first nonlinear Dirac equation. We consider approximate Dirac-harmonic maps {(ϕn,ψn)}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{(\phi _n,\psi _n)\}$$\end{document}, 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.


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.
Let us recall the precise definiton. Let M be a compact Riemann surface, equipped with a Riemannian metric h and with a fixed spin structure, M be the spinor bundle over M and ·, · M be the metric on M induced by the Riemannian metric h. Choosing a local orthonormal basis e α , α = 1, 2 on M, the usual Dirac operator is defined as / ∂ := e α · ∇ e α , where ∇ is the spin connection on M. The usual Dirac operator / ∂ on a surface can be seen as the Cauchy-Riemann operator. Consider R 2 with the Euclidean metric dx 2 + dy 2 . Let e 1 = ∂ ∂ x and e 2 = ∂ ∂ y be the standard orthonormal frame. A spinor field is simply a map ψ : R 2 → 2 = C 2 , and the action of e 1 and e 2 on spinors can be identified with multiplication with matrices If ψ := ψ 1 ψ 2 : R 2 → C 2 is a spinor field, then the Dirac operator is For more details on spin geometry and Dirac operators, one can refer to [14]. Let φ be a smooth map from M to another compact Riemannian manifold (N , g) with dimension n ≥ 2. Let φ T N be the pull-back bundle of T N by φ and then we get the twisted bundle M ⊗ φ T N. Naturally, there is a metric ·, · M⊗φ T N on M ⊗ φ T N which is induced from the metrics on M and φ T N. Also we have a natural connection ∇ on M ⊗ φ T N which is induced from the connections on M and φ T N. Let ψ be a section of the bundle M ⊗ φ T N. In local coordinates, it can be written as where each ψ i is a usual spinor on M and ∂ y i is the nature local basis on N . Then ∇ becomes where i jk are the Christoffel symbols of the Levi-Civita connection of N . The Dirac operator along the map φ is defined by / Dψ := e α · ∇ e α ψ. We consider the following functional The functional L(φ, ψ) is conformally invariant. That is , for any conformal diffeomorphism f : M → M, setting Then L( φ, ψ) = L(φ, ψ). For the proof, one can refer to [3]. Here λ is the conformal factor of the conformal map f, i.e. f * h = λ 2 h. Critical points (φ, ψ) are called Dirac-harmonic maps from M to N . The Euler-Lagrange equations of the functional L are / Dψ = 0, (1.4) where R(φ, ψ) is defined by Here R m li j stands for the Riemann curvature tensor of the target manifold (N , g). One can refer to [2,3].
Here P(ξ ; ·) denotes the shape operator, defined by P(ξ ; X ), Y = A(X, Y ), ξ for X, Y ∈ (T N), and Re(z) denotes the real part of z ∈ C. We refer to [2,3,5,11,24,30,33] for more details. Denote In this paper, with applications for the Dirac-harmonic heat flow in mind, we want to consider pairs (φ, ψ) that satisfy the Euler-Lagrange equations up to an error term in L 1 .
Here is the precise Thus, (φ, ψ) is a Dirac-harmonic map if and only if τ (φ, ψ) = h(φ, ψ) = 0. In the sequel, we shall need to assume that the error terms are in stronger spaces than L 1 , however. See for instance Theorem 1.2.
As for harmonic maps, the conformal invariance of the energy functional L leads to non-compactness of the set of Dirac-harmonic maps in dimension 2. 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. At such points, finitely many bubbles (i.e. nonconstant Dirac-harmonic spheres) separate, and the total energies from these bubbles account for the total loss of Dirichlet energies during the process of convergence. Moreover, the image of the remaining the base map and those of the bubbles are connected in the target manifold. This is called the no neck property.
In this paper, 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 .
Denote the energy of φ on ⊂ M by and the energy of the pair (φ, ψ) on ⊂ M by We shall often omit the domain M from the notation and simply write defining the blow-up set  (1.11) and the image φ(M) ) is a connected set. Remark 1.3 From the proof of Theorem 4.1 in Sect. 4, it is easy to see that also the following identities hold: This is due to the fact that both M |∇ψ| 4 3 d M and L(φ, ψ) are conformally invariant [3].
As an application of Theorem 1.2, we study the asymptotic behavior at infinite time for the Dirac-harmonic map flow in dimension 2.
For that purpose, we first review the heat flow for Dirac-harmonic maps as introduced and studied in [4,12] (a different flow has been introduced and studied in [1]). One tries to find (1.14) with the following boundary-initial data: are given maps and B = B ± is the chiral boundary operator defined as follows: where − → n is the outward unit normal vector field on ∂ M, and G is the chiral operator satisfying: for any X ∈ (T M).
In dimension dim M = 2, [4] established the short-time existence for (1.14) with smooth initial-boundary data (1.15). Later, [12] showed that, under some smallness assumption for φ 0 H 1 + Bχ L 2 , there exists a unique global weak solution to (1.14) with initial-boundary data (1.15), which has at most finitely many singular times and enjoys the property It follows from (1.19) that there exists a sequence t n ↑ ∞ such that (φ n , ψ n ) : is an approximate Diracharmonic map with boundary-data (1.20) which satisfies the assumptions of Theorem 1.2. In fact, h(φ n , ψ n ) = 0 and Thus, as an immediate corollary, we obtain and Remark 1.5 In this Theorem 1.4, we only consider the interior blow-up phenomenon for the Dirac-harmonic maps flow. The boundary blow-up case is treated in a subsequent paper [13].
This paper is organized as follows. In Sect. 2, we shall prove some basic lemmas, called small energy regularity, Pohozaev's identity and removable singularity, so that the expert will readily know what we are talking about, and we shall recall some known results for later use. In Sect. 3, we shall establish the three circle theorem for approximate Dirac-harmonic maps which ensures the exponential decay of the tangential energy. Our main result Theorem 1.2 will be proved in Sect. 4.

Some basic lemmas
In this section, we will prove some basic lemmas and recall some known results which will be used in this paper.
Firstly, we prove a small energy regularity theorem. (
Proof Proposition 2.2 is proved in [3]. For the reader's convenience, we recall it here. On the one hand, we have On the other hand, by a direct computation, we get where R M is the curvature operator of the connection on the spinor bundle M. For this curvature, we have (see [3,6,14]) Thus we obtain Then, the conclusion of the proposition follows immediately.

Lemma 2.3
Let D ⊂ R 2 be the unit disk and (φ, ψ) be a smooth approximate Diracharmonic map, then for any where (r, θ) are polar coordinates in D centered at 0, φ r = dφ( ∂ ∂r ), ψ r = ∇ ∂ ∂r ψ and Proof Multiplying the equation (1.7) by r φ r and integrating over D t , by Proposition 2.2 and the fact that On the one hand, integrating by parts, we have On the other hand, by Lemma 2.6, we get Combining the preceding equations, we get the conclusion of the lemma.

Corollary 2.4 Under the assumption of Lemma
Proof By Lemma 2.3, for any 0 < s < 1 2 , we have It is easy to see that where we used the fact that Multiplying (2.7) by 1 s and integrating from t to 2t, we get Thirdly, we state an interior removable singularity result.
Proof By a standard argument as in Lemma A.2 in [9], it is easy to see that (φ, ψ) is a weak solution of (2.12) and (2.13). It is known that the equation of φ can be written as an elliptic system with an anti-symmetric potential [5,24,30]: . Then it follows from Riviere's regularity result and its extensions (see [26][27][28]) that φ ∈ W 2, p (D) for any 1 < p < 2 which implies φ ∈ W 1,4 (D). Applying a simple argument to the Dirac equation for ψ gives that ψ ∈ W 1,2 (D). This indicate that ψ ∈ L 8 (D). Then by (2.12), we have φ ∈ L 2 (D) which implies the conclusion of the theorem.
In the end of this section, we recall two lemmas which are used in this paper.

Lemma 2.6 ([5]) For any
where ψ, ω := h i j ψ i , ω j . ψ) is a smooth Dirac-harmonic map from the standard sphere S 2 to N satisfying

Proposition 2.7 ([2]) Let N be a compact Riemannian manifold. Then there exists a constant
then both φ and ψ are trivial.

Three circle theorem for approximate Dirac-harmonic maps
In this section, we will extend the three circle theorem for Dirac-harmonic maps in [18] to the case of approximate Dirac-harmonic maps. 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]. Let us first state the three circle theorem for harmonic functions (see [18,22,25]).
Next, we prove an L 2 interior estimate for the following integro-differential equations.
Then there exists a positive constant ρ 0 such that if ρ ≤ ρ 0 , there holds Proof The proof is similar to Lemma 3.1 in [18] where Take a cut-off function η(x) = η(|x|) with compact support in B σ satisfying η(x) ≡ 1 in B σ and |∇η| ≤ 4 (1−σ ) and | η| ≤ 16 (1−σ ) 2 . Computing directly, we get By the standard elliptic estimate and Sobolev embedding, we have Similarly, we can compute By the first order elliptic estimate, we have Taking ρ 0 sufficiently small, we get We now introduce seminorms, and define for j = 0, 1, 2 Multiplying (3.6) by (1 − σ ) 2 and noting that 1 − σ = 1−σ 2 , we have We claim now that j satisfy an interpolation inequality for any > 0, where C > 0 is a universal constant. In fact, by the definition of 1 , for any γ > 0, we have where the second inequality is derived from the interpolation Theorem 7.27 (or Theorem 7.28) in [8]. By letting γ → 0, we obtain (3.8). Using (3.8) in (3.7), we then obtain this is Choosing a new cut-off function η in (3.6) and using (3.9), we get Then it is easy to see that the lemma follows from (3.9) and (3.10).
Denote P i := D e (i+1)L r 2 \D e i L r 2 and where L > 0 is the constant in Theorem 3.1.
We have the following three circle theorem:  (3.11) and for any e (i−1)L r 2 ≤ r ≤ 1 2 e (i+2)L r 2 , there hold

Moreover, for any e
But, u k does not satisfy at least one of the conclusions in (a), (b) and (c). If (a) does not hold, then we have If (b) does not hold, then we have If (c) does not hold, then we have In all those three cases, we may get the same conclusion that Without loss of generality, we assume F 2 (u k , v k ) = 1 (if not, we consider u k = ,v k ) ). Then we obtain By Lemma 3.2, we have u k W 2,2 (P 2 ) + v k W 1,2 (P 2 ) ≤ C. So, there exists a subsequence of (u k , v k ) (we still denote it by (u k , v k )), such that It is easy to see that u is a harmonic function and v is a holomorphic function in D e (i+2)L \D e (i−1)L and they satisfy is equipped with the metric g = dt 2 + dθ 2 , which is conformal to the standard Euclidean metric ds 2 on R 2 . In fact, Then we know that u • f is a harmonic function and e According to Theorem 3.1, we know Thus, (3.14) But, letting k → ∞ in (3.13) which implies (F 1 (u, v) + F 3 (u, v)). (3.15) This contradiction finishes the proof.
As a direct corollary of the three circle theorem, we can get the following decay lemma.

16)
and Then we have Then using the (a) and (b) of Theorem 3.3, by iterating, we obtain (3.19) which implies (3.17) immediately. If J = ∅, without loss of generality, we may assume

So, we have
Then for each j m , m = 1, ..., k, we have ≤ Ce j m L r 2 = C(e l L r 2 )e −L(l− j m ) .

Corollary 3.5 Under the assumptions of Lemma 3.4, we have
Proof By Lemma 3.2, Lemma 3.4 and a standard scaling argument, we get ≤ C (e l L r 2 )

Energy identity and no neck result
In this section, we will prove our main result Theorem 1.2. We first consider the following simpler case of a single interior blow-up point.
By Theorem 2.1, for any t ∈ (r n R, δ), we obtain For simplicity, we will denote φ n , ψ n , τ n , h n by φ, ψ, τ and h respectively. We define φ * (r ) and ψ * (r ) as follows: Next, we use the same method as in [18] to compute the equation for (φ − φ * , ψ − ψ * ). Here, for reader's convenience, we repeat this process again.
By equation (1.7), we have , e α · ψ); ψ)) + τ dθ Computing directly, we have where A i may differ from line to line and just stands for an expression satisfying Moreover, (4.12) implies for any t ∈ ( 1 2 r n R, 2δ). Then, we get Using the same method, we get and for any t ∈ ( 1 2 r n R, 2δ) by (4.12). Without loss of generality, we may assume δ = e m n L r n R for some positive integer m n which tends to ∞ as n → ∞. Substituting u = φ − φ * and v = ψ − ψ * in Corollary 3.5, we obtain the energy decay in the θ -direction,  Taking a cut-off function η ∈ C ∞ 0 (D δ ), such that 0 ≤ η ≤ 1, η ≡ 1 in D 1 2 δ \D 2r n R and |∇η| ≤ C δ in D δ \D 1 2 δ and |∇η| ≤ C r n R in D 2r n R \D r n R , by the elliptic estimates for first order equations and Sobolev embedding, we obtain where the last inequality follows from (4.22). This is E(ψ; D δ (x n )\D r n R (x n )) ≤ C( + δ). (4.23) This is (4.2) and we finished the proof of Theorem 4.1.
Proof of Theorem 1.2 It is easy to see that Theorem 1.2 is a consequence of Theorem 4.1, the removable singularity Theorem 2.5 and the standard argument in [7].