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 $\{(\phi _n,\psi _n)\}$, 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.

1 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, $\Sigma M$ be the spinor bundle over M and $\langle \cdot ,\cdot \rangle _{\Sigma M}$ be the metric on $\Sigma M$ induced by the Riemannian metric h. Choosing a local orthonormal basis ${e_\alpha ,\alpha =1,2}$ on M, the usual Dirac operator is defined as , where $\nabla$ is the spin connection on $\Sigma M$. The usual Dirac operator on a surface can be seen as the Cauchy-Riemann operator. Consider $\mathbb {R}^2$ with the Euclidean metric $dx^2+dy^2$. Let $e_1=\frac{\partial }{\partial x}$ and $e_2=\frac{\partial }{\partial y}$ be the standard orthonormal frame. A spinor field is simply a map $\psi :\mathbb {R}^2\rightarrow \Delta _2=\mathbb {C}^2$, and the action of $e_1$ and $e_2$ on spinors can be identified with multiplication with matrices

$$\begin{aligned} e_1=\begin{pmatrix}0 &{}\quad 1 \\ -1 &{}\quad 0\end{pmatrix}, \quad e_2=\begin{pmatrix}0 &{}\quad i \\ i &{}\quad 0\end{pmatrix}. \end{aligned}$$

If $\psi :=\begin{pmatrix}\psi _1 \\ \psi _2\end{pmatrix}:\mathbb {R}^2\rightarrow \mathbb {C}^2$ is a spinor field, then the Dirac operator is

(1.1)

where

$$\begin{aligned} \frac{\partial }{\partial z}=\frac{1}{2}\left( \frac{\partial }{\partial x}-i\frac{\partial }{\partial y}\right) , \quad \frac{\partial }{\partial \overline{z}}=\frac{1}{2}\left( \frac{\partial }{\partial x}+i\frac{\partial }{\partial y}\right) . \end{aligned}$$

For more details on spin geometry and Dirac operators, one can refer to [14].

Let $\phi$ be a smooth map from M to another compact Riemannian manifold (N, g) with dimension $n\ge 2$. Let $\phi ^\star TN$ be the pull-back bundle of TN by $\phi$ and then we get the twisted bundle $\Sigma M\otimes \phi ^\star TN$. Naturally, there is a metric $\langle \cdot ,\cdot \rangle _{\Sigma M\otimes \phi ^\star TN}$ on $\Sigma M\otimes \phi ^\star TN$ which is induced from the metrics on $\Sigma M$ and $\phi ^\star TN$. Also we have a natural connection $\widetilde{\nabla }$ on $\Sigma M\otimes \phi ^\star TN$ which is induced from the connections on $\Sigma M$ and $\phi ^\star TN$. Let $\psi$ be a section of the bundle $\Sigma M\otimes \phi ^\star TN$. In local coordinates, it can be written as

$$\begin{aligned} \psi =\psi ^i\otimes \partial _{y^i}(\phi ), \end{aligned}$$

where each $\psi ^i$ is a usual spinor on M and ${\partial _{y^i}}$ is the nature local basis on N. Then $\widetilde{\nabla }$ becomes

$$\begin{aligned} \widetilde{\nabla }\psi =\nabla \psi ^i\otimes \partial _{y^i}(\phi ) +(\Gamma ^i_{jk}\nabla \phi ^j)\psi ^k\otimes \partial _{y^i}(\phi ), \end{aligned}$$

(1.2)

where $\Gamma ^i_{jk}$ are the Christoffel symbols of the Levi-Civita connection of N. The Dirac operator along the map $\phi$ is defined by .

We consider the following functional

The functional $L(\phi ,\psi )$ is conformally invariant. That is , for any conformal diffeomorphism $f:M\rightarrow M$, setting

$$\begin{aligned} \widetilde{\phi }=\phi \circ f \qquad and \qquad \widetilde{\psi }=\lambda ^{-1/2}\psi \circ f. \end{aligned}$$

Then $L(\widetilde{\phi },\widetilde{\psi })=L(\phi ,\psi )$. For the proof, one can refer to [3]. Here $\lambda$ is the conformal factor of the conformal map f, i.e. $f^*h=\lambda ^2h$. Critical points $(\phi ,\psi )$ are called Dirac-harmonic maps from M to N.

The Euler-Lagrange equations of the functional L are

$$\begin{aligned} \left( \Delta \phi ^i+\Gamma ^i_{jk}h^{\alpha \beta }\phi ^j_\alpha \phi ^k_\beta \right) \frac{\partial }{\partial y^i}(\phi (x))=R(\phi ,\psi ), \end{aligned}$$

(1.3)

(1.4)

where $R(\phi ,\psi )$ is defined by

$$\begin{aligned} R(\phi ,\psi )=\frac{1}{2}R^m_{lij}(\phi (x))\langle \psi ^i, \nabla \phi ^l\cdot \psi ^j\rangle \frac{\partial }{\partial y^m}(\phi (x)). \end{aligned}$$

Here $R^m_{lij}$ stands for the Riemann curvature tensor of the target manifold (N, g). One can refer to [2, 3].

By the Nash embedding theorem, we embed N into $\mathbb {R}^K$. Then, the critical points $(\phi ,\psi )$ satisfy the Euler-Lagrange equations

$$\begin{aligned} \Delta \phi =A(\phi )(d\phi ,d\phi )+Re(P(\mathcal {A}(d\phi (e_{\alpha }),e_{\alpha }\cdot \psi );\psi )), \end{aligned}$$

(1.5)

(1.6)

where is the usual Dirac operator, A is the second fundamental form of N in $\mathbb {R}^K$, and

$$\begin{aligned} \mathcal {A}(d\phi (e_\alpha ),e_\alpha \cdot \psi )&:=(\nabla \phi ^i\cdot \psi ^j)\otimes A(\partial _{y^i},\partial _{y^j}),\\ Re(P(\mathcal {A}(d\phi (e_{\alpha }),e_{\alpha }\cdot \psi );\psi ))&:=P(A(\partial _{y^l},\partial _{y^j});\partial _{y^i})Re(\langle \psi ^i,d\phi ^l\cdot \psi ^j\rangle ). \end{aligned}$$

Here $P(\xi ;\cdot )$ denotes the shape operator, defined by $\langle P(\xi ;X),Y\rangle =\langle A(X,Y),\xi \rangle$ for $X,Y\in \Gamma (TN)$, and Re(z) denotes the real part of $z\in \mathbb {C}$. We refer to [2, 3, 5, 11, 24, 30, 33] for more details.

Denote

$$\begin{aligned} W^{2,2}(M,N):= & {} \left\{ \ \phi \in W^{2,2}(M,\mathbb {R}^K)\ with \ \phi (x)\in N\ for\ a.e.\ x\in M \ \right\} ,\\ W^{1,4/3}(M,\Sigma M\otimes \phi ^\star TN):= & {} \big \{\ \psi \in W^{1,4/3}(M,\Sigma M\otimes \mathbb {R}^K)\text { with } \psi (x)\in \Sigma M\otimes \phi ^\star TN\\&\text { for a.e. } x\in M \ \big \}. \end{aligned}$$

In this paper, with applications for the Dirac-harmonic heat flow in mind, we want to consider pairs $(\phi ,\psi )$ that satisfy the Euler-Lagrange equations up to an error term in $L^1$. Here is the precise

Definition 1.1

$(\phi ,\psi )\in W^{2,2}(M,N)\times W^{1,\frac{4}{3}}(M,\Sigma M\times \phi ^\star TN)$ is called an approximate Dirac-harmonic map if there exist $(\tau (\phi ,\psi ),h(\phi ,\psi ))\in L^1(M)$ such that

$$\begin{aligned} \tau (\phi ,\psi )= & {} \Delta \phi - A(d\phi ,d\phi )-Re\left( P(\mathcal {A}(d\phi (e_\alpha ),e_\alpha \cdot \psi );\psi )\right) , \end{aligned}$$

(1.7)

(1.8)

Thus, $(\phi ,\psi )$ is a Dirac-harmonic map if and only if $\tau (\phi ,\psi )=h(\phi ,\psi )=0$. 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 $\phi$ on $\Omega \subset M$ by

$$\begin{aligned} E(\phi ;\Omega )=\frac{1}{2}\int _\Omega |\nabla \phi |^2dM, \end{aligned}$$

the energy of $\psi$ on $\Omega \subset M$ by

$$\begin{aligned} E(\psi ;\Omega )=\int _\Omega |\psi |^4dM, \end{aligned}$$

and the energy of the pair $(\phi ,\psi )$ on $\Omega \subset M$ by

$$\begin{aligned} E(\phi ,\psi ;\Omega )=\int _\Omega (|\nabla \phi |^2+|\psi |^4)dM. \end{aligned}$$

We shall often omit the domain M from the notation and simply write $E(\phi )=E(\phi ;M), E(\psi )=E(\psi ;M)$ and $E(\phi ,\psi )=E(\phi ,\psi ;M)$.

Our first main result is

Theorem 1.2

For a sequence of smooth approximate Dirac-harmonic maps $\{(\phi _n,\psi _n)\}$ from a closed Riemann surface M to a compact Riemannian manifold N with uniformly bounded energy

$$\begin{aligned} E(\phi _n,\psi _n)\le \Lambda <\infty \end{aligned}$$

and

$$\begin{aligned} \Vert \tau (\phi _n,\psi _n)\Vert _{L^2}+\Vert h(\phi _n,\psi _n)\Vert _{L^{4}}\le \Lambda , \end{aligned}$$

defining the blow-up set

$$\begin{aligned} \mathcal {S}:=\cap _{r>0}\big \{x\in M|\liminf _{n\rightarrow \infty }\int _{D(x,r)}(|d\phi _n|^2+|\psi _n|^4) \ge \frac{\epsilon _0^2}{2}\big \}, \end{aligned}$$

(1.9)

where $\epsilon _0>0$ is as in Theorem 2.1, then $\mathcal {S}$ is a finite set $\{p_1,\ldots ,p_I\}$. There exists an approximate Dirac-harmonic map $(\phi ,\psi )$ so that, up to a subsequence, still denoted by $\{(\phi _n,\psi _n)\}$, converges weakly in $W^{2,2}_{loc}(M{\setminus } \mathcal {S})\times W^{1,2}_{loc}(M{\setminus } \mathcal {S})$ to $(\phi ,\psi )$ and there are a finite set of Dirac-harmonic spheres $(\sigma _i^l,\xi _i^l):S^2\rightarrow N, i=1,\ldots ,I$; $l=1,...,L_i$ such that

$$\begin{aligned} \lim _{n\rightarrow \infty }E(\phi _n)= & {} E(\phi )+\sum _{i=1}^I\sum _{l=1}^{L_i}E(\sigma _i^l),\end{aligned}$$

(1.10)

$$\begin{aligned} \lim _{n\rightarrow \infty }E(\psi _n)= & {} E(\psi )+\sum _{i=1}^I\sum _{l=1}^{L_i}E(\xi _i^l), \end{aligned}$$

(1.11)

and the image $\phi (M)\cup _{i=1}^I\cup _{l=1}^{L_i}(\sigma ^l_i(S^2))$ 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:

$$\begin{aligned} \lim _{n\rightarrow \infty }\int _M|\nabla \psi _n|^{\frac{4}{3}}dM&=\int _M|\nabla \psi |^{\frac{4}{3}}dM+\sum _{i=1}^I\sum _{l=1}^{L_i} \int _{S^2}|\nabla \xi ^i_l|^{\frac{4}{3}}dS^2,\end{aligned}$$

(1.12)

$$\begin{aligned} \lim _{n\rightarrow \infty }L(\phi _n,\psi _n)&=L(\phi ,\psi )+\sum _{i=1}^I\sum _{l=1}^{L_i} L\left( \sigma _i^l,\xi _i^l\right) . \end{aligned}$$

(1.13)

This is due to the fact that both $\int _M|\nabla \psi |^{\frac{4}{3}}dM$ and $L(\phi ,\psi )$ 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 $(\phi ,\psi ):M\times [0,\infty )\rightarrow N\times \phi ^\star TN$ that solves

(1.14)

with the following boundary-initial data:

$$\begin{aligned} {\left\{ \begin{array}{ll} \phi (x,t)=\varphi (x),\quad &{}on\quad \partial M\times [0,\infty );\\ \phi (x,0)=\phi _0(x),\quad &{}in\quad M;\\ \mathcal {B}\psi (x,t)=\mathcal {B}\chi (x),\quad &{}on\quad \partial M\times [0,\infty );\\ \phi _0(x)=\varphi (x),\quad &{}on\quad \partial M, \end{array}\right. } \end{aligned}$$

(1.15)

where $\tau (\phi )=\Delta \phi -A(\phi )(d\phi ,d\phi )$ is the tension field of $\phi , M$ is a compact Riemannian spin manifold with smooth boundary $\partial M, \phi _0\in W^{1,2}(M,N), \varphi \in C^{2+\alpha }(\partial M;N), \chi \in C^{1+\alpha }(\partial M;\Sigma M\otimes \phi ^\star TN)$ are given maps and $\mathcal {B}=\mathcal {B}^{\pm }$ is the chiral boundary operator defined as follows:

$$\begin{aligned}&\displaystyle \mathcal {B}^{\pm }: L^2(M,\Sigma M\otimes \phi ^\star TN|_{\partial M}) \rightarrow L^2(M,\Sigma M\otimes \phi ^\star TN|_{\partial M}) \end{aligned}$$

(1.16)

$$\begin{aligned}&\displaystyle \psi \mapsto \frac{1}{2} \left( Id \pm \overrightarrow{n}\cdot G \right) \cdot \psi , \end{aligned}$$

(1.17)

where $\overrightarrow{n}$ is the outward unit normal vector field on $\partial M$, and G is the chiral operator satisfying:

$$\begin{aligned} G^2=Id,\quad G^*=G,\quad \nabla G=0,\quad GX\cdot =-X\cdot G, \end{aligned}$$

(1.18)

for any $X\in \Gamma (TM)$.

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 $\Vert \phi _0\Vert _{H^{1}}+\Vert \mathcal {B}\chi \Vert _{L^{2}}$, 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

$$\begin{aligned}&E(\phi (t),\psi (t);M)+\int _0^t\int _{M\times \{s\}}|\partial _t \phi |^2dMds\nonumber \\ {}&\quad \le C(M,E(\phi _0),\Vert \mathcal {B}\psi _0\Vert _{L^2(\partial M)}),\quad \forall \ 0\le t<\infty . \end{aligned}$$

(1.19)

It follows from (1.19) that there exists a sequence $t_n\uparrow \infty$ such that $(\phi _n,\psi _n):=(\phi (\cdot ,t_n),\psi (\cdot ,t_n)) \in W^{2,2}(M,N)\times W^{1,\frac{4}{3}}(M,\Sigma M\times \phi ^\star TN)$ is an approximate Dirac-harmonic map with boundary-data

$$\begin{aligned} {\left\{ \begin{array}{ll} \phi (x)=\varphi (x),\quad &{}on\quad \partial M;\\ \mathcal {B}\psi (x)=\mathcal {B}\chi (x),\quad &{}on\quad \partial M, \end{array}\right. } \end{aligned}$$

(1.20)

which satisfies the assumptions of Theorem 1.2. In fact, $h(\phi _n,\psi _n)=0$ and

$$\begin{aligned} \tau (\phi _n,\psi _n):=\partial _t\phi (\cdot ,t_n)\ \text { satisfying } \ \Vert \tau (\phi _n,\psi _n)\Vert _{L^2}\rightarrow 0. \end{aligned}$$

Thus, as an immediate corollary, we obtain

Theorem 1.4

For $\mathrm{dim}\ M = 2$ and $\phi _0\in H^1(M,N), \varphi \in C^{2+\alpha }(\partial M,N), \chi \in C^{1+\alpha }(\partial M,\Sigma M\otimes \varphi ^\star TN)$, let $(\phi ,\psi ):M\times [0,\infty )\rightarrow N\times \phi ^\star TN$ be a global weak solution of (1.14) and (1.15), which has finitely many singular times and satisfies (1.19). Moreover, we assume that $(\phi ,\psi )$ blows up at infinite time and at interior points. Then there exist $t_n\uparrow \infty$, a Dirac-harmonic map $(\phi _\infty ,\psi _\infty )\in C^{2+\alpha }(M,N)\times C^{1+\alpha }(M,\Sigma M\otimes \phi _\infty ^{*}TN)$ with boundary data $\phi _\infty |_{\partial M}=\varphi$ and $\mathcal {B}\psi _\infty |_{\partial M}=\mathcal {B}\chi$, and a nonnegative integer I and finitely many points $\{p_1,...,p_I\}\in M$ such that

• (1) $(\phi _n,\psi _n):=(\phi (\cdot ,t_n),\psi (\cdot ,t_n))\rightharpoonup (\phi _\infty ,\psi _\infty )\ in\ W^{1,2}(M,N)\times W^{1,\frac{4}{3}}(M,\Sigma M\times \phi ^\star TN)$;

• (2) $(\phi _n,\psi _n)\rightarrow (\phi _\infty ,\psi _\infty )\ in\ W_{loc}^{1,2}(M{\setminus } \{p_1,...,p_L\})\times L_{loc}^{4}(M{\setminus } \{p_1,...,p_I\})$;

• (3) For $1\le i\le I$, there exist a positive integer $L_i$ and $L_i$ nontrivial Dirac-harmonic spheres $(\sigma _i^l,\xi _i^l):S^2\rightarrow N, i=1,...,I$; $l=1,...,L_i$ such that

  $$\begin{aligned} \lim _{n\rightarrow \infty }E(\phi _n)= & {} E(\phi _\infty )+\sum _{i=1}^I\sum _{l=1}^{L_i}E(\sigma _i^l),\end{aligned}$$

  (1.21)
  $$\begin{aligned} \lim _{n\rightarrow \infty }E(\psi _n)= & {} E(\psi _\infty )+\sum _{i=1}^I\sum _{l=1}^{L_i}E(\xi _i^l), \end{aligned}$$

  (1.22)

  and

  $$\begin{aligned} \lim _{n\rightarrow \infty }\Vert \phi (\cdot ,t_n)-\phi _\infty (\cdot )-\sigma _i^l(\cdot )\Vert _{L^\infty (M)}=0. \end{aligned}$$

  (1.23)

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.

2 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.

Theorem 2.1

There is a small constant $\epsilon _0>0$ such that if $(\phi ,\psi )\in W^{2,p}(D,N)\times W^{1,q}(D,\Sigma D\otimes \phi ^\star TN)$ is an approximate Dirac-harmonic map from the unit disc D in $\mathbb {R}^2$ to a compact Riemannian manifold (N, g) with $\tau \in L^p, h\in L^q$ for some $\frac{4}{3}\le p\le 2, \frac{8}{5}\le q\le 2$, and satisfies

$$\begin{aligned} E(\phi ,\psi ;D)=\int _{D}{(|d\phi |^2+|\psi |^4)}dx<\epsilon _0^2, \end{aligned}$$

(2.1)

then

$$\begin{aligned} \Vert \phi -\overline{\phi }\Vert _{W^{2,p}\left( D_\frac{1}{2}\right) }\le & {} C(\Vert d\phi \Vert _{L^2(D)}+\Vert \tau \Vert _{L^p(D)}),\\ \Vert \psi \Vert _{W^{1,q}\left( D_\frac{1}{2}\right) }\le & {} C(\Vert \psi \Vert _{L^4(D)}+\Vert h\Vert _{L^q(D)}), \end{aligned}$$

where $\overline{\phi }:=\frac{1}{|D_{1/2}|}\int _{D_{1/2}}\phi dx$ and $C>0$ is a constant depending only on $p,\ q,\ \Lambda ,\ N$.

Moreover, by the Sobolev embedding $W^{2,p}(\mathbb {R}^2)\subset C^0(\mathbb {R}^2)$, we have

$$\begin{aligned} \Vert \phi \Vert _{Osc(D_{1/2})}= & {} \sup _{x,y\in D_{1/2}}|\phi (x)-\phi (y)|\nonumber \\\le & {} C(\Lambda ,N)(\Vert \nabla \phi \Vert _{L^2(D)}+\Vert \tau (u)\Vert _{L^p(D)}). \end{aligned}$$

(2.2)

Proof

Without loss of generality, we assume $\frac{1}{|D_{1/2}|}\int _{D_{1/2}}\phi dx=0$.

Choosing a cut-off function $\eta \in C_0^\infty (D)$ satisfying $0\le \eta \le 1,\eta |_{D_{3/4}}\equiv 1,|\nabla \eta |+|\nabla ^2\eta |\le C$, by the standard theory of first order elliptic equations, for any $1<q<2$, we have

Taking $q=\frac{8}{5}$ and $\epsilon _0>0$ sufficiently small, by Sobolev embedding, we obtain

$$\begin{aligned} \Vert \eta \psi \Vert _{L^{8}(D)}\le C\Vert \eta \psi \Vert _{W^{1,8/5}(D)}\le C(\Vert \psi \Vert _{L^q(D)}+\Vert h\Vert _{L^q(D)}). \end{aligned}$$

(2.3)

Computing directly, we get

$$\begin{aligned} |\Delta (\eta \phi )|&=|\eta \Delta \phi +2\nabla \eta \nabla \phi +\phi \Delta \eta |\nonumber \\&\le C\left( |\phi |+|d\phi |+|d\phi ||\eta d\phi |+|\psi |^2|\eta d\phi |+|\tau |\right) \nonumber \\&\le C|d\phi ||d(\eta \phi )|+ C\left( |\phi |+|d\phi |+\eta |\psi |^2|d\phi |+|\tau |\right) . \end{aligned}$$

(2.4)

By standard elliptic estimates,

$$\begin{aligned} \Vert \eta \phi \Vert _{W^{2,4/3}(D)}&\le C\Vert |d\phi ||d(\eta \phi )|\Vert _{L^{4/3}(D)}\\&\quad +C\left( \Vert d\phi \Vert _{L^{4/3}(D)}+\Vert |\eta \psi |^2|d\phi |\Vert _{L^{4/3}(D)}+\Vert |\tau |\Vert _{L^{4/3}(D)}\right) \\&\le C\Vert d(\eta \phi )\Vert _{L^{4}(D)}\Vert d\phi \Vert _{L^2(D)}\\&\quad +C\left( \Vert d\phi \Vert _{L^{4/3}(D)}+\Vert \eta \psi \Vert ^2_{L^{8}(D)}\Vert d\phi \Vert _{L^2(D)}+\Vert |\tau |\Vert _{L^{4/3}(D)}\right) \\&\le C\epsilon _0\Vert d(\eta \phi )\Vert _{L^{4}(D)}+C(\Vert d\phi \Vert _{L^2(D)}+\Vert |\tau |\Vert _{L^{4/3}(D)}). \end{aligned}$$

Taking $\epsilon _0>0$ sufficiently small, we have

$$\begin{aligned} \Vert d\phi \Vert _{L^{4}(D_{3/4})}\le C\Vert \eta \phi \Vert _{W^{2,4/3}(D)}\le C(\Vert d\phi \Vert _{L^{2}(D)}+\Vert \tau \Vert _{L^{4/3}(D)}). \end{aligned}$$

(2.5)

By the interior elliptic estimates, for any $\frac{4}{3}\le p\le 2$ we have

$$\begin{aligned} \Vert \phi \Vert _{W^{2,p}(D_{1/2})}&\le C(\Vert \Delta \phi \Vert _{L^{p}(D_{3/4})}+\Vert d\phi \Vert _{L^{p}(D_{3/4})})\\&\le C\left( \Vert d\phi \Vert ^2_{L^{4}(D_{3/4})}+\Vert \psi \Vert ^2_{L^{8}(D_{3/4})}\Vert d\phi \Vert _{L^4(D_{3/4})}\right. \\&\left. \quad +\Vert |\tau |\Vert _{L^{p}(D)}+\Vert d\phi \Vert _{L^{p}(D_{3/4})}\right) \\&\le C(\Vert d\phi \Vert _{L^2(D)}+\Vert \tau \Vert _{L^{p}(D)}). \end{aligned}$$

By the interior elliptic estimates for first order equations, for any $\frac{8}{5}\le q\le 2$, we get

$\square$

Secondly, by a direct computation, we have the following proposition.

Proposition 2.2

Suppose $\phi \in C^2(M,N), \psi \in C^2(M,\Sigma M\otimes \phi ^\star TN)$. Let $\{e_\alpha \}_{\alpha =1}^2$ be a unit normal basis of TM and $e_\beta \in \Gamma (TM)$ a section satisfying

$$\begin{aligned}{}[e_\beta ,e_\alpha ]=0,\ \alpha =1,2, \end{aligned}$$

then

(2.6)

where $[\cdot ,\cdot ]$ is the Lie bracket, $\phi _\beta =d\phi (e_\beta )$ and $\psi _\beta =\widetilde{\nabla }_{e_\beta }\psi$.

Proof

Proposition 2.2 is proved in [3]. For the reader’s convenience, we recall it here.

On the one hand, we have

$$\begin{aligned} \langle Re\left( P(\mathcal {A}(d\phi (e_\alpha ),e_\alpha \cdot \psi );\psi )\right) ,\phi _\beta \rangle&=\left\langle \frac{1}{2}R^m_{lij}(\phi )\langle \psi ^i,\nabla \phi ^l\cdot \psi ^j\rangle \partial _{y^m}, \phi ^p_\beta \partial _{y^p}\right\rangle \\&=\frac{1}{2}R_{mlij}\langle \psi ^i,\nabla \phi ^l\cdot \psi ^j\rangle \phi ^m_\beta . \end{aligned}$$

On the other hand, by a direct computation, we get

$$\begin{aligned} \widetilde{\nabla }_{e_\alpha }\widetilde{\nabla }_{e_\beta }\psi -\widetilde{\nabla }_{e_\beta }\widetilde{\nabla }_{e_\alpha }\psi = R^{\Sigma M}(e_\alpha ,e_\beta )\psi ^i\otimes \partial {y^i}+R^m_{lij}\phi ^i_\alpha \phi ^j_\beta \psi ^l\otimes \partial {y^m}, \end{aligned}$$

where $R^{\Sigma M}$ is the curvature operator of the connection on the spinor bundle $\Sigma M$. For this curvature, we have (see [3, 6, 14])

$$\begin{aligned} e_\alpha \cdot R^{\Sigma M}(e_\alpha ,X)\psi ^i=\frac{1}{2}Ric(X), \ \forall X\in \Gamma (M). \end{aligned}$$

Thus we obtain

$$\begin{aligned} \langle \psi ,e_\alpha \cdot R^{\Sigma M}(e_\alpha ,e_\beta )\psi ^i\otimes \partial {y^i}\rangle&= g_{ij}\langle \psi ^j,e_\alpha \cdot R^{\Sigma M}(e_\alpha ,e_\beta )\psi ^i\rangle \\&= \frac{1}{2}g_{ij}\langle \psi ^j,Ric(e_\beta )\cdot \psi ^i\rangle =0. \end{aligned}$$

Therefore,

Then, the conclusion of the proposition follows immediately. $\square$

Thirdly, we compute Pohozaev’s identity for approximate Dirac-harmonic maps.

Lemma 2.3

Let $D\subset \mathbb {R}^2$ be the unit disk and $(\phi ,\psi )$ be a smooth approximate Dirac-harmonic map, then for any $0<t<\frac{1}{2}$, we have

(2.7)

where $(r,\theta )$ are polar coordinates in D centered at $0, \phi _r=d\phi (\frac{\partial }{\partial r}), \psi _r=\widetilde{\nabla }_{\frac{\partial }{\partial r}}\psi$ and $\psi _\theta =\widetilde{\nabla }_{\frac{\partial }{\partial \theta }}\psi$.

Proof

Multiplying the equation (1.7) by $r\phi _r$ and integrating over $D_t$, by Proposition 2.2 and the fact that $r\phi _r=x^\beta \frac{\partial \phi }{\partial x^\beta }=x^\beta d\phi (\frac{\partial }{\partial x^\beta })$, we get

On the one hand, integrating by parts, we have

$$\begin{aligned} \mathbb {I}&=\int _{\partial D_t}r|\phi _r|^2-\int _{D_t}\nabla \phi \nabla (r\phi _r)dx\\&= \int _{\partial D_t}r|\phi _r|^2-\int _{D_t}|\nabla \phi |^2dx-\frac{1}{2}\int _{D_t}r\frac{\partial }{\partial r}|\nabla \phi |^2 dx\\&= r\int _{\partial D_t}\left( |\phi _r|^2-\frac{1}{2}|\nabla \phi |^2\right) . \end{aligned}$$

On the other hand, by Lemma 2.6, we get

(2.8)

and

(2.9)

Combining the preceding equations, we get the conclusion of the lemma. $\square$

Corollary 2.4

Under the assumption of Lemma 2.3, if $\Vert \tau (\phi ,\psi )\Vert _{L^2(D)}+\Vert h(\phi ,\psi )\Vert _{L^4(D)}\le C$, then for any $0<t<\frac{1}{4}$, we have

$$\begin{aligned}&\int _{ D_{2t}{\setminus } D_t}\left( |\phi _r|^2-\frac{1}{2}|\nabla \phi |^2\right) dx\nonumber \\&\quad \le C\left( \Vert r^{-1}\frac{\partial \psi }{\partial \theta }\Vert _{L^{\frac{4}{3}}(D_{2t}{\setminus } D_t)}+\Vert r^{-1}\frac{\partial \phi }{\partial \theta }\Vert _{L^{2}(D_{2t}{\setminus } D_t)}\right) +Ct. \end{aligned}$$

(2.10)

Proof

By Lemma 2.3, for any $0<s<\frac{1}{2}$, we have

$$\begin{aligned} s\int _{\partial D_s}\left( |\phi _r|^2-\frac{1}{2}|\nabla \phi |^2\right) :=\mathbb {J}_1+\cdots +\mathbb {J}_4. \end{aligned}$$

(2.11)

It is easy to see that

and

where we used the fact that

$$\begin{aligned} |\psi _r|\le & {} C\left( |\frac{\partial \psi }{\partial r}|+|\psi ||\frac{\partial \phi }{\partial r}|\right) ,\\ |\psi _\theta |\le & {} C\left( |\frac{\partial \psi }{\partial \theta }|+|\psi ||\frac{\partial \phi }{\partial \theta }|\right) . \end{aligned}$$

Multiplying (2.7) by $\frac{1}{s}$ and integrating from t to 2t, we get

$$\begin{aligned} \int _{ D_{2t}{\setminus } D_t}\left( |\phi _r|^2-\frac{1}{2}|\nabla \phi |^2\right)&\le \int _t^{2t}\frac{1}{2s}\int _{\partial D_s}\left\langle \psi ,r^{-1}\frac{\partial }{\partial \theta }\cdot \psi _\theta \right\rangle d\theta ds+Ct\\&\le C\Vert r^{-1}\psi _\theta \Vert _{L^{\frac{4}{3}}(D_{2t}{\setminus } D_t)}\Vert \psi \Vert _{L^{4}(D_{2t}{\setminus } D_t)}+Ct\\&\le C\left( \Vert r^{-1}\frac{\partial \psi }{\partial \theta }\Vert _{L^{\frac{4}{3}}(D_{2t}{\setminus } D_t)}+\Vert r^{-1}\frac{\partial \phi }{\partial \theta }\Vert _{L^{2}(D_{2t}{\setminus } D_t)}\right) +Ct. \end{aligned}$$

$\square$

Thirdly, we state an interior removable singularity result.

Theorem 2.5

Let $(\phi ,\psi )\in W^{2,2}_{loc}(D{\setminus }\{0\})\times W^{1,2}_{loc}(D{\setminus }\{0\})$ be an approximate Dirac-harmonic map from $D{\setminus }\{0\}$ to N with finite energy

$$\begin{aligned} \Vert d\phi \Vert _{L^2(D)}+\Vert \psi \Vert _{L^4(D)}\le C \end{aligned}$$

that satisfies

$$\begin{aligned} \tau =f\in L^2(D),\quad x\in D{\setminus }\{0\}, \end{aligned}$$

(2.12)

$$\begin{aligned} h=g\in L^2(D),\quad x\in D{\setminus }\{0\}, \end{aligned}$$

(2.13)

then $(\phi ,\psi )$ can be extended to a field in $W^{2,2}(D)\times W^{1,2}(D)$.

Proof

By a standard argument as in Lemma A.2 in [9], it is easy to see that $(\phi ,\psi )$ is a weak solution of (2.12) and (2.13). It is known that the equation of $\phi$ can be written as an elliptic system with an anti-symmetric potential [5, 24, 30]:

$$\begin{aligned} \Delta \phi = \Omega \cdot \nabla \phi +f, \end{aligned}$$

with $\Omega \in L^2(B_1,so(N)\otimes \mathbb {R}^2)$ satisfying $|\Omega |\le C (|\nabla \phi |+|\psi |^2)$. Then it follows from Riviere’s regularity result and its extensions (see [26,27,28]) that $\phi \in W^{2,p}(D)$ for any $1<p<2$ which implies $\phi \in W^{1,4}(D)$. Applying a simple argument to the Dirac equation for $\psi$ gives that $\psi \in W^{1,2}(D)$. This indicate that $\psi \in L^{8}(D)$. Then by (2.12), we have $\Delta \phi \in L^2(D)$ which implies the conclusion of the theorem. $\square$

In the end of this section, we recall two lemmas which are used in this paper.

Lemma 2.6

([5]) For any $\psi ,\omega \in W^{1,3/4}(M,\Sigma M\otimes \phi ^\star TN)$, we have

(2.14)

where $\langle \psi ,\omega \rangle :=h_{ij}\langle \psi ^i,\omega ^j\rangle$.

Proposition 2.7

([2]) Let N be a compact Riemannian manifold. Then there exists a constant $\epsilon _1=\epsilon _1(N)>0$ such that if $(\phi ,\psi )$ is a smooth Dirac-harmonic map from the standard sphere $S^2$ to N satisfying

$$\begin{aligned} \int _{S^2}(|d\phi |^2+|\psi |^4)<\epsilon _1, \end{aligned}$$

then both $\phi$ and $\psi$ are trivial.

3 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]).

Theorem 3.1

There exists a constant $L>0$, such that if u is a nontrivial smooth harmonic function defined in $[(i-1)L,(i+2)L]\times S^1$ that satisfies

$$\begin{aligned} \int _{\{iL\}\times S^1}u d\theta =\int _{\{(i+1)L\}\times S^1}u d\theta =0, \end{aligned}$$

then

$$\begin{aligned} \Vert u\Vert ^2_{L^2([iL,(i+1)L]\times S^1)}< & {} \frac{1}{2}\left( e^{-L}\Vert u\Vert ^2_{L^2([(i-1)L,iL]\times S^1)}\right. \nonumber \\&\left. +\, e^{-L}\Vert u\Vert ^2_{L^2([(i+1)L,(i+2)L]\times S^1)}\right) . \end{aligned}$$

(3.1)

Next, we prove an $L^2$ interior estimate for the following integro-differential equations.

Lemma 3.2

Suppose $u\in W^{2,2}(D_{4}{\setminus } D_{1}), v\in W^{1,2}(D_{4}{\setminus } D_{1})$ and satisfies

$$\begin{aligned} \Delta u= & {} A^1u+A^2\nabla u+A^3v+\frac{1}{2\pi }\int _0^{2\pi }A^4u+A^5\nabla u+A^6vd\theta +f_1, \end{aligned}$$

(3.2)

(3.3)

where

$$\begin{aligned} \sum _{i=1}^6\left( \Vert A^i\Vert _{L^{4}(D_{4}{\setminus } D_{1})}+\Vert B^i\Vert _{L^{4}(D_{4}{\setminus } D_{1})}\right) \le \rho \quad and \quad \sum _{i=1}^2\Vert f_i\Vert _{L^2(D_{4}{\setminus } D_{1})}\le C. \end{aligned}$$

(3.4)

Then there exists a positive constant $\rho _0$ such that if $\rho \le \rho _0$, there holds

$$\begin{aligned}&\Vert u\Vert _{W^{2,2}(D_{3}{\setminus } D_{2})}+\Vert v\Vert _{W^{1,2}(D_{3}{\setminus } D_{2})}\nonumber \\&\quad \le C\left( \Vert u\Vert _{L^2(D_{4}{\setminus } D_{1})}+\Vert v\Vert _{L^2(D_{4}{\setminus } D_{1})}+\Vert f_1\Vert _{L^2(D_{4}{\setminus } D_{1})}+\Vert f_2\Vert _{L^2(D_{4}{\setminus } D_{1})}\right) . \end{aligned}$$

(3.5)

Proof

The proof is similar to Lemma 3.1 in [18] where $f_1=f_2=0$.

Denote $B_{\sigma }=D_{3+\sigma }{\setminus } D_{2-\sigma }, 0<\sigma <1$. Let $\sigma '=\frac{\sigma +1}{2}$. Take a cut-off function $\eta (x)=\eta (|x|)$ with compact support in $B_{\sigma '}$ satisfying $\eta (x)\equiv 1$ in $B_{\sigma }$ and $|\nabla \eta |\le \frac{4}{(1-\sigma )}$ and $|\Delta \eta |\le \frac{16}{(1-\sigma )^2}$. Computing directly, we get

$$\begin{aligned} \Delta (\eta u)&=\eta \Delta u+2\nabla \eta \nabla u+\Delta \eta u\\&=(2\nabla \eta +\eta A^2)\nabla u +(\Delta \eta +\eta A^1)u+\eta A^3v+\eta f_1\\&\quad +\eta \cdot \frac{1}{2\pi }\int _0^{2\pi }A^4u+A^5\nabla u+A^6vd\theta . \end{aligned}$$

By the standard elliptic estimate and Sobolev embedding, we have

$$\begin{aligned} \Vert \eta u\Vert _{W^{2,2}(D_4)}&\le C\big (\Vert A^1\eta u\Vert _{L^2(D_4)}+\Vert A^2\eta \nabla u\Vert _{L^2(D_4)}\\&\quad +\Vert A^3\eta v\Vert _{L^2(D_4)}+\Vert \eta f_1\Vert _{L^2(D_4)}\\&\quad +\Vert \nabla \eta \nabla u\Vert _{L^2(D_4)}+\Vert \Delta \eta u\Vert _{L^2(D_4)}\\&\quad +\Vert \eta \cdot \frac{1}{2\pi }\int _0^{2\pi }A^4u+A^5\nabla u+A^6vd\theta \Vert _{L^2(D_4)}\big )\\&\le C\left( \Vert A^1\Vert _{L^2(D_{4}{\setminus } D_{1})}+\Vert A^4\Vert _{L^2(D_{4}{\setminus } D_{1})}\right. \\&\left. \quad +\Vert A^2\Vert _{L^4(D_{4}{\setminus } D_{1})}+\Vert A^5\Vert _{L^4(D_{4}{\setminus } D_{1})}\right) \Vert \eta u\Vert _{W^{2,2}(D_4)}\\&\quad +C(\Vert A^3\Vert _{L^4(D_{4}{\setminus } D_{1})}+\Vert A^6\Vert _{L^4(D_{4}{\setminus } D_{1})})\Vert \eta v\Vert _{W^{1,2}(D_4)} +C(\Vert A^2\nabla \eta u\Vert _{L^2(D_4)} \\&\quad +\Vert A^5\nabla \eta u\Vert _{L^2(D_4)})+C(\Vert \nabla \eta \nabla u\Vert _{L^2(D_4)}+\Vert \Delta \eta u\Vert _{L^2(D_4)}+\Vert \eta f_1\Vert _{L^2(D_4)})\\&\le C\rho _0\Vert \eta u\Vert _{W^{2,2}(D_4)} +C\rho _0\Vert \eta v\Vert _{W^{1,2}(D_4)}\\&\quad +C(\Vert A^2\Vert _{L^4(D_{4}{\setminus } D_{1})}+\Vert A^5\Vert _{L^4(D_{4}{\setminus } D_{1})})\Vert \nabla \eta u\Vert _{W^{1,2}(D_4)}\\&\quad +C\left( \frac{\Vert \nabla u\Vert _{L^2(B_{\sigma '})}}{1-\sigma }+\frac{\Vert u\Vert _{L^2(B_{\sigma '})}}{(1-\sigma )^2}+\Vert \eta f_1\Vert _{L^2(D_4)}\right) \\&\le C\rho _0(\Vert \eta u\Vert _{W^{2,2}(D_4)} +\Vert \eta v\Vert _{W^{1,2}(D_4)})\\&\quad +C\left( \frac{\Vert \nabla u\Vert _{L^2(B_{\sigma '})}}{1-\sigma }+\frac{\Vert u\Vert _{L^2(B_{\sigma '})}}{(1-\sigma )^2}+\Vert \eta f_1\Vert _{L^2(D_4)}\right) . \end{aligned}$$

Similarly, we can compute

By the first order elliptic estimate, we have

$$\begin{aligned} \Vert \eta v\Vert _{W^{1,2}(D_4)}&\le C\bigg (\Vert B^1\eta u\Vert _{L^2(D_4)}+\Vert B^2\eta \nabla u\Vert _{L^2(D_4)}+\Vert B^3\eta v\Vert _{L^2(D_4)}\\&\quad +\Vert \eta f_2\Vert _{L^2(D_4)} +\Vert \nabla \eta v\Vert _{L^2(D_4)}\\&\quad +\Vert \eta \cdot \frac{1}{2\pi }\int _0^{2\pi }B^4u+B^5\nabla u+B^6vd\theta \Vert _{L^2(D_4)}\bigg )\\&\le C(\Vert B^1\Vert _{L^4(D_{4}{\setminus } D_{1})}+\Vert B^4\Vert _{L^4(D_{4}{\setminus } D_{1})}\\&\quad +\Vert B^2\Vert _{L^{4}(D_{4}{\setminus } D_{1})}+\Vert B^5\Vert _{L^{4}(D_{4}{\setminus } D_{1})})\Vert \eta u\Vert _{W^{2,2}(D_4)}\\&\quad +C(\Vert B^3\Vert _{L^4(D_{4}{\setminus } D_{1})}+\Vert B^6\Vert _{L^4(D_{4}{\setminus } D_{1})})\Vert \eta v\Vert _{W^{1,2}(D_4)}\\&\quad +C(\Vert B^2\nabla \eta u\Vert _{L^2(D_4)}+\Vert B^5\nabla \eta u\Vert _{L^2(D_4)})\\&\quad +C(\Vert \nabla \eta v\Vert _{L^2(D_4)}+\Vert \eta f_2\Vert _{L^2(D_4)})\\&\le C\rho _0\Vert \eta u\Vert _{W^{2,2}(D_4)} +C\rho _0\Vert \eta v\Vert _{W^{1,2}(D_4)}\\&\quad +C(\Vert B^2\Vert _{L^4(D_{4}{\setminus } D_{1})}+\Vert B^5\Vert _{L^4(D_{4}{\setminus } D_{1})})\Vert \nabla \eta u\Vert _{W^{1,2}(D_4)}\\&\quad +C\left( \frac{\Vert v\Vert _{L^2(D_{4}{\setminus } D_{1})}}{1-\sigma }+\Vert \eta f_2\Vert _{L^2(D_4)}\right) \\&\le C\rho _0(\Vert \eta u\Vert _{W^{2,2}(D_4)} +\Vert \eta v\Vert _{W^{1,2}(D_4)})\\&\quad +C\left( \frac{\Vert \nabla u\Vert _{L^2(B_{\sigma '})}}{1-\sigma }+\frac{\Vert u\Vert _{L^2(B_{\sigma '})}}{(1-\sigma )^2}+\frac{\Vert v\Vert _{L^2(D_{4}{\setminus } D_{1})}}{1-\sigma }+\Vert f_2\Vert _{L^2(D_{4}{\setminus } D_{1})}\right) . \end{aligned}$$

Taking $\rho _0$ sufficiently small, we get

$$\begin{aligned} \Vert \eta u\Vert _{W^{2,2}(B_1)}+\Vert \eta v\Vert _{W^{1,2}(B_1)}&\le C\Bigg (\frac{\Vert \nabla u\Vert _{L^2(B_{\sigma '})}}{1-\sigma }+\frac{\Vert u\Vert _{L^2(B_{\sigma '})}}{(1-\sigma )^2}+\frac{\Vert v\Vert _{L^2(D_{4}{\setminus } D_{1})}}{1-\sigma }\nonumber \\&\quad +\Vert f_1\Vert _{L^2(D_{4}{\setminus } D_{1})}+\Vert f_2\Vert _{L^2(D_{4}{\setminus } D_{1})}\Bigg ). \end{aligned}$$

(3.6)

We now introduce seminorms, and define for $j=0,1,2$

$$\begin{aligned} \Xi _j=\sup _{0\le \sigma \le 1}(1-\sigma )^j\Vert D^ju\Vert _{L^2(B_{\sigma })}. \end{aligned}$$

Multiplying (3.6) by $(1-\sigma )^2$ and noting that $1-\sigma '=\frac{1-\sigma }{2}$, we have

$$\begin{aligned} \Xi _2\le C\left( \Xi _1+\Xi _0 +\Vert v\Vert _{L^2(D_{4}{\setminus } D_{1})}+\Vert f_1\Vert _{L^2(D_{4}{\setminus } D_{1})}+\Vert f_2\Vert _{L^2(D_{4}{\setminus } D_{1})}\right) . \end{aligned}$$

(3.7)

We claim now that $\Xi _j$ satisfy an interpolation inequality

$$\begin{aligned} \Xi _1\le \epsilon \Xi _2+\frac{C}{\epsilon }\Xi _0 \end{aligned}$$

(3.8)

for any $\epsilon >0$, where $C>0$ is a universal constant. In fact, by the definition of $\Xi _1$, for any $\gamma >0$, we have

$$\begin{aligned} \Xi _1\le & {} (2-\sigma _\gamma )\Vert Du\Vert _{L^2(B_{\sigma _\gamma })}+\gamma \\\le & {} \epsilon (2-\sigma _\gamma )^2\Vert D^2u\Vert _{L^2(B_{\sigma _\gamma })}+\frac{C}{\epsilon }\Vert u\Vert _{L^2(B_{\sigma _\gamma })}+\gamma , \end{aligned}$$

where the second inequality is derived from the interpolation Theorem 7.27 (or Theorem 7.28) in [8].

By letting $\gamma \rightarrow 0$, we obtain (3.8). Using (3.8) in (3.7), we then obtain

$$\begin{aligned} \Xi _2\le C\left( \Vert u\Vert _{L^2(D_{4}{\setminus } D_{1})}+\Vert v\Vert _{L^2(D_{4}{\setminus } D_{1})}+\Vert f_1\Vert _{L^2(D_{4}{\setminus } D_{1})}+\Vert f_2\Vert _{L^2(D_{4}{\setminus } D_{1})}\right) , \end{aligned}$$

this is

$$\begin{aligned} \Vert D^2u\Vert _{L^2(B_{\sigma })}\le & {} \frac{C}{(1-\sigma )^2} \left( \Vert u\Vert _{L^2(D_{4}{\setminus } D_{1})}+\Vert v\Vert _{L^2(D_{4}{\setminus } D_{1})}\right. \nonumber \\&\left. +\Vert f_1\Vert _{L^2(D_{4}{\setminus } D_{1})}+\Vert f_2\Vert _{L^2(D_{4}{\setminus } D_{1})}\right) . \end{aligned}$$

Taking $\sigma =\frac{1}{2}$, it follows

$$\begin{aligned} \Vert u\Vert _{W^{2,2}(B_{1/2})}\le & {} C\left( \Vert u\Vert _{L^2(D_{4}{\setminus } D_{1})}+\Vert v\Vert _{L^2(D_{4}{\setminus } D_{1})}\right. \nonumber \\&\left. +\Vert f_1\Vert _{L^2(D_{4}{\setminus } D_{1})}+\Vert f_2\Vert _{L^2(D_{4}{\setminus } D_{1})}\right) . \end{aligned}$$

(3.9)

Choosing a new cut-off function $\eta$ in (3.6) and using (3.9), we get

$$\begin{aligned} \Vert v\Vert _{W^{1,2}(B_{1/4})}\le & {} C\left( \Vert u\Vert _{L^2(D_{4}{\setminus } D_{1})}+\Vert v\Vert _{L^2(D_{4}{\setminus } D_{1})}\right. \nonumber \\&\left. +\Vert f_1\Vert _{L^2(D_{4}{\setminus } D_{1})}+\Vert f_2\Vert _{L^2(D_{4}{\setminus } D_{1})}\right) . \end{aligned}$$

(3.10)

Then it is easy to see that the lemma follows from (3.9) and (3.10). $\square$

Denote $P_i:=D_{e^{(i+1)L}r_2}{\setminus } D_{e^{iL}r_2}$ and

$$\begin{aligned} F_i(u,v):=\int _{P_i}\frac{1}{|x|^2}|u|^2dx+\int _{P_i}\frac{1}{|x|}|v|^2dx, \end{aligned}$$

where $L>0$ is the constant in Theorem 3.1.

We have the following three circle theorem:

Theorem 3.3

Suppose $u\in W^{2,2}(P_{i-1}\cup P_{i}\cup P_{i+1}), v\in W^{1,2}(P_{i-1}\cup P_{i}\cup P_{i+1})$ satisfy equations (3.2) and (3.3). Then there exists a positive constant $\rho _0$, such that if $0<\rho _1<\rho _0$ and

$$\begin{aligned} \max _{i-1,i,i+1}\left( \Vert |x|f_1\Vert ^2_{L^2(P_j)}+\Vert |x|^{\frac{1}{2}}f_2\Vert ^2_{L^2(P_j)} \right) \le \rho _1 F_i(u,v), \end{aligned}$$

(3.11)

and for any $e^{(i-1)L}r_2\le r\le \frac{1}{2}e^{(i+2)L}r_2$, there hold

$$\begin{aligned}&\Vert |x|^{\frac{3}{2}}(|A^1|+|A^4|)\Vert _{L^{4}(D_{2r}{\setminus } D_r)}\nonumber \\&\quad +\Vert |x|(|A^3|+|A^6|+|B^1|+|B^4|)\Vert _{L^{4}(D_{2r}{\setminus } D_r)}\nonumber \\&\quad +\Vert |x|^{\frac{1}{2}}(|A^2|+|A^5|+|B^3|+|B^6|)\Vert _{L^{4}(D_{2r}{\setminus } D_r)}\nonumber \\&\quad +\Vert |B^2|+|B^5|\Vert _{L^{4}(D_{2r}{\setminus } D_r)}\le \rho _1, \end{aligned}$$

(3.12)

and

$$\begin{aligned}&\left| \int _{0}^{2\pi }u(e^{iL}r_2,\theta )d\theta \right| ^2+ \left| \int _{0}^{2\pi }u(e^{(i+1)L}r_2,\theta )d\theta \right| ^2\\&\quad + \left| \int _{0}^{2\pi }v(e^{iL}r_2,\theta )d\theta \right| ^2+ \left| \int _{0}^{2\pi }v(e^{(i+1)L}r_2,\theta )d\theta \right| ^2\le \rho _1 F_i(u,v) \end{aligned}$$

then, there hold

• $(a)\ F_{i+1}(u,v)\le e^{-L}F_{i}(u,v)$ implies $F_{i}(u,v)\le e^{-L}F_{i-1}(u,v)$;

• $(b)\ F_{i-1}(u,v)\le e^{-L}F_{i}(u,v)$ implies $F_{i}(u,v)\le e^{-L}F_{i+1}(u,v)$;

• (c) either $F_{i}(u,v)\le e^{-L}F_{i-1}(u,v)$ or $F_{i}(u,v)\le e^{-L}F_{i+1}(u,v)$.

Proof

Since $F_i(u,v)$, (3.11) and (3.12) are scaling invariant, we may assume $r_2=1$ and $i=2$. If the conclusion does not hold, there exist sequences $\rho _{1k}\rightarrow 0, A^j_{k}, B^j_{k}$ (j=1,...,6), $f_{1k}, f_{2k}, u_k, v_k$ such that $u_k$ and $v_k$ satisfy

and

$$\begin{aligned}&\left| \int _{0}^{2\pi }u_k(e^{iL},\theta )d\theta \right| ^2+ \left| \int _{0}^{2\pi }u_k(e^{(i+1)L},\theta )d\theta \right| ^2\\&\quad + \left| \int _{0}^{2\pi }u_k(e^{iL},\theta )d\theta \right| ^2 + \left| \int _{0}^{2\pi }u_k(e^{(i+1)L},\theta )d\theta \right| ^2\le \rho _{1k}F_i(u_k,v_k). \end{aligned}$$

Moreover, for any $e^{(i-1)L}\le r\le e^{(i+2)L}, A^j_{k}, B^j_{k}, f_{1k}, f_{2k}$ satisfy

$$\begin{aligned}&\Vert |x|^{\frac{3}{2}}(|A_k^1|+|A_k^4|)\Vert _{L^{4}(D_{2r}{\setminus } D_r)}\\&\quad +\Vert |x|(|A_k^3|+|A_k^6|+|B_k^1|+|B_k^4|)\Vert _{L^{4}(D_{2r}{\setminus } D_r)}\\&\quad +\Vert |x|^{\frac{1}{2}}(|A_k^2|+|A_k^5|+|B_k^3|+|B_k^6|)\Vert _{L^{4}(D_{2r}{\setminus } D_r)}\\&\quad +\Vert |B_k^2|+|B_k^5|\Vert _{L^{4}(D_{2r}{\setminus } D_r)}\le \rho _{1k} \end{aligned}$$

and

$$\begin{aligned} \max _{i-1,i,i+1}(\Vert |x|f_{1k}\Vert ^2_{L^2(P_j)}+\Vert |x|^{\frac{1}{2}}f_{2k}\Vert ^2_{L^2(P_j)} )\le \rho _{1k} F_i(u,v). \end{aligned}$$

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

$$\begin{aligned} F_{2}(u_k,v_k)\ge e^{L}F_{3}(u_k,v_k)\quad and \quad F_{2}(u_k,v_k)\ge e^{-L}F_{1}(u_k,v_k); \end{aligned}$$

If (b) does not hold, then we have

$$\begin{aligned} F_{2}(u_k,v_k)\ge e^{L}F_{1}(u_k,v_k)\quad and \quad F_{2}(u_k,v_k)\ge e^{-L}F_{3}(u_k,v_k); \end{aligned}$$

If (c) does not hold, then we have

$$\begin{aligned} F_{2}(u_k,v_k)\ge e^{-L}\max \{F_{1}(u_k,v_k),F_{3}(u_k,v_k)\}; \end{aligned}$$

In all those three cases, we may get the same conclusion that

$$\begin{aligned} 2F_{2}(u_k,v_k)\ge e^{-L}(F_{1}(u_k,v_k)+F_{3}(u_k,v_k)). \end{aligned}$$

(3.13)

Without loss of generality, we assume $F_{2}(u_k,v_k)=1$ (if not, we consider $\widetilde{u_k}=\frac{u_k}{F_2(u_k,v_k)}$ and $\widetilde{v_k}=\frac{v_k}{F_2(u_k,v_k)}$). Then we obtain

$$\begin{aligned} \Vert u_k\Vert _{L^2(P_{1}\cup P_2\cup P_3)}+\Vert v_k\Vert _{L^2(P_{1}\cup P_2\cup P_3)}\le C. \end{aligned}$$

By Lemma 3.2, we have $\Vert u_k\Vert _{W^{2,2}( P_2)}+\Vert v_k\Vert _{W^{1,2}( P_2)}\le C$. So, there exists a subsequence of $(u_k,v_k)$ (we still denote it by $(u_k,v_k)$), such that

$$\begin{aligned} u_k\rightharpoonup u, \quad v_k\rightharpoonup v&\text{ weakly } \text{ in } L^2(P_1\cup P_2\cup P_3);\\ u_k\rightarrow u, \quad v_k\rightarrow v&\text{ strongly } \text{ in } L^2( P_2). \end{aligned}$$

It is easy to see that u is a harmonic function and v is a holomorphic function in $D_{e^{(i+2)L}}{\setminus } D_{e^{(i-1)L}}$ and they satisfy

$$\begin{aligned} \int _{\partial D_{e^{iL}}}u=\int _{\partial D_{e^{(i+1)L}}}u=\int _{\partial D_{e^{iL}}}v=\int _{\partial D_{e^{(i+1)L}}}v=0. \end{aligned}$$

Let $f:\mathbb {R}^1\times \mathbb {S}^1\rightarrow \mathbb {R}^2, f(t,\theta )=(e^{t},\theta ) (t,\theta )\in \mathbb {R}^1\times \mathbb {S}^1$ where $\mathbb {R}^1\times \mathbb {S}^1$ is equipped with the metric $g=dt^2+d\theta ^2$, which is conformal to the standard Euclidean metric $ds^2$ on $\mathbb {R}^2$. In fact,

$$\begin{aligned} (f^{-1})^*g=\frac{1}{r^2}ds^2. \end{aligned}$$

Then we know that $u\circ f$ is a harmonic function and $e^{\frac{t}{2}}v\circ f$ is a holomorphic function in $[L,4L]\times S^1$.

According to Theorem 3.1, we know

$$\begin{aligned}&\Vert u\circ f\Vert ^2_{L^2([2L,3L]\times S^1)}\\&\quad <\frac{1}{2}\left( e^{-L}\Vert u\circ f\Vert ^2_{L^2([L,2L]\times S^1)}+e^{-L}\Vert u\circ f\Vert ^2_{L^2([3L,4L]\times S^1)}\right) \end{aligned}$$

and

$$\begin{aligned}&\Vert e^{\frac{t}{2}}v\circ f\Vert ^2_{L^2([2L,3L]\times S^1)}\\&\quad <\frac{1}{2}\left( e^{-L}\Vert e^{\frac{t}{2}}v\circ f\Vert ^2_{L^2([L,2L]\times S^1)}+e^{-L}\Vert e^{\frac{t}{2}}v\circ f\Vert ^2_{L^2([3L,4L]\times S^1)}\right) \end{aligned}$$

which implies

$$\begin{aligned} \Vert \frac{1}{|x|}u\Vert ^2_{L^2(P_2)}<\frac{1}{2}\left( e^{-L}\Vert \frac{1}{|x|}u\Vert ^2_{L^2(P_{1})} +e^{-L}\Vert \frac{1}{|x|}u\Vert ^2_{L^2(P_{3})}\right) \end{aligned}$$

and

$$\begin{aligned} \Vert \frac{1}{\sqrt{|x|}}v\Vert ^2_{L^2(P_2)} <\frac{1}{2}\left( e^{-L}\Vert \frac{1}{\sqrt{|x|}}v\Vert ^2_{L^2(P_{1})} +e^{-L}\Vert \frac{1}{\sqrt{|x|}}v\Vert ^2_{L^2(P_{3})}\right) . \end{aligned}$$

Thus,

$$\begin{aligned} 2F_{2}(u,v)< e^{-L}(F_{1}(u,v)+F_{3}(u,v)). \end{aligned}$$

(3.14)

But, letting $k\rightarrow \infty$ in (3.13) which implies

$$\begin{aligned} 2F_{2}(u,v)\ge e^{-L}(F_{1}(u,v)+F_{3}(u,v)). \end{aligned}$$

(3.15)

This contradiction finishes the proof. $\square$

As a direct corollary of the three circle theorem, we can get the following decay lemma.

Lemma 3.4

Let $\rho _1>0$ be the constant in Theorem 3.3. Let $u\in W^{2,2}(D_{e^{(l+1)L}r_2}{\setminus } D_{r_2}), v\in W^{1,2}(D_{e^{(l+1)L}r_2}{\setminus } D_{r_2}), f_i\in L^{2}(D_{e^{(l+1)L}r_2}{\setminus } D_{r_2}), i=1,2$, and some integer $l>1$, satisfying equations (3.2), (3.3) and for any $r_2\le r\le \frac{1}{2} e^{(l+1)L}r_2$, there hold

$$\begin{aligned}&\Vert |x|^{\frac{3}{2}}(|A^1|+|A^4|)\Vert _{L^{4}(D_{2r}{\setminus } D_r)}\nonumber \\&\quad +\Vert |x|(|A^3|+|A^6|+|B^1|+|B^4|)\Vert _{L^{4}(D_{2r}{\setminus } D_r)}\nonumber \\&\quad +\Vert |x|^{\frac{1}{2}}(|A^2|+|A^5|+|B^3|+|B^6|)\Vert _{L^{4}(D_{2r}{\setminus } D_r)}\nonumber \\&\quad +\Vert |B^2|+|B^5|\Vert _{L^{4}(D_{2r}{\setminus } D_r)}\le \rho _1, \end{aligned}$$

(3.16)

and

$$\begin{aligned} \int _{\partial D_r}u=\int _{\partial D_r}v=0. \end{aligned}$$

Then we have

$$\begin{aligned} F_{i}(u,v)\le C \bigg (e^{lL}r_2+F_{0}(u,v)+F_{l}(u,v)\big )\big ( e^{-(l-i)L}+e^{-iL}\bigg ). \end{aligned}$$

(3.17)

Proof

Denote the set of $j\ (0<j<l)$ for which

$$\begin{aligned} \max _{j-1,j,j+1}\left( \Vert |x|f_1\Vert ^2_{L^2(P_i)}+\Vert |x|^{\frac{1}{2}}f_2\Vert ^2_{L^2(P_i)} \right) >\rho _1 F_j(u,v) \end{aligned}$$

(3.18)

by $J:=\{j_1,...,j_k\}$. If $J=\emptyset$, according to (c) of Theorem 3.3, we get

$$\begin{aligned} F_{i}(u,v)\le e^{-L}F_{i-1}(u,v)\ or\ F_{i}(u,v)\le e^{-L}F_{i+1}(u,v). \end{aligned}$$

Then using the (a) and (b) of Theorem 3.3, by iterating, we obtain

$$\begin{aligned} F_{i}(u,v)\le e^{-Li}F_{0}(u,v)\ or\ F_{i}(u,v)\le e^{-L(l-i)}F_{l}(u,v). \end{aligned}$$

So, we have

$$\begin{aligned} F_{i}(u,v)\le \frac{1}{2}(e^{-Li}F_{0}(u,v)+ e^{-L(l-i)}F_{l}(u,v)), \end{aligned}$$

(3.19)

which implies (3.17) immediately.

If $J\ne \emptyset$, without loss of generality, we may assume

$$\begin{aligned} 0<j_1<j_2<\cdots<j_k<l. \end{aligned}$$

Then for each $j_m, m=1,...,k$, we have

$$\begin{aligned} F_{j_m}(u,v)&\le C\max _{j_m-1,j_m,j_m+1}\left( \Vert |x|f_1\Vert ^2_{L^2(P_i)}+\Vert |x|^{\frac{1}{2}}f_2\Vert ^2_{L^2(P_i)} \right) \\&\le C e^{j_mL}r_2=C (e^{lL}r_2)e^{-L(l-j_m)}. \end{aligned}$$

By the choice of $j_m$, the condition (3.11) holds for $j_m<i<j_{m+1}, m=1,...,k-1$. Similar to deriving (3.19), we obtain

$$\begin{aligned} F_{i}(u,v)\le & {} e^{-L(i-j_m)}F_{j_m}(u,v)\ or\ F_{i}(u,v)\\\le & {} e^{-L(j_{m+1}-i)}F_{j_{m+1}}(u,v). \end{aligned}$$

Thus, we know for $j_1\le i\le j_k$, there exists $m\in \{1,...,k-1\}$ such that $j_m\le i\le j_{m+1}$. Then we get

$$\begin{aligned} F_{i}(u,v)&\le \frac{1}{2}( e^{-L(i-j_m)}F_{j_m}(u,v)+ e^{-L(j_{m+1}-i)}F_{j_{m+1}}(u,v))\nonumber \\&\le C (e^{lL}r_2)\big (e^{-L(i-j_m)}e^{-L(l-j_m)}+e^{-L(j_{m+1}-i)} e^{-L(l-j_{m+1})}\big )\nonumber \\&\le C (e^{lL}r_2)e^{-(l-i)L}. \end{aligned}$$

(3.20)

So, if $j_1=1$ and $j_k=l-1$, then the inequality (3.19) follows immediately. If not, assuming $j_1>1$, similar to deriving (3.20), we have, for $1\le i\le j_1$,

$$\begin{aligned} F_{i}(u,v)&\le \frac{1}{2}(e^{-Li}F_{0}(u,v)+ e^{-L(j_1-i)}F_{j_1}(u,v))\\&\le C \big (e^{-Li}F_{0}(u,v)+(e^{lL}r_2)e^{-(l-i)L}\big ). \end{aligned}$$

Similarly, if $j_k<l-1$, then for $j_k\le i\le l$, we have

$$\begin{aligned} F_{i}(u,v)&\le \frac{1}{2}\bigg (e^{-L(l-i)}F_{l}(u,v)+ e^{-L(i-j_k)}F_{j_k}(u,v)\bigg )\\&\le C \bigg (e^{-L(l-i)}F_{l}(u,v)+(e^{lL}r_2)e^{-(l-i)L}\bigg ). \end{aligned}$$

Combining the preceding estimates proves the lemma. $\square$

Corollary 3.5

Under the assumptions of Lemma 3.4, we have

$$\begin{aligned}&\Vert \nabla u\Vert _{L^2(P_i)}+\Vert \nabla v\Vert _{L^{\frac{4}{3}}(P_i)} \nonumber \\&\le C \bigg ((e^{lL}r_2)^{\frac{1}{2}}+F^{1/2}_{0}(u,v)+F^{1/2}_{l}(u,v)\big )\big ( e^{-\frac{1}{2}(l-i)L}+e^{-\frac{1}{2}iL}\bigg ). \end{aligned}$$

(3.21)

Proof

By Lemma 3.2, Lemma 3.4 and a standard scaling argument, we get

$$\begin{aligned} \Vert \nabla u\Vert _{L^2(P_i)}+\Vert \nabla v\Vert _{L^{\frac{4}{3}}(P_i)}&\le C\bigg (F^{1/2}_{i-1}(u,v)+F^{1/2}_i(u,v)\\&\quad +F^{1/2}_{i+1}(u,v)+e^{iL}r_2\Vert f_1\Vert _{L^2(P_{i-1}\cup P_i\cup P_{i+1})}\\&\quad +(e^{iL}r_2)^{\frac{1}{2}}\Vert f_2\Vert _{L^2(P_{i-1}\cup P_i\cup P_{i+1})}\bigg ) \nonumber \\&\le C \bigg ((e^{lL}r_2)^{\frac{1}{2}}+F^{1/2}_{0}(u)+F^{1/2}_{l}(u)\big )\big ( e^{-\frac{1}{2}(l-i)L}+e^{-\frac{1}{2}iL}\bigg ). \end{aligned}$$

$\square$

4 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.

Theorem 4.1

Let $(\phi _n,\psi _n)$ be a sequence of smooth approximate Dirac-harmonic maps from $D_1(0)$ to N with

• (a) $\ \Vert \phi _n\Vert _{W^{1,2}(D)}+\Vert \psi _n\Vert _{L^{4}(D)}+\Vert \tau _n\Vert _{L^{2}(D)} +\Vert h_n\Vert _{L^{4}(D)}\le \Lambda ,$

• (b) $\ (\phi _n,\psi _n)\rightarrow (\phi ,\psi ) \text{ strongly } \text{ in } W_{loc}^{1,2}(D{\setminus }\{0\},\mathbb {R}^N)\times W_{loc}^{1,\frac{4}{3}}(D{\setminus }\{0\},\mathbb {R}^N)\ as\ n\rightarrow \infty$,

where $\tau _n:=\tau (\phi _n,\psi _n)$ and $h_n:=h(\phi _n,\psi _n)$. Then there exist a subsequence of $(\phi _n,\psi _n)$ (still denoted by $(\phi _n,\psi _n)$) and a nonnegative integer Q such that, for any $i=1,...,Q$, there exist point $x^i_n$, positive numbers $\lambda ^i_n$ and a nonconstant Dirac-harmonic sphere $(\sigma ^i,\xi ^i):S^2\rightarrow N$ such that:

• (1) $\ x^i_n\rightarrow 0,\ \lambda ^i_n\rightarrow 0$, as $n\rightarrow \infty$;

• (2) $\ \lim _{n\rightarrow \infty }\big (\frac{\lambda ^i_n}{\lambda ^j_n}+\frac{\lambda ^j_n}{\lambda ^i_n} +\frac{|x^i_n-x^j_n|}{\lambda ^i_n+\lambda ^j_n}\big )=\infty$ for any $i\ne j$;

• (3) $\ (\sigma ^i,\xi ^i)$ is the weak limit of $(\phi _n(x^i_n+\lambda ^i_nx),\sqrt{\lambda ^i_n}\psi _n(x^i_n+\lambda ^i_nx))$ in $W^{1,2}_{loc}(\mathbb {R}^2)\times W^{1,\frac{4}{3}}_{loc}(\mathbb {R}^2)$;

• (4) Energy identity: we have

  $$\begin{aligned} \lim _{n\rightarrow \infty }E(\phi _n)= & {} E(\phi )+\sum _{i=1}^{Q}E(\sigma ^i), \end{aligned}$$

  (4.1)
  $$\begin{aligned} \lim _{n\rightarrow \infty }E(\psi _n)= & {} E(\psi )+\sum _{i=1}^{Q}E(\xi ^i), \end{aligned}$$

  (4.2)

• (5) No neck property: The image

  $$\begin{aligned} \phi (D)\cup \bigcup _{i=1}^Q\sigma ^i(S^2) \end{aligned}$$

  (4.3)

  is a connected set.

Proof

Assume 0 is the only blow-up point of the sequence $\{(\phi _n,\psi _n)\}$ in D, i.e.

$$\begin{aligned} \liminf _{n\rightarrow \infty }E(\phi _n,\psi _n;D_r)\ge \frac{\epsilon _0^2}{2} \text{ for } \text{ all } r>0. \end{aligned}$$

(4.4)

By the standard argument of blow-up analysis we can assume that, for any n, there exist sequences $x_n\rightarrow 0$ and $r_n\rightarrow 0$ such that

$$\begin{aligned} E(\phi _n,\psi _n;D_{r_n}(x_n))=\sup _{\begin{array}{c} x\in D,r\le r_n\\ D_r(x)\subset D \end{array}}E(\phi _n,\psi _n;D_r(x))=\frac{\epsilon _0^2}{4}. \end{aligned}$$

(4.5)

Denoting

$$\begin{aligned} \phi '_n(x):=\phi _n(x_n+r_nx),\psi '_n(x):=\sqrt{r_n}\psi _n(x_n+r_nx) \end{aligned}$$

(4.6)

then we have

$$\begin{aligned} \tau (\phi '_n,\psi '_n)=\Delta \phi '_n- A(d\phi '_n,d\phi '_n)-Re\left( P(\mathcal {A}(d\phi '_n(e_\alpha ),e_\alpha \cdot \psi '_n);\psi '_n)\right) , \end{aligned}$$

(4.7)

(4.8)

where $\tau (\phi '_n,\psi '_n)=r_n^2\tau (\phi _n,\psi _n)$ and $h(\phi '_n,\psi '_n)=r_n^{3/2}h(\phi _n,\psi _n)$. Noting that for any $D_{R}(y)\subset \mathbb R^2$ with $R>0$, there holds

$$\begin{aligned} E(\phi '_n,\psi '_n;D_{R}(y))\le & {} E(\phi _n,\psi _n;D_{\frac{1}{2}}(x_n))\le \Lambda < \infty ,\\ E(\phi '_n,\psi '_n;D)= & {} E(\phi _n,\psi _n;D_{r_n}(x_n))=\frac{\epsilon _0^2}{4}, \end{aligned}$$

for n large enough. By the small energy regularity Theorem 2.1, the removable singularity Theorem 2.5 and conformal invariance of Dirac-harmonic maps in dimension two, we can take a subsequence, still denoted by $(\phi '_n,\psi '_n)$, that strongly converges to a nonconstant Dirac-harmonic sphere. This is the first bubble.

By the standard induction argument in [7], we only need to prove the theorem in the case where there is only one bubble. Under this assumption, we have the following:

Claim

For any $\epsilon >0$, there exist $\delta >0$ and $R>0$ such that

$$\begin{aligned} E(\phi _n,\psi _n;D_{8t}(x_n){\setminus } D_{t}(x_n))\le \epsilon ^2 \text{ for } \text{ any } t\in \bigg (\frac{1}{2}r_nR,2\delta \bigg ) \end{aligned}$$

(4.9)

when n is large enough.

Proof

In fact, if (4.9) is not true, then we can find $\overline{\epsilon }>0, t_n\rightarrow 0$, such that $\lim _{n\rightarrow \infty }\frac{t_n}{r_n}=\infty$ and

$$\begin{aligned} E(\phi _n,\psi _n;D_{8t_n}(x_n){\setminus } D_{t_n}(x_n))\ge \overline{\epsilon }>0. \end{aligned}$$

(4.10)

Setting

$$\begin{aligned} u_n(x):=\phi _n(x_n+t_nx),v_n(x):=\sqrt{t_n}\psi _n(x_n+t_nx), \end{aligned}$$

then it is easy to see that 0 is an energy concentration point for $(u_n,v_n)$. We have to consider the following two cases:

$\mathbf (a)$ :

$(u_n,v_n)$ has no other energy concentration points except 0.

By Theorem 2.1, passing to a subsequence, we may assume that $(u_n,v_n)$ converges to a Dirac-harmonic map $(\sigma ,\xi ):\mathbb R^2\rightarrow N$ strongly in $W^{1,2}_{loc}(\mathbb R^2)\times L^4_{loc}(\mathbb R^2)$ as $n\rightarrow \infty$. In particular, we have

$$\begin{aligned} \lim _{n\rightarrow \infty }E(u_n,v_n;D_8{\setminus } D_1)=E(\sigma ,\xi ;D_8{\setminus } D_1)\ge \overline{\epsilon }. \end{aligned}$$

According to the standard theory of Dirac-harmonic maps, we know that $(\sigma ,\xi )$ is a nontrivial Dirac-harmonic sphere. This is the second bubble. This is a contradiction to the “one bubble” assumption.

$\mathbf (b)$ :

$(u_n,v_n)$ has another energy concentration point $p\ne 0$.

Without loss of generality, we may assume that p is the only blow-up point in $D_r(p)$ for some small $r>0$. By the standard theory of blow-up analysis, there exist $x_n'\rightarrow p$ and $r_n'\rightarrow 0$ such that

$$\begin{aligned} E(u_n,v_n;D_{r_n'}(x_n'))=\sup _{\begin{array}{c} x\in D_r(p),s\le r_n\\ D_s(x)\subset D_r(p) \end{array}}E(u_n,v_n;D_s(x))=\frac{\epsilon _0^2}{4}. \end{aligned}$$

(4.11)

From the process of constructing the first bubble, we know that there exists a nontrivial Dirac-harmonic sphere $(\widetilde{\sigma },\widetilde{\xi })$ such that

$$\begin{aligned} (u_n(x_n'+r_n'x),r_n'^{1/2}v_n(x_n'+r_n'x))\rightarrow (\widetilde{\sigma },\widetilde{\xi }) \text{ strongly } \text{ in } W^{1,2}_{loc}(\mathbb R^2)\times L^4_{loc}(\mathbb R^2) \end{aligned}$$

as $n\rightarrow \infty$. This is

$$\begin{aligned}&(\phi _n(x_n+t_nx_n'+t_nr_n'x),(t_nr_n')^{1/2}\psi _n(x_n+t_nx_n'+t_nr_n'x))\rightarrow (\widetilde{\sigma },\widetilde{\xi })\\&\text{ strongly } \text{ in } W^{1,2}_{loc}(\mathbb R^2)\times L^4_{loc}(\mathbb R^2) \end{aligned}$$

as $n\rightarrow \infty$. By (4.11), $(\widetilde{\sigma },\widetilde{\xi })$ is nontrivial. Therefore, we again get the second bubble contradicting the “one bubble” assumption. So, we proved Claim (4.9).

By Theorem 2.1, for any $t\in (r_nR,\delta )$, we obtain

$$\begin{aligned}&\Vert |x|^{\frac{3}{4}}\nabla \phi _n\Vert _{L^8(D_{2t}(x_n){\setminus } D_{t}(x_n))}+\Vert |x|^{\frac{3}{8}}\psi _n\Vert _{L^{16}(D_{2t}(x_n){\setminus } D_{t}(x_n))}\nonumber \\&\quad \le C(\Vert \nabla \phi _n\Vert _{L^2(D_{4t}(x_n){\setminus } D_{t/2}(x_n))}+\Vert \psi _n\Vert _{L^4(D_{4t}(x_n){\setminus } D_{t/2}(x_n))}+t\Vert \tau _n\Vert _{L^2(D_{4t}(x_n){\setminus } D_{t/2}(x_n))}\nonumber \\&\qquad +\sqrt{t}\Vert h_n\Vert _{L^2(D_{4t}(x_n){\setminus } D_{t/2}(x_n))})\nonumber \\&\quad \le C(\epsilon +\sqrt{\delta }). \end{aligned}$$

(4.12)

For simplicity, we will denote $\phi _n, \psi _n, \tau _n, h_n$ by $\phi , \psi , \tau$ and h respectively.

We define $\phi ^*(r)$ and $\psi ^*(r)$ as follows:

$$\begin{aligned} \phi ^*(r)=\frac{1}{2\pi }\int _0^{2\pi }\phi (r,\theta ) d\theta \quad and \quad \psi ^*(r)=\frac{1}{2\pi }\int _0^{2\pi }\psi (r,\theta ) d\theta . \end{aligned}$$

(4.13)

Next, we use the same method as in [18] to compute the equation for $(\phi -\phi ^*,\psi -\psi ^*)$. Here, for reader’s convenience, we repeat this process again.

By equation (1.7), we have

$$\begin{aligned} \Delta \phi ^*(t)= & {} \frac{1}{2\pi }\int _0^{2\pi }A(\phi )(d\phi ,d\phi ) +Re(P(\phi )\left( \mathcal {A}(\phi )(d\phi (e_\alpha ),e_\alpha \cdot \psi );\psi \right) )+\tau d\theta \\= & {} \mathrm {I}+\mathrm {II}+\frac{1}{2\pi }\int _0^{2\pi }\tau d\theta . \end{aligned}$$

Computing directly, we have

$$\begin{aligned} \mathrm {I}= & {} \frac{1}{2\pi }\int _0^{2\pi }A(\phi )(d\phi ,d\phi )-A(\phi ^*)(d\phi ,d\phi )\\&+A(\phi ^*)(d\phi ,d\phi )-A(\phi ^*)(d\phi ^*,d\phi ^*)+A(\phi ^*)(d\phi ^*,d\phi ^*)d\theta \\= & {} A(\phi ^*)(d\phi ^*,d\phi ^*)+\frac{1}{2\pi }\int _0^{2\pi }A^4(\phi -\phi ^*)+A^5\nabla (\phi -\phi ^*)d\theta , \end{aligned}$$

and

$$\begin{aligned} \mathrm {II}= & {} \frac{1}{2\pi }Re\int _0^{2\pi } P(\phi )\left( \mathcal {A}(\phi )(d\phi (e_\alpha ),e_\alpha \cdot \psi );\psi \right) \\&-P(\phi ^*)\left( \mathcal {A}(\phi )(d\phi (e_\alpha ),e_\alpha \cdot \psi );\psi \right) \\&+P(\phi ^*)\left( \mathcal {A}(\phi )(d\phi (e_\alpha ),e_\alpha \cdot \psi );\psi \right) \\&-P(\phi ^*)\left( \mathcal {A}(\phi ^*)(d\phi (e_\alpha ),e_\alpha \cdot \psi );\psi \right) \\&+P(\phi ^*)\left( \mathcal {A}(\phi ^*)(d\phi (e_\alpha ),e_\alpha \cdot \psi );\psi \right) \\&-P(\phi ^*)\left( \mathcal {A}(\phi ^*)(d\phi ^*(e_\alpha ),e_\alpha \cdot \psi );\psi \right) \\&+P(\phi ^*)\left( \mathcal {A}(\phi ^*)(d\phi ^*(e_\alpha ),e_\alpha \cdot \psi );\psi \right) \\&-P(\phi ^*)\left( \mathcal {A}(\phi ^*)(d\phi ^*(e_\alpha ),e_\alpha \cdot \psi ^*);\psi \right) \\&+P(\phi ^*)\left( \mathcal {A}(\phi ^*)(d\phi ^*(e_\alpha ),e_\alpha \cdot \psi ^*);\psi \right) \\&-P(\phi ^*)\left( \mathcal {A}(\phi ^*)(d\phi ^*(e_\alpha ),e_\alpha \cdot \psi ^*);\psi ^*\right) \\&+P(\phi ^*)\left( \mathcal {A}(\phi ^*)(d\phi ^*(e_\alpha ),e_\alpha \cdot \psi ^*);\psi ^*\right) d\theta \\= & {} Re \left( P(\phi ^*)\left( \mathcal {A}(\phi ^*)(d\phi ^*(e_\alpha ),e_\alpha \cdot \Psi ^*);\psi ^*\right) \right) \\&+ \frac{1}{2\pi }\int _0^{2\pi }A^4(\phi -\phi ^*)+A^5\nabla (\phi -\phi ^*)\\&+\frac{1}{2\pi }Re\int _0^{2\pi }A^6(\psi -\psi ^*)d\theta , \end{aligned}$$

where $A^i$ may differ from line to line and just stands for an expression satisfying

$$\begin{aligned} |A^4|&\le C(N)(|d\phi |^2+|d\phi ||\psi |^2),\\ |A^5|&\le C(N)(|d\phi |+|\psi |^2),\\ |A^6|&\le C(N)|d\phi ||\psi |. \end{aligned}$$

Moreover, (4.12) implies

$$\begin{aligned} \Vert |x|^{\frac{3}{2}}A^4\Vert _{L^4(D_{2t}{\setminus } D_{t})}+\Vert |x|A^6\Vert _{L^4(D_{2t}{\setminus } D_{t})}+\Vert |x|^{\frac{1}{2}}A^5\Vert _{L^4(D_{2t}{\setminus } D_{t})}\le C\epsilon \end{aligned}$$

for any $t\in (\frac{1}{2}r_nR,2\delta )$. Then, we get

$$\begin{aligned} \Delta (\phi -\phi ^*)= & {} A(\phi )(d\phi ,d\phi )-A(\phi ^*)(d\phi ^*,d\phi ^*)\\&+Re P(\phi )\left( \mathcal {A}(\phi )(d\phi (e_\alpha ),e_\alpha \cdot \psi );\psi \right) \\&-Re P(\phi ^*)\left( \mathcal {A}(\phi ^*)(d\phi ^*(e_\alpha ),e_\alpha \cdot \psi ^*);\psi ^*\right) \\&-\frac{1}{2\pi }\int _0^{2\pi }A^4(\phi -\phi ^*)+A^5\nabla (\phi -\phi ^*)+Re\big (A^6(\psi -\psi ^*)\big ) d\theta \\&+\tau -\frac{1}{2\pi }\int _0^{2\pi }\tau d\theta . \end{aligned}$$

Using the same method, we get

$$\begin{aligned} \Delta (\phi -\phi ^*)= & {} A^1(\phi -\phi ^*)+A^2\nabla (\phi -\phi ^*)+Re(A^3(\psi -\psi ^*))\nonumber \\&+\frac{1}{2\pi }\int _0^{2\pi }A^4(\phi -\phi ^*)+A^5\nabla (\phi -\phi ^*)+Re(A^6(\psi -\psi ^*))d\theta \nonumber \\&+\tau -\frac{1}{2\pi }\int _0^{2\pi }\tau d\theta , \end{aligned}$$

(4.14)

and

(4.15)

where $A^i,\ B^i,\ i=1,...,6$ satisfy

$$\begin{aligned}&|A^1|+|A^4|\le C(N)(|d\phi |^2+|d\phi ||\psi |^2),\\&|A^2|+|A^5|+|B^3|+|B|^6\le C(N)(|d\phi |+|\psi |^2),\\&|A^3|+|A^6|+|B^1|+|B^4|\le C(N)|d\phi ||\psi |,\\&|B^2|+|B^5|\le C(N)|\psi |. \end{aligned}$$

and

$$\begin{aligned}&\Vert |x|^{\frac{3}{2}}(|A^1|+|A^4|)\Vert _{L^{4}(D_{2r}{\setminus } D_r)}\nonumber \\&\quad +\Vert |x|\left( |A^3|+|A^6|+|B^1|+|B^4|\right) \Vert _{L^{4}(D_{2r}{\setminus } D_r)}\nonumber \\&\quad +\Vert |x|^{\frac{1}{2}}(|A^2|+|A^5|+|B^3|+|B^6|)\Vert _{L^{4}(D_{2r}{\setminus } D_r)}\nonumber \\&\quad +\Vert |B^2|+|B^5|\Vert _{L^{4}(D_{2r}{\setminus } D_r)}\le C\epsilon \end{aligned}$$

(4.16)

for any $t\in (\frac{1}{2}r_nR,2\delta )$ by (4.12).

Without loss of generality, we may assume $\delta =e^{m_nL}r_nR$ for some positive integer $m_n$ which tends to $\infty$ as $n\rightarrow \infty$. Substituting $u=\phi -\phi ^*$ and $v=\psi -\psi ^*$ in Corollary 3.5, we obtain the energy decay in the $\theta$-direction,

$$\begin{aligned} \Vert r^{-1}\frac{\partial \phi }{\partial \theta }\Vert _{L^2(P_i)}+\Vert r^{-1}\frac{\partial \psi }{\partial \theta }\Vert _{L^{\frac{4}{3}}(P_i)}&\le \Vert \nabla u\Vert _{L^2(P_i)}+\Vert \nabla v\Vert _{L^{\frac{4}{3}}(P_i)} \nonumber \\&\le C \bigg ((e^{iL}r_nR)^{\frac{1}{2}}+F^{1/2}_{0}(u,v)+F^{1/2}_{m_n}(u,v)\bigg )\nonumber \\&\quad \bigg ( e^{-\frac{1}{2}(m_n-i)L}+e^{-\frac{1}{2}iL}\bigg )\nonumber \\&\le C(\sqrt{\epsilon }+\sqrt{\delta })\bigg ( e^{-\frac{1}{2}(m_n-i)L}+e^{-\frac{1}{2}iL}\bigg ), \end{aligned}$$

(4.17)

where the last inequality follows from Poincaré’s inequality and the assumption (4.9).

By Corollary 2.4, we get

$$\begin{aligned} \Vert \frac{\partial \phi }{\partial r}\Vert _{L^2(P_i)}&\le \Vert r^{-1}\frac{\partial \phi }{\partial \theta }\Vert ^{\frac{1}{2}}_{L^2(P_i)}+C\Vert r^{-1}\frac{\partial \psi }{\partial \theta }\Vert ^{\frac{1}{2}}_{L^{\frac{4}{3}}(P_i)} +C\sqrt{e^{iL}r_nR}\nonumber \\&\le C\left( \epsilon ^{\frac{1}{4}}+\delta ^{\frac{1}{4}}\right) \big ( e^{-\frac{1}{4}(m_n-i)L}+e^{-\frac{1}{4}iL}\big ). \end{aligned}$$

(4.18)

Therefore,

$$\begin{aligned} \Vert \nabla \phi \Vert _{L^2(P_i)}&\le C(\epsilon ^{\frac{1}{4}}+\delta ^{\frac{1}{4}})\big ( e^{-\frac{1}{4}(m_n-i)L}+e^{-\frac{1}{4}iL}\big ). \end{aligned}$$

(4.19)

Then, by Theorem 2.1, we have

$$\begin{aligned} E(\phi ;D_\delta (x_n){\setminus } D_{r_nR}(x_n))&\le \sum _{i=0}^{m_n-1}\Vert \nabla \phi \Vert ^2_{L^2(P_i)}\le C(\epsilon ^{\frac{1}{2}}+\delta ^{\frac{1}{2}}) \end{aligned}$$

(4.20)

and

$$\begin{aligned} Osc_{D_\delta (x_n){\setminus } D_{r_nR}(x_n)}\phi _n&\le C\sum _{i=0}^{m_n-1}\left( \Vert \nabla \phi \Vert _{L^2(P_i)}+e^{iL}r_nR\Vert \tau \Vert _{L^2(P_i)}\right) \nonumber \\&\le C\left( \epsilon ^{\frac{1}{4}}+\delta ^{\frac{1}{4}}\right) . \end{aligned}$$

(4.21)

So, we have proved (4.1) and (4.3).

Combining this with equation (1.8), we get

$$\begin{aligned} \Vert \frac{\partial \psi }{\partial r}\Vert _{L^{\frac{4}{3}}(P_i)}&\le \Vert r^{-1}\frac{\partial \psi }{\partial \theta }\Vert _{L^{\frac{4}{3}}(P_i)}+C\Vert \nabla \phi \Vert _{L^2(P_i)}\Vert \psi \Vert _{L^4(P_i)}+\Vert h\Vert _{L^{\frac{4}{3}}(P_i)}\\&\le C(\epsilon ^{\frac{1}{4}}+\delta ^{\frac{1}{4}})\big ( e^{-\frac{1}{4}(m_n-i)L}+e^{-\frac{1}{4}iL}\big )+Ce^{iL}r_nR\Vert h\Vert _{L^4(P_i)}\\&\le C(\epsilon ^{\frac{1}{4}}+\delta ^{\frac{1}{4}})\big ( e^{-\frac{1}{4}(m_n-i)L}+e^{-\frac{1}{4}iL}\big ). \end{aligned}$$

Thus,

$$\begin{aligned} \Vert \nabla \psi \Vert _{L^{\frac{4}{3}}(P_i)}&\le C\left( \epsilon ^{\frac{1}{4}}+\delta ^{\frac{1}{4}}\right) \big ( e^{-\frac{1}{4}(m_n-i)L}+e^{-\frac{1}{4}iL}\big ). \end{aligned}$$

(4.22)

Taking a cut-off function $\eta \in C^\infty _0(D_\delta )$, such that $0\le \eta \le 1, \eta \equiv 1$ in $D_{\frac{1}{2}\delta }{\setminus } D_{2r_nR}$ and

$$\begin{aligned} |\nabla \eta |\le \frac{C}{\delta } \ in\ D_{\delta }{\setminus } D_{\frac{1}{2}\delta }\ and\ |\nabla \eta |\le \frac{C}{r_nR} \ in\ D_{2r_nR}{\setminus } D_{r_nR}, \end{aligned}$$

by the elliptic estimates for first order equations and Sobolev embedding, we obtain

where the last inequality follows from (4.22). This is

$$\begin{aligned} E(\psi ;D_\delta (x_n){\setminus } D_{r_nR}(x_n))\le C(\epsilon +\delta ). \end{aligned}$$

(4.23)

This is (4.2) and we finished the proof of Theorem 4.1. $\square$

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]. $\square$