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Regularity of Dirac-harmonic maps with \(\lambda \)-curvature term in higher dimensions

  • Jürgen Jost
  • Lei Liu
  • Miaomiao ZhuEmail author
Open Access
Article

Abstract

In this paper, we will study the partial regularity for stationary Dirac-harmonic maps with \(\lambda \)-curvature term. For a weakly stationary Dirac-harmonic map with \(\lambda \)-curvature term \((\phi ,\psi )\) from a smooth bounded open domain \(\Omega \subset {\mathbb {R}}^m\) with \(m\ge 2\) to a compact Riemannian manifold N, if \(\psi \in W^{1,p}(\Omega )\) for some \(p>\frac{2m}{3}\), we prove that \((\phi , \psi )\) is smooth outside a closed singular set whose \((m-2)\)-dimensional Hausdorff measure is zero. Furthermore, if the target manifold N does not admit any harmonic sphere \(S^l\), \(l=2,\ldots , m-1\), then \((\phi ,\psi )\) is smooth.

Mathematics Subject Classification

53C43 58E20 

1 Introduction

Variational problems from fields of theoretical physics, like quantum field or string theory, usually come in some particular dimension, with some finite dimensional, but non-compact symmetry group. These include harmonic maps coming from the nonlinear sigma model in dimension 2 or Yang–Mills fields in dimension 4. Typically, they then represent borderline cases of the Palais–Smale condition, and therefore, standard PDE methods for proving the regularity of solutions may not apply. In those dimensions, geometric analysis can usually identify a particular blow-up behavior, that is, a special scheme for the emergence and the control of singularities. That is, minimizing sequences can develop singularities, but in the limit, these singularities can be described as regular solutions on some blown-up domain.

The mathematical aspects, however, are also of much interest and subtlety in higher dimensions. In those dimensions, solutions can really become singular. Again, this has been widely explored in geometric analysis. For instance, the equations for minimal submanifolds in Euclidean or Riemannian spaces loose the conformal invariance, and completely new phenomena emerge, in particular around the Bernstein problem, and this has been a key trigger for the development of geometric measure theory. For harmonic mappings, see [16, 17, 19, 37], and for Yang–Mills, Rivière has carried out the systematic investigation in dimensions larger than 4, see [34] and the references therein. In those cases, the best analytical results that can be obtained are usually partial regularity results, that is, one can control the Hausdorff dimension of the singular set and often also the structure of the singularities.

Here, we engage in such an investigation for Dirac-harmonic maps, a variational problem motivated by the supersymmetric non-linear sigma model of quantum field theory. They arise again naturally in dimension 2, where we again find conformal invariance and can perform a—rather subtle—blow-up analysis. Dirac-harmonic maps were first introduced and studied in [8, 9] in dimension 2. In light of the above, it seems worthwhile to also investigate them in higher dimensions, and again, we expect that the analytical behavior will be rather different. Such an analysis has been started by Wang and Xu [42]. In particular, they derived a monotonicity formula and controlled the singular set as for harmonic maps. In fact, since Dirac-harmonic maps generalize harmonic maps in the sense that they couple a harmonic map type field with a nonlinear Dirac field, one should naturally expect that the structure of harmonic map regularity theory can serve as a guideline. Nevertheless, as it turns out already in dimension 2, while the results are indeed roughly similar to those known for harmonic maps, their proofs can become considerably more difficult. This forces the development of new techniques, some of which then in turn also lead to deeper insights for harmonic maps. Here, we take a step further by implementing the important analysis of Lin [27] who could show regularity in the absence of obstructions, represented by harmonic spheres in a certain range of dimensions. Also, we consider a model that is more general than that in [42], but which is important from the original perspective of quantum field theory, that of Dirac-harmonic maps with curvature term. While the curvature term usually only comes with a constant factor in the literature, we find that we can also admit a field-dependent, variable factor, without impeding the analysis.

We now recall the technical details of the models, and then state our main results at the end of this introduction. Let (Mg) be an m-dimensional compact spin Riemannian manifold, \(\Sigma M\) the spinor bundle over M and \(\langle \cdot ,\cdot \rangle _{\Sigma M}\) the metric on \(\Sigma M\). Choosing a local orthonormal basis \({e_\alpha ,\alpha =1, \ldots ,m}\) on M, the usual Dirac operator is defined as Open image in new window, where \(\nabla \) is the spin connection on \(\Sigma M\) and \(\cdot \) is the Clifford multiplication. For more details on spin geometry and Dirac operators, one can refer to [26].

Let \(\phi \) be a smooth map from M to another compact Riemannian manifold (Nh) with dimension \(n\ge 2\). If \(\phi ^{*}TN\) is the pull-back bundle of TN by \(\phi \), we get the twisted bundle \(\Sigma M\otimes \phi ^{*}TN\). Naturally, there is a metric \(\langle \cdot ,\cdot \rangle _{\Sigma M\otimes \phi ^{*}TN}\) on \(\Sigma M\otimes \phi ^{*}TN\) which is induced from the metrics on \(\Sigma M\) and \(\phi ^{*}TN\). Also we have a natural connection \(\widetilde{\nabla }\) on \(\Sigma M\otimes \phi ^{*}TN\) which is induced from the connections on \(\Sigma M\) and \(\phi ^{*}TN\). Let \(\psi \) be a section of the bundle \(\Sigma M\otimes \phi ^{*}TN\). In local coordinates \(\{y^i\}\), it can be written as
$$\begin{aligned} \psi =\psi ^i\otimes \partial _{y^i}(\phi ), \end{aligned}$$
where each \(\psi ^i \in \Gamma (\Sigma M)\) is a usual spinor and \(\{\partial _{y^i}\}\) is a local basis on N. Then \(\widetilde{\nabla }\) becomes
$$\begin{aligned} \widetilde{\nabla }\psi =\nabla \psi ^i\otimes \partial _{y^i}(\phi ) +(\Gamma ^i_{jk}\nabla \phi ^j)\psi ^k\otimes \partial _{y^i}(\phi ), \end{aligned}$$
(1.1)
where \(\Gamma ^i_{jk}\) are the Christoffel symbols of the Levi–Civita connection of N. The Dirac operator along the map \(\phi \) is defined byNow, consider the action functional introduced in [8, 9]Critical points \((\phi ,\psi )\) of L are called Dirac-harmonic maps from M to N.
In local coordinates, the Euler–Lagrange equations of the functional L are given as followswhere \(R(\phi ,\psi )\) is defined by
$$\begin{aligned} R(\phi ,\psi )=\frac{1}{2}R^m_{lij}(\phi (x))Re\langle \psi ^i,\nabla \phi ^l\cdot \psi ^j\rangle \frac{\partial }{\partial y^m}(\phi (x)). \end{aligned}$$
Here Re(z) denotes the real part of \(z\in \mathbb {C}\) and \(R^m_{lij}\) stands for the Riemann curvature tensor of the target manifold (Ng). See [8, 9] for details.

Dirac-harmonic maps are motivated from the supersymmetric nonlinear sigma model from quantum field theory [12, 20]. They have been investigated extensively in recent years. This subject generalizes the theory of harmonic maps and harmonic spinors. The regularity problem for harmonic maps has been extensively studied in the literature, see e.g. [3, 6, 13, 18, 29, 37] for the classical regularity theory of minimizing harmonic maps and stationary harmonic maps. Based on the geometric analysis techniques developed for harmonic maps and more generally critical elliptic systems with an antisymmetric structure [33, 35], regularity issues for Dirac-harmonic maps in dimension two were systematically studied in [8, 11, 39, 42, 45]. In higher dimensions, Wang-Xu [42] introduced the notion of stationary Dirac-harmonic maps and derived a monotonicity formula for stationary Dirac-harmonic maps, based on which some partial regularity results were obtained. They proved the singular set has Hausdorff dimension at most \(m-2\). In this paper, we give conditions on the target manifold under which the dimension can be reduced further. Moreover, we prove these properties hold for a general case, i.e. Dirac-harmonic maps with \(\lambda \)-curvature term. The blow-up analysis for Dirac-harmonic maps has been investigated in [8, 28, 44, 46]. To study the existence problem, a heat flow approach was introduced in [10] and further explored in [22, 23, 24].

Usually, the supersymmetric nonlinear sigma model of quantum field theory includes an additional curvature term in addition to (1.2). This leads us to consider the following functionalCritical points \((\phi ,\psi )\) of \(L_c\) are called Dirac-harmonic maps with curvature term from M to N. They were first proposed and studied by Chen–Jost–Wang [7], where a type of Liouville theorem was proved. The regularity for weak solutions in dimension two was considered in [4]. The blow-up theory, including the energy identity and bubble tree convergence, for a sequence of Dirac-harmonic maps with curvature term from a closed Riemann surface with uniformly bounded energy has been systematically investigated in [21]. For the regularity problem of a similar model with a different type of curvature term, i.e., Dirac-harmonic maps with Ricci type spinor potential, we refer to Xu–Chen [43].
In this paper, we shall consider the following functional:where \(\lambda \) is a smooth function on N. Since (Nh) ia a compact Riemannian manifold, we define a nonnegative constant:1
$$\begin{aligned} \Lambda _1:= \mathrm{max}_{y\in N} \ |\lambda (y)| + |\nabla \lambda (y)|. \end{aligned}$$
(1.6)
The critical points \((\phi ,\psi )\) of \(L_\lambda \) are called Dirac-harmonic maps with \(\lambda \)-curvature term from M to N. Thus, \((\phi ,\psi )\) is a Dirac-harmonic map iff \(\lambda \equiv 0\) and it is a Dirac-harmonic map with curvature term iff \(\lambda \equiv 1\).
By the Nash embedding theorem, we embed N isometrically into \(\mathbb {R}^K\). Following Wang-Xu’s analysis set up for Dirac-harmonic maps in higher dimensions in [42], we denote
$$\begin{aligned} W^{1,2}(M,N):= & {} \left\{ \phi \in W^{1,2}(M,{\mathbb {R}}^K)|\phi (x)\in N,\ a.e. x\in M\right\} ,\\ S^{1,4/3}(\Sigma M\otimes \phi ^{*}TN):= & {} \left\{ \psi \in \Gamma (\Sigma M\otimes \phi ^{*}TN)|\int _M|(\psi |^4+|\nabla \psi |^{\frac{4}{3}})<\infty \right\} . \end{aligned}$$
Here \(\psi \in \Gamma (\Sigma M\otimes \phi ^{*}TN)\) should be understood as a K-tuple of spinors \((\psi ^1, \ldots ,\psi ^K)\) satisfying
$$\begin{aligned} \sum _{i=1}^K\psi ^i\nu _i=0 \end{aligned}$$
for any normal vector \(\nu =(\nu _1, \ldots ,\nu _K)\in {\mathbb {R}}^K\).

In the sequel, for simplicity, we shall consider the case that \(M=\Omega \) is a bounded open domain of \({\mathbb {R}}^m\) with smooth boundary and equipped with the Euclidean metric. Then, the spinor bundle \(\Sigma M\) over M can be identified with \(\Sigma =\Omega \times \mathbb {C}^L\), \(L=\mathrm {rank}_{\mathbb {C}}\Sigma \). See [26].

Definition 1.1

We call \((\phi ,\psi )\in W^{1,2}(\Omega ,N)\times S^{1,\frac{4}{3}}(\mathbb {C}^L\otimes \phi ^{*}TN)\) a weakly Dirac-harmonic map with \(\lambda \)-curvature term if it is a critical point of \(L_\lambda \) over the Sobolev space \(W^{1,2}(\Omega ,N)\times S^{1,4/3}(\mathbb {C}^L\otimes \phi ^{*}TN)\).

Our first main result is the following small regularity theorem.

Theorem 1.2

For \(m\ge 2\), there exists an \(\epsilon _0=\epsilon _0(m,\Lambda _1, N)>0\) such that if \((\phi ,\psi )\in W^{1,2}(\Omega ,N)\times S^{1,\frac{4}{3}}( \mathbb {C}^L\otimes \phi ^{*}TN)\) is a weakly Dirac-harmonic map with \(\lambda \)-curvature term satisfying
$$\begin{aligned} \sup _{x\in B_{r_0}(x_0),0<r\le r_0}r^{2-m}\int _{B_r(x)}(|\nabla \phi |^2+|\psi |^4)dvol_g\le \epsilon ^2_0, \end{aligned}$$
(1.7)
then \((\phi ,\psi )\in C^\infty (B_{\frac{r_0}{2}}(x_0))\), and it satisfies
$$\begin{aligned}&\Vert \nabla \phi \Vert _{L^\infty (B_{r_0/2}(x_0))}+\Vert \psi \Vert ^2_{L^\infty (B_{r_0/2}(x_0))}\nonumber \\&\quad \le Cr_0^{-\frac{m}{2}}(\Vert \nabla \phi \Vert _{L^2(B_{r_0}(x_0))}+\Vert \psi \Vert ^2_{M^{4,2}(B_{r_0}(x_0))}),\end{aligned}$$
(1.8)
where \(C=C(m,\Lambda _1, N)>0\) and \(\Lambda _1\) is as in (1.6).

When \(\lambda =0\), the conclusion in the above theorem has been proven in [42]. When \(m=2\) and \(\lambda =1\), one can refer to [4].

Similarly to the classical regularity theory of harmonic maps, in order to study the partial regularity in higher dimensions, we need to introduce the notion of stationary solutions.

Definition 1.3

A weakly Dirac-harmonic map with \(\lambda \)-curvature term \((\phi ,\psi )\in W^{1,2}(\Omega ,N)\times S^{1,\frac{4}{3}}( \mathbb {C}^L\otimes \phi ^{*}TN)\) is called stationary if it is also a critical point of \(L_{\lambda }\) with respect to the domain variations, i.e. for any \(Y\in C^\infty _0(\Omega ,{\mathbb {R}}^n)\), it holdswhere \(\phi _t(x)=\phi (x+tY(x))\) and \(\psi _t(x)=\psi (x+tY(x))\).
We would like to remark that for the cases \(\lambda =0, 1\), the above definition has been introduced in [5, 42], respectively, where the following monotonicity formula was derived: for any \(x_0\in \Omega \) and \(0<r_1\le r_2<dist(x_0,\partial \Omega )\),
$$\begin{aligned}&r_2^{2-m}\int _{B_{r_2}(x_0)}(|\nabla \phi |^2+\frac{\lambda }{6}R_{ikjl}\langle \psi ^i,\psi ^j\rangle \langle \psi ^k,\psi ^l\rangle )dx\\&\quad -\,r_1^{2-m}\int _{B_{r_1}(x_0)}(|\nabla \phi |^2+\frac{\lambda }{6}R_{ikjl}\langle \psi ^i,\psi ^j\rangle \langle \psi ^k,\psi ^l\rangle )dx\\&\qquad = \int _{r_1}^{r_2}r^{2-m}\int _{\partial B_r}(2|\frac{\partial \phi }{\partial r}|^2+Re\langle \psi ,\partial _r\cdot \psi _r\rangle )dH^{n-1}dr \end{aligned}$$
where \(\partial _r=\frac{\partial }{\partial r}=\frac{\partial }{\partial |x-x_0|}\) and \(\psi _r=\widetilde{\nabla }_{\partial r}\psi \). The second term of the right hand side of the above equation does not have a fixed sign, which makes the use of this formula difficult. This is why in [42], some additional condition on the spinors was imposed in order to get the partial regularity of stationary Dirac-harmonic maps.

In this paper, we shall impose the same extra condition for the spinor as in Wang-Xu [42] and get the following partial regularity theorem for stationary Dirac-harmonic maps with \(\lambda \)-curvature term. For similar results for stationary harmonic maps and stationary Dirac-harmonic maps, we refer to [3, 13, 42].

Theorem 1.4

For \(m\ge 2\), let \((\phi ,\psi )\in W^{1,2}(\Omega ,N)\times S^{1,\frac{4}{3}}( \mathbb {C}^L\otimes \phi ^{*}TN)\) be a weakly stationary Dirac-harmonic map with \(\lambda \)-curvature term. Suppose \(\Vert \psi \Vert _{ W^{1,p}(\Omega )}<\infty \) for some \(p>\frac{2m}{3}\), then there exists a closed subset \(S(\phi )\subset \Omega \), with \(H^{m-2}(S(\phi ))=0\), such that \((\phi ,\psi )\in C^\infty (\Omega {\setminus } S(\phi ))\).

Furthermore, we have

Theorem 1.5

Under the same assumption as in the above theorem, if N does not admit harmonic spheres \(S^2\), then the Hausdorff dimension of the singular set \(S(\phi )\) is \(d_h\le m-3\). Moreover, if N does not admit harmonic spheres \(S^l\), \(l=2, \ldots ,m-1\), then \((\phi ,\psi )\) is smooth.

Similarly to the (approximate) harmonic maps (see [30, 31]), one can also consider detailed properties of the stratification of the singular set. This will be addressed in future works.

To prove Theorem 1.2, we firstly use the idea of Wang in [41] to improve the regularity of the spinor \(\psi \) and then apply regularity results for elliptic system with an antisymmetric structure (see e.g. Theorem 5.2 in Appendix) to handle the map \(\phi \). For Theorem 1.4, since \(\nabla \psi \in L^p\) for some \(p>\frac{2m}{3}\), it follows from using Theorem 1.2, the monotonicity formula and applying similar arguments as in Wang-Xu [42]. As for our last Theorem 1.5, thanks to the observation in Proposition 4.5 for some formulas of the spinors, following Lin’s scheme in [27], we consider the concentration set of a blow-up sequence of Dirac-harmonic maps with \(\lambda \)-curvature term. The proof is based on the analysis of defect measures by geometric measure theory.

The rest of the paper is organized as follows. In Sect. 2, we first derive the Euler–Lagrange equation for stationary Dirac-harmonic maps with \(\lambda \)-curvature term. Secondly, we establish the monotonicity formula crucial to prove Theorems 1.4 and 1.5. In Sect. 3, we prove the small regularity Theorem 1.2 and then Theorem 1.4 follows immediately by applying some monotonicity formula argument. In Sect. 4, we use the blow-up analysis to prove Theorem 1.4. For the reader’s convenience, we will state some well-known regularity results and estimates for some first and second order elliptic systems in Sect. 5.

2 Euler–Lagrange equations and monotonicity formula

In this section, we will derive the Euler–Lagrange equation and the monotonicity formula for Dirac-harmonic maps with \(\lambda \)-curvature term.

First, similarly to the cases \(\lambda =0, 1\) considered in [7, 9], respectively, the Euler–Lagrange equations of the functional \(L_\lambda \) can be derived in terms of local coordinates as follows:

Lemma 2.1

Let \((\phi ,\psi )\) be a Dirac-harmonic map with \(\lambda \)-curvature term from M to N. Then, in local coordinates, \((\phi ,\psi )\) satisfies
$$\begin{aligned} \tau (\phi )= & {} \frac{1}{2}R^m_{lij}(\phi )\langle \psi ^i,\nabla \phi ^l\cdot \psi ^j\rangle \frac{\partial }{\partial y^m}(\phi ) -\frac{\lambda }{12}h^{mp}R_{ikjl;p}\langle \psi ^i,\psi ^j\rangle \langle \psi ^k,\psi ^l\rangle \frac{\partial }{\partial y^m}(\phi )\nonumber \\&-\,\frac{\nabla ^N\lambda (\phi )}{12}R_{ikjl}\langle \psi ^i,\psi ^j\rangle \langle \psi ^k,\psi ^l\rangle , \end{aligned}$$
(2.1)
where \(\tau (\phi )=\left( -\Delta \phi ^i+\Gamma ^i_{jk}g^{\alpha \beta }\phi ^j_\alpha \phi ^k_\beta \right) \frac{\partial }{\partial y^i}(\phi (x))\) is the tension field of \(\phi \) and \(\nabla ^N\lambda =h^{mp}\frac{\partial \lambda }{\partial y^p}\frac{\partial }{\partial y^m}\) is the gradient vector field on N.

Proof

By the computation of Section II in [7], we obtain the \(\psi \)-equation for \(L_\lambda \),andwhere \(\phi _t\) is the variation of \(\phi \) with \(\phi _0=\phi \) and \(\frac{d}{d t}|_{t=0}=\xi \).
We just need to compute the last term:
$$\begin{aligned}&-\,\frac{d}{d t}|_{t=0}\int _M\frac{\lambda (\phi _t)}{12}R_{ijkl}\langle \psi ^i,\psi ^k\rangle \langle \psi ^j,\psi ^l\rangle dvol_g\\&\quad =-\int _M\frac{\lambda (\phi )}{12}R_{ijkl,m}\langle \psi ^i,\psi ^k\rangle \langle \psi ^j,\psi ^l\rangle \xi ^m dvol_g\\&\qquad -\,\frac{1}{12}\int _MR_{ijkl}\langle \psi ^i,\psi ^k\rangle \langle \psi ^j,\psi ^l\rangle \frac{\partial \lambda }{\partial y^m} \xi ^m dvol_g. \end{aligned}$$
Thus,
$$\begin{aligned}&-\frac{d L_\lambda (\phi _t)}{d t}|_{t=0}\\&\quad =\frac{1}{2}\int _M\big (-2h_{im}\tau ^i(\phi ) +\langle \psi ^i,\nabla \phi ^l\cdot \psi ^j\rangle R_{mlij}\\&\qquad -\,\frac{\lambda (\phi )}{6}R_{ijkl;m}\langle \psi ^i,\psi ^k\rangle \langle \psi ^j,\psi ^l\rangle \\&\qquad -\,\frac{1}{6}R_{ijkl}\langle \psi ^i,\psi ^k\rangle \langle \psi ^j,\psi ^l\rangle \frac{\partial \lambda }{\partial y^m}\big ) \xi ^m. \end{aligned}$$
The conclusion of the lemma follows immediately. \(\square \)
By the Nash embedding theorem, we embed N isometrically into \(\mathbb {R}^N\), denoted by \(f:N\rightarrow \mathbb {R}^K\). Since \(\lambda \in C^\infty (N)\), there exists an extended function \(\lambda \in C_0^\infty ({\mathbb {R}}^K)\) (for simplicity, we still denote it by \(\lambda \)), such that
$$\begin{aligned} \Vert \lambda \Vert _{C^2({\mathbb {R}}^K)}\le C(N)\Vert \lambda \Vert _{C^2(N)}. \end{aligned}$$
Set
$$\begin{aligned} \phi '=f\circ \phi \quad and \quad \psi '=f_*\psi . \end{aligned}$$
If we identify \(\phi \) with \(\phi '\) and \(\psi \) with \(\psi '\), similarly to the case of \(\lambda =1\) and \(\mathrm{dim}\ M =2\) considered in [4, 21], we can get the following extrinsic form of the Euler–Lagrange equation:

Lemma 2.2

Let \((\phi ,\psi )\in W^{1,2}(\Omega ,N)\times S^{1,\frac{4}{3}}( \mathbb {C}^L\otimes \phi ^{*}TN)\) be a weakly Dirac-harmonic map with \(\lambda \)-curvature term. Then, \((\phi ,\psi )\) satisfieswhereand \(B:=(\frac{\partial \lambda }{\partial z^1}, \ldots ,\frac{\partial \lambda }{\partial z^K})\), \(B^\top \) is the tangential part of B along the map \(\phi \), \(P(\cdot ;\cdot )\) is the shape operator, i.e.
$$\begin{aligned} \langle P(\xi ;X),Y\rangle =\langle A(X,Y),\xi \rangle \end{aligned}$$
for any \(X,Y\in \Gamma (TN)\),\(\xi \in \Gamma (T^\bot N)\), A is the second fundamental form of N in \({\mathbb {R}}^K\) and

Proof

The proof here is almost the same as the computations in the case of \(\lambda =1\) (see Section 3 in [21] where the inner product for the spinors was taken to be Hermitian as in this paper and hence one needs to take the real parts for certain terms. See also Lemma 3.5 in [4]). We omit the details here. \(\square \)

Secondly, we will derive some useful formulae (i.e. Lemma 2.3 and Lemma 2.4) for stationary Dirac-harmonic maps with \(\lambda \)-curvature term which are just Lemma 4.2 and Lemma 4.4 in [42] for \(\lambda =0\) and Proposition 5.3 and Proposition 5.5 in [5] for \(\lambda =1\).

Lemma 2.3

Let \((\phi ,\psi )\in W^{1,2}(\Omega ,N)\times S^{1,\frac{4}{3}}( \mathbb {C}^L\otimes \phi ^{*}TN)\) be a weakly Dirac-harmonic map with \(\lambda \)-curvature term. Then \((\phi ,\psi )\) is stationary if and only if for any \(Y\in C^\infty _0(\Omega ,{\mathbb {R}}^n)\), there holds
$$\begin{aligned}&\int _\Omega \left( \left\langle \frac{\partial \phi }{\partial x^\alpha },\frac{\partial \phi }{\partial x^\beta }\right\rangle -\frac{1}{2}|\nabla \phi |^2\delta _{\alpha \beta }+\frac{1}{2}Re\left\langle \psi ,\frac{\partial }{\partial x^\alpha }\cdot \widetilde{\nabla }_{\frac{\partial }{\partial x^\beta }}\psi \right\rangle \right. \nonumber \\&\quad \left. -\,\frac{\lambda }{12}R_{ikjl}\langle \psi ^i,\psi ^j\rangle \langle \psi ^k,\psi ^l\rangle \delta _{\alpha \beta }\right) \frac{\partial Y^\beta }{\partial x^\alpha }=0. \end{aligned}$$
(2.5)

Proof

Let \(t\in {\mathbb {R}}\) small enough and \(y=F_t(x):=x+tY(x)\) and \(x=F_t^{-1}(y)\). On one hand, by Lemma 4.2 in [42], we haveOn the other hand, we have
$$\begin{aligned}&\frac{d}{dt}|_{t=0}\frac{1}{2}\int _{\Omega }\frac{\lambda }{6}R_{ikjl}\langle \psi _t^i,\psi _t^j\rangle \langle \psi _t^k,\psi _t^l\rangle dx\nonumber \\&\quad =\frac{d}{dt}|_{t=0}\frac{1}{2}\int _{\Omega }\frac{\lambda }{6}R_{ikjl}\langle \psi ^i,\psi ^j\rangle \langle \psi ^k,\psi ^l\rangle JacF_t^{-1}dx\nonumber \\&\quad =-\frac{\lambda }{12}\int _{\Omega }R_{ikjl}\langle \psi ^i,\psi ^j\rangle \langle \psi ^k,\psi ^l\rangle div(Y)dx, \end{aligned}$$
(2.7)
where we used the fact that
$$\begin{aligned}\frac{d}{dt}|_{t=0}JacF_t^{-1}=-div(Y). \end{aligned}$$
Combining (2.6) with (2.7), we will get the conclusion of the lemma. \(\square \)

Now, we can derive the monotonicity formula for weakly stationary Dirac-harmonic maps with \(\lambda \)-curvature term (see [5, 42] for the cases of \(\lambda =0, 1\))

Lemma 2.4

Let \((\phi ,\psi )\in W^{1,2}(\Omega ,N)\times S^{1,\frac{4}{3}}(\mathbb {C}^L\otimes \phi ^{*}TN)\) be a weakly stationary Dirac-harmonic map with \(\lambda \)-curvature term. Then for any \(x_0\in \Omega \) and \(0<r_1\le r_2<dist(x_0,\partial \Omega )\), there holds
$$\begin{aligned}&r_2^{2-m}\int _{B_{r_2}(x_0)}\left( |\nabla \phi |^2+\frac{\lambda }{6}R_{ikjl}\langle \psi ^i,\psi ^j\rangle \langle \psi ^k,\psi ^l\rangle \right) dx\\&\quad -\,r_1^{2-m}\int _{B_{r_1}(x_0)}\left( |\nabla \phi |^2+\frac{\lambda }{6}R_{ikjl}\langle \psi ^i,\psi ^j\rangle \langle \psi ^k,\psi ^l\rangle \right) dx\\&\qquad = \int _{ B_{r_2}(x_0)\setminus B_{r_1}(x_0)}|x-x_0|^{2-m}(2|\frac{\partial \phi }{\partial r}|^2+Re\langle \psi ,\partial _r\cdot \psi _r\rangle )dx \end{aligned}$$
where \(\partial _r=\frac{\partial }{\partial r}=\frac{\partial }{\partial |x-x_0|}\) and \(\psi _r=\widetilde{\nabla }_{\partial r}\psi \).

Proof

For simplicity, we assume \(x_0=0\in \Omega \). For any \(\epsilon >0\) and \(0<r<dist(0,\partial \Omega )\), let \(\varphi _\epsilon (x)=\varphi _\epsilon (|x|)\in C_0^\infty (B_r)\) be such that \(0\le \varphi _\epsilon (x)\le 1\) and \(\varphi _\epsilon (x)|_{B_{(1-\epsilon )r}}=1\). Taking \(Y(x)=x\varphi _\epsilon (x)\) into the formula (2.5) and noting that
$$\begin{aligned} \frac{\partial Y^\beta }{\partial x^\alpha }=\varphi _\epsilon (x)\delta _{\alpha ,\beta }+\frac{x^\alpha x^\beta }{|x|}\varphi _\epsilon '(x), \end{aligned}$$
we haveUsing the Eq. (2.2) and letting \(\epsilon \rightarrow 0\), we get
$$\begin{aligned}&(2-m)\int _{B_r}\left( |\nabla \phi |^2+\frac{\lambda }{6}R_{ikjl}\langle \psi ^i,\psi ^j\rangle \langle \psi ^k,\psi ^l\rangle \right) \\&\quad +\,r\int _{\partial B_r}\left( |\nabla \phi |^2+\frac{\lambda }{6}R_{ikjl}\langle \psi ^i,\psi ^j\rangle \langle \psi ^k,\psi ^l\rangle \right) \\&\qquad = r\int _{\partial B_r}\left( 2|\frac{\partial \phi }{\partial r}|^2+Re\langle \psi ,\partial r\cdot \widetilde{\nabla }_{\partial r}\psi \rangle \right) , \end{aligned}$$
which yields
$$\begin{aligned}&\frac{d}{dr}\left( r^{2-m}\int _{B_r}(|\nabla \phi |^2+\frac{\lambda }{6}R_{ikjl}\langle \psi ^i,\psi ^j\rangle \langle \psi ^k,\psi ^l\rangle )dx\right) \\&\quad =r^{2-m}\int _{\partial B_r}\left( 2|\frac{\partial \phi }{\partial r}|^2+Re\langle \psi ,\partial r\cdot \widetilde{\nabla }_{\partial r}\psi \rangle \right) . \end{aligned}$$
The conclusion of the lemma follows by integrating r from \(r_1\) to \(r_2\). \(\square \)

The following corollary is a small extension of the case of \(\lambda =0\) considered in [42]:

Corollary 2.5

Let \((\phi ,\psi )\in W^{1,2}(\Omega ,N)\times S^{1,\frac{4}{3}}(\mathbb {C}^L\otimes \phi ^{*}TN)\) be a weakly stationary Dirac-harmonic map with \(\lambda \)-curvature term. If we assume
$$\begin{aligned} \Vert \psi \Vert _{W^{1,p}(\Omega )}\le \Lambda \end{aligned}$$
for some \(\frac{2m}{3}<p<m\), then for any \(x_0\in \Omega \) and \(0<r_1\le r_2<\min \{dist(x_0,\partial \Omega ),1\}\), there holds
$$\begin{aligned}&r_1^{2-m}\int _{B_{r_1}(x_0)}|\nabla \phi |^2dx\\&\quad \le r_2^{2-m}\int _{B_{r_2}(x_0)}|\nabla \phi |^2dx\\&\qquad +\,C(m)\Vert \psi \Vert _{L^{\frac{mp}{m-p}}(B_{r_2}(x_0))}\Vert \nabla \psi \Vert _{L^p(B_{r_2}(x_0))} r_2^{3-\frac{2m}{p}}\\&\qquad +\,C(m,N)\Lambda _1\Vert \psi \Vert ^4_{L^{\frac{mp}{m-p}}(B_{r_2}(x_0))}r_2^{6-\frac{4m}{p}}, \end{aligned}$$
where \(\Lambda _1\) is as defined in (1.6).

Proof

By Lemma 2.4, we know
$$\begin{aligned} r_1^{2-m}\int _{B_{r_1}(x_0)}|\nabla \phi |^2dx&\le r_2^{2-m}\int _{B_{r_2}(x_0)}(|\nabla \phi |^2+\frac{\lambda }{6}R_{ikjl}\langle \psi ^i,\psi ^j\rangle \langle \psi ^k,\psi ^l\rangle )dx\\&\quad +\, \int _{ B_{r_2}(x_0)\setminus B_{r_1}(x_0)}|x-x_0|^{2-m}|\psi ||\nabla \psi | dx\\&\quad -\,r_1^{2-m}\int _{B_{r_1}(x_0)}\frac{\lambda }{6}R_{ikjl}\langle \psi ^i,\psi ^j\rangle \langle \psi ^k,\psi ^l\rangle dx, \end{aligned}$$
which implies
$$\begin{aligned} r_1^{2-m}\int _{B_{r_1}(x_0)}|\nabla \phi |^2dx&\le r_2^{2-m}\int _{B_{r_2}(x_0)}|\nabla \phi |^2dx+C(N)\Lambda _1r_2^{2-m}\int _{B_{r_2}(x_0)}|\psi |^4dx\nonumber \\&\quad +\, \int _{ B_{r_2}(x_0)\setminus B_{r_1}(x_0)}|x-x_0|^{2-m}|\psi ||\nabla \psi | dx\nonumber \\&\quad +C(N)\Lambda _1r_1^{2-m}\int _{B_{r_1}(x_0)}|\psi |^4dx. \end{aligned}$$
(2.8)
By Sobolev’s embedding and Young’s inequality, we have
$$\begin{aligned}&\int _{B_r(x_0)}|x-x_0|^{2-m}|\psi ||\nabla \psi |dx\nonumber \\&\quad \le \Vert \psi \Vert _{L^{\frac{mp}{m-p}}(B_r(x_0))}\Vert \nabla \psi \Vert _{L^p(B_r(x_0))}\Vert |x-x_0|^{2-m}\Vert _{L^{\frac{mp}{mp-2m+p}}(B_r(x_0))}\nonumber \\&\quad \le C(m)\Vert \psi \Vert _{L^{\frac{mp}{m-p}}(B_r(x_0))}\Vert \nabla \psi \Vert _{L^p(B_r(x_0))} r^{3-\frac{2m}{p}} \end{aligned}$$
(2.9)
and
$$\begin{aligned} r_1^{2-m}\int _{B_{r_1}(x_0)}|\psi |^4dx+r_2^{2-m}\int _{B_{r_2}(x_0)}|\psi |^4dx\le C(m)\Vert \psi \Vert ^4_{L^{\frac{mp}{m-p}}(B_{r_2}(x_0))} r_2^{6-\frac{4m}{p}}. \end{aligned}$$
(2.10)
\(\square \)

Then the conclusion of the corollary follows immediately from (2.8).

3 Proof of Theorem 1.2 and 1.4

In this section, we will prove our main results: Theorem 1.2 and 1.4.

For Theorem 1.2, we will firstly use the idea in [41] to raise the integrability of \(\psi \). Let us recall the definition of Morrey spaces (see [15]). For \(p\ge 1\), \(0<\mu \le m\) and a domain \(U\subset {\mathbb {R}}^m\), the Morrey space \(M^{p,\mu }(U)\) is defined by
$$\begin{aligned} M^{p,\mu }(U):=\left\{ f\in L^p_{loc}(U)|\ \Vert f\Vert _{M^{p,\mu }(U)}<\infty \right\} \end{aligned}$$
where
$$\begin{aligned} \Vert f\Vert ^p_{M^{p,\mu }(U)}:=\sup _{B_r\subset U}r^{\mu -m}\int _{B_r}|f|^p. \end{aligned}$$

Lemma 3.1

For any \(4<p<\infty \) and \(m\ge 2\), there exists a positive constant \(\epsilon _1=\epsilon _1(p, m, N)>0\) and \(C=C(m, p, N)>0\), such that if \((\phi ,\psi )\) is a weak solution of (2.4) and
$$\begin{aligned} \Vert \nabla \phi \Vert _{M^{2,2}(B_1)}+\Vert |\lambda ||\psi |^2\Vert _{M^{2,2}(B_1)}\le \epsilon _1, \end{aligned}$$
then \(\psi \in L^p(B_{1/2})\) and satisfies the estimate
$$\begin{aligned} \Vert \psi \Vert _{L^p(B_{1/2})}\le C(m, p, N)\Vert \psi \Vert _{M^{4,2}(B_{1})}. \end{aligned}$$
(3.1)

The idea of proving this lemma is similar to Lemma 2.2 in [41] which has been applied to some other Dirac type equation in dimension 2 in [4, 39]. Recently, [25] (Lemma 6.1) proved a similar lemma for a more general equation in higher dimensions which can be used in our case.

Proof

By (2.4), it is easy to see that \(\psi \) satisfies the equation of the form (5.1) in the Appendix with
$$\begin{aligned}|A|\le C(N)(|\nabla \phi |+|\lambda ||\psi |^2),\ B\equiv 0,\end{aligned}$$
the conclusion of the lemma follows from Lemma 5.1 in the Appendix (or Lemma 6.1 in [25]) immediately. \(\square \)

Combining Lemma 3.1 with Theorem 5.2, we can now prove Theorem 1.2.

Proof of Theorem 1.2

Without loss of generality, we may assume \(r_0=1\). By assumption (1.7), it is easy to see that
$$\begin{aligned} \Vert \nabla \phi \Vert _{M^{2,2}(B_{1}(x_0))}+\Vert |\psi |^2\Vert _{M^{2,2}(B_{1}(x_0))}\le \epsilon _0. \end{aligned}$$
If \(\epsilon _0\le \frac{\epsilon _1}{1+\Lambda _1}\), by Lemma 3.1, we have \(\psi \in L^{\frac{4mq}{2+q}}(B_{\frac{3}{4}}(x_0))\) for any \(2<q<\infty \) and
$$\begin{aligned} \Vert \psi \Vert _{L^{\frac{4mq}{2+q}}(B_{\frac{3}{4}}(x_0))}\le C(m,q,N)\Vert \psi \Vert _{M^{4,2}(B_{1}(x_0))}. \end{aligned}$$
Thus
$$\begin{aligned} G(\psi )\in L^{\frac{mq}{2+q}}(B_{\frac{3}{4}}(x_0)). \end{aligned}$$
By slightly modifying the extrinsic equations for Dirac-harmonic maps (i.e., the case of \(\lambda =0\)) considered in [11, 39, 45] (see equations (3.6) and (3.8) in [39]), it is easy to see that the equation (2.3) for the map can be written as the following form
$$\begin{aligned} \Delta \phi =\widehat{\Omega }\cdot \nabla \phi + f \end{aligned}$$
with an antisymmetric potential \(\widehat{\Omega }\) satisfying
$$\begin{aligned} |\widehat{\Omega }|\le \widetilde{C}(N)(|\nabla \phi |+|\psi |^2), \end{aligned}$$
and with an error term f satisfying
$$\begin{aligned} |f|=|G(\psi )|\le C(\Lambda _1, N)|\psi |^4. \end{aligned}$$
Take \(\epsilon _0=\min \{\frac{\epsilon _1}{1+\Lambda _1},\frac{\epsilon }{\widetilde{C}(N)}\}\), where \(\epsilon \) is the constant in Theorem 5.2 in the Appendix. By Theorem 5.2, we know \(\nabla \phi \in M^{q,2}(B_{\frac{5}{8}}(x_0))\) for any \(2<q<\infty \) and
$$\begin{aligned} \Vert \nabla \phi \Vert _{M^{q,2}(B_{\frac{5}{8}}(x_0))}&\le C(m,q,N)\left( \Vert \nabla \phi \Vert _{L^2(B_{\frac{3}{4}}(x_0))}+\Vert G(\psi )\Vert _{L^{\frac{mq}{2+q}}(B_{\frac{3}{4}}(x_0))}\right) \\&\le C\left( m,q,\Lambda _1, N\right) \left( \Vert \nabla \phi \Vert _{L^2(B_1(x_0))}+\Vert \psi \Vert _{M^{4,2}(B_1(x_0))}\right) , \end{aligned}$$
which implies \(|\Delta \phi |\in L^q(B_{\frac{5}{8}}(x_0))\) and for some \(q>m\). The elliptic theory tells us \(\phi \in W^{2,q}(B_{\frac{1}{2}}(x_0))\). Thus \(\phi \in C^{1,\alpha }(B_{\frac{1}{2}}(x_0))\) for some \(\alpha >0\). Then by (2.4) and the standard first order elliptic estimates Lemma 5.3, we get \(\psi \in W^{1,q}(B_{\frac{1}{2}}(x_0))\) which yields \(\psi \in C^{1,\alpha }(B_{\frac{1}{2}}(x_0))\) and (1.8) holds. The higher order regularity then follows from the classical Schauder estimates for the Laplace and Dirac equation (see Lemma 5.4 in the Appendix) and a standard bootstrap argument. \(\square \)

Now, we prove our main Theorem 1.4.

Proof of Theorem 1.4

Without loss of generality, we assume \(\lambda \ne 0\) (for \(\lambda =0\), one can see [42]). Let \(\epsilon _0>0\) be the constant in Theorem 1.2. Define
$$\begin{aligned} S(\phi ):= \left\{ x\in \Omega :\liminf _{r\searrow 0}r^{2-m}\int _{B_r(x)}|\nabla \phi |^2\ge \frac{\epsilon _0^2}{2^m}\right\} . \end{aligned}$$
(3.2)
It is well known that \(H^{n-2}(S(\phi ))=0\). Next, we show \(S(\phi )\) is a closed set and \((\phi ,\psi )\in C^\infty (\Omega \setminus S(\phi ))\).
For any \(x_0\in \Omega {\setminus } S(\phi )\) and \(\epsilon >0\), there exists \(0<r_0<\epsilon \) such that,
$$\begin{aligned} (2r_0)^{2-m}\int _{B_{2r_0}(x_0)}|\nabla \phi |^2dx <\frac{\epsilon _0^2}{2^m}. \end{aligned}$$
(3.3)
Therefore,
$$\begin{aligned} \sup _{z\in B_{r_0}(x_0)}r_0^{2-m}\int _{B_{r_0}(z)}|\nabla \phi |^2dx <\frac{\epsilon _0^2}{4}. \end{aligned}$$
(3.4)
By Corollary 2.5, for any \(0<r_0<\frac{1}{2}\min \{dist(x_0,\partial \Omega ),1\}\), we have
$$\begin{aligned}&\sup _{z\in B_{r_0}(x_0),0<r\le r_0}r^{2-m}\int _{B_{r}(z)}(|\nabla \phi |^2+|\psi |^4)dx \nonumber \\&\le \sup _{z\in B_{r_0}(x_0)}r_0^{2-m}\int _{B_{r_0}(z)}|\nabla \phi |^2dx+C(m)\Vert \psi \Vert _{L^{\frac{mp}{m-p}}(B_{2r_0}(x_0))}\Vert \nabla \psi \Vert _{L^p(B_{2r_0}(x_0))} r_0^{3-\frac{2m}{p}} \nonumber \\&\quad +\,C(m,N)(1+\Lambda _1)\Vert \psi \Vert ^4_{L^{\frac{mp}{m-p}}(B_{2r_0}(x_0))}r_0^{6-\frac{4m}{p}}\nonumber \\&\le \frac{\epsilon _0^2}{4}+C(m,p,\Omega ,N)(\Lambda ^2+(1+\Lambda _1)\Lambda ^4)r_0^{3-\frac{2m}{p}}, \end{aligned}$$
(3.5)
where the last inequality follows from Sobolev’s embedding \(W^{1,p}(\Omega )\hookrightarrow L^{\frac{mp}{m-p}}(\Omega )\).
Taking \(\epsilon \le \left( \frac{\epsilon _0^2}{4C(m,p,\Omega ,N)(\Lambda ^2+(1+\Lambda _1)\Lambda ^4)}\right) ^{\frac{2m}{p}-3}\), we get
$$\begin{aligned} \sup _{z\in B_{r_0}(x_0),0<r\le r_0}r^{2-m}\int _{B_{r}(z)}(|\nabla \phi |^2+|\psi |^4)dx \le \frac{\epsilon _0^2}{2}. \end{aligned}$$
(3.6)
Then Theorem 1.2 tells us that \((\phi ,\psi )\in C^\infty (B_{r_0/2}(x_0))\) which implies \(B_{r_0/4}(x_0)\subset \Omega {\setminus } S(\phi )\). We finished the proof. \(\square \)

4 Proof of Theorem 1.5

In this section, we consider a weakly converging sequence of stationary Dirac-harmonic maps with \(\lambda \)-curvature term.

Let \(\{(\phi _n,\psi _n)\}\) be a sequence of stationary Dirac-harmonic maps with \(\lambda \)-curvature term with bounded energy
$$\begin{aligned} E(\phi _n,\psi _n)=\int _\Omega (|\nabla \phi _n|^2+|\psi _n|^4)\le \Lambda . \end{aligned}$$
Additionally, we assume
$$\begin{aligned} \Vert \psi _n\Vert _{ W^{1,p}(\Omega )}\le \Lambda \end{aligned}$$
for some \(p>\frac{2m}{3}\). Similar to harmonic maps [36], define the energy concentration set \(\Sigma \) as follows
$$\begin{aligned} \Sigma =\left\{ x\in \Omega |\liminf _{r\searrow 0}\liminf _{n\rightarrow \infty }r^{2-m}\int _{B_r(x)}|\nabla \phi _n|^2dx\ge \epsilon _0\right\} . \end{aligned}$$
(4.1)
Suppose \((\phi _n,\psi _n)\rightharpoonup (\phi ,\psi )\) weakly in \(W^{1,2}(\Omega ,N)\times L^4(\mathbb {C}^L\otimes \phi _n^*TN)\) and
$$\begin{aligned} \mu _n:=|\nabla \phi _n|^2dx\rightarrow \mu =|\nabla \phi |^2dx+\nu \end{aligned}$$
in the sense of Radon measures.

Without loss of generality, we assume \(B_1(0)\subseteq \Omega \). Then, we have

Lemma 4.1

Let \(\{(\phi _n,\psi _n)\}\) be a sequence of stationary Dirac-harmonic maps with \(\lambda \)-curvature term with bounded energy and \(\Vert \psi _n\Vert _{W^{1,p}}\le \Lambda \) for some \(p>\frac{2m}{3}\). Denote
$$\begin{aligned} \Sigma =\left\{ x\in B_1|\liminf _{r\searrow 0}\liminf _{n\rightarrow \infty }r^{2-m}\int _{B_r(x)}|\nabla \phi _n|^2dx\ge \frac{\epsilon _0^2}{2^m}\right\} , \end{aligned}$$
(4.2)
where \(\epsilon _0\) is the constant in Theorem 1.2, then \(\Sigma \) is closed in \(B_1\) and \(H^{m-2}(\Sigma )\le C(\epsilon _0,m,\Lambda )\). Moreover,
$$\begin{aligned} \Sigma =\mathrm {spt}(\nu )\cup sing(\phi ), \end{aligned}$$
(4.3)
where \(\mathrm {sing}(\phi )\) denoted the singular set of \(\phi \), i.e. for any \(x_0\in \mathrm {sing}(\phi )\), \(\phi \) is not smooth at \(x_0\).

Proof

For \(x_0\in B_1\setminus \Sigma \), by the proof of Theorem 1.4, there exists a positive constant \(r_0>0\) and a subsequence of \(\{n\}\) (also denoted by \(\{n\}\)), such that, for any n, there holds
$$\begin{aligned} (2r_0)^{2-m}\int _{B_{2r_0}(x)}|\nabla \phi _n|^2dx<\frac{\epsilon _0^2}{2^m}, \end{aligned}$$
which implies (similar to deriving (3.6))
$$\begin{aligned} \sup _{z\in B_{r_0}(x),0<r\le r_0}r^{2-m}\int _{B_{r}(z)}(|\nabla \phi _n|^2+|\psi _n|^4)dx<\frac{\epsilon _0^2}{2}. \end{aligned}$$
By Theorem 1.2, we know
$$\begin{aligned} r_0\Vert \nabla \phi _n\Vert _{L^\infty (B_{r_0/2}(x_0))}+\sqrt{r_0}\Vert \psi _n\Vert _{L^\infty (B_{r_0/2}(x_0))}\le C(m,r_0,\epsilon _0,\Lambda _1,N). \end{aligned}$$
(4.4)
Then, it is easy to see that there exists a small positive constant \(r_1=r_1(m,r_0,\epsilon _0,\lambda _1, N)\), such that, whenever \(r\le r_1\),
$$\begin{aligned} \sup _{x\in B_{r_0/4}(x_0)}r^{2-m}\int _{B_{r}(x)}|\nabla \phi _n|^2dx<\frac{\epsilon _0^2}{2^m}. \end{aligned}$$
Thus, \(B_{r_0/4}(x_0)\subset B_1{\setminus }\Sigma \). So, \(\Sigma \) is a closed set.

It is standard to get \(H^{m-2}(\Sigma )\le C(\epsilon _0,m,\Lambda )\) by a covering lemma (cf. [27]).

For (4.3), on the one hand, let \(x_0\in B_1{\setminus } \Sigma \). Then (4.4) holds and by standard elliptic estimates, we have
$$\begin{aligned} \Vert \phi _n\Vert _{C^{1+\alpha }(B_{r_0/4}(x_0))}+\Vert \psi _n\Vert _{C^\alpha (B_{r_0/2}(x_0))}\le C, \end{aligned}$$
(4.5)
for some \(0<\alpha <1\). So, there exists a subsequence of \(\{\phi _n,\psi _n\}\) (also denoted by \(\{\phi _n,\psi _n\}\)) such that \(\phi _n\rightarrow \phi \) strongly in \(W^{1,2}\) and \(\phi \in C^\infty (B_{r_0/8}(x_0))\) which imply that \(x_0\notin \mathrm {sing}(\phi )\) and \(x_0\notin \mathrm {spt} \nu \) since \(\nu \equiv 0\) on \(B_{r_0/8}(x_0)\).
On the other hand, let \(x_0\in \Sigma \), by the definition, for any \(r>0\) small enough, when n is sufficient large, we have
$$\begin{aligned} \frac{\mu _n(B_r(x_0))}{r^{m-2}}\ge \frac{\epsilon _0^2}{2^{m+1}}. \end{aligned}$$
Letting \(n\rightarrow \infty \),
$$\begin{aligned} \frac{\mu (B_r(x_0))}{r^{m-2}}\ge \frac{\epsilon _0^2}{2^{m+1}} \end{aligned}$$
for a.e. \(r>0\). Suppose \(x_0\notin \mathrm {sing}(\phi )\), then
$$\begin{aligned} r^{2-m}\int _{B_r(x_0)}|\nabla \phi |^2dx\le \frac{\epsilon _0^2}{2^{m+2}} \end{aligned}$$
whenever \(r>0\) is small enough. Then we have
$$\begin{aligned} \frac{\nu (B_r(x_0))}{r^{m-2}}\ge \frac{\epsilon _0^2}{2^{m+2}} \end{aligned}$$
for all small positive \(r>0\) and \(x_0\in \mathrm {spt}\nu \). This finishes the proof of lemma. \(\square \)

Lemma 4.2

Under the assumption of the preceding lemma, the limit
$$\begin{aligned} \lim _{r\rightarrow 0}\frac{\nu (B_r(x))}{r^{m-2}} \end{aligned}$$
(4.6)
exists for \(H^{m-2}\) a.e. \(x\in \Sigma \). If we denote it by \(\theta _\nu (x)\), then
$$\begin{aligned} \frac{\epsilon _0^2}{2^m}\le \theta _\nu (x)\le C(m,p,\Omega ,\Lambda _1,\Lambda ,N)\delta _0^{2-m}, \end{aligned}$$
where \(\delta _0:=dist(B_1(0),\partial \Omega )\).

Proof

For any \(x\in \Omega \) and any two sequence \(s_i\rightarrow 0\), \(t_i\rightarrow 0\), by Corollary 2.5 and Sobolev’s embedding \(W^{1,p}(\Omega )\hookrightarrow L^{\frac{mp}{m-p}}(\Omega )\), we have
$$\begin{aligned} \frac{\mu _n(B_{s_i}(x))}{s_i^{m-2}}\le \frac{\mu _n(B_{t_j}(x))}{t_j^{m-2}}+C(m,p,\Omega ,\Lambda _1,\Lambda ,N)(t_j)^{3-\frac{2m}{p}} \end{aligned}$$
(4.7)
for \(s_i\le t_j\). Letting firstly \(i\rightarrow \infty \) and secondly \(j\rightarrow \infty \), we get
$$\begin{aligned} \limsup _{r\rightarrow 0}\frac{\mu (B_{r}(x))}{r^{m-2}}\le \liminf _{r\rightarrow 0}\frac{\mu (B_{r}(x))}{r^{m-2}}, \end{aligned}$$
which implies that
$$\begin{aligned} \lim _{r\searrow 0}\frac{\mu (B_{r}(x))}{r^{m-2}} \end{aligned}$$
exists for any \(x\in \Omega \).
Noting that for \(H^{m-2}\) a.e. \(x\in \Omega \),
$$\begin{aligned} \lim _{r\rightarrow 0}r^{2-m}\int _{B_r(x)}|\nabla \phi |^2dx=0, \end{aligned}$$
(4.8)
therefore,
$$\begin{aligned} \lim _{r\rightarrow 0}\frac{\nu (B_{r}(x))}{r^{m-2}}= \lim _{r\rightarrow 0}\frac{\mu (B_{r}(x))}{r^{m-2}}. \end{aligned}$$
Obviously, from (4.7), we can get
$$\begin{aligned} r^{2-m}\mu (B_r(x))\le C(\Lambda )\delta _0^{2-m}+C(m,p,\Omega ,\Lambda _1,\Lambda ,N)(\delta _0)^{3-\frac{2m}{p}}\le C(m,p,\Omega ,\Lambda _1,\Lambda ,N)\delta _0^{2-m}. \end{aligned}$$
This implies \(\mu \lfloor \Sigma \) is absolutely continuous with respect to \(H^{m-2}\lfloor \Sigma \) and the Radon-Nikodym theorem tells us that there exists a measurable function \(\theta (x)\) such that
$$\begin{aligned} \mu \lfloor \Sigma =\theta (x)H^{m-2}\lfloor \Sigma . \end{aligned}$$
Noting that for \(H^{m-2}\) a.e. \(x\in \Sigma \),
$$\begin{aligned} 2^{2-m}\le \liminf _{r\rightarrow 0}\frac{H^{m-2}(\Sigma \cap B_r(x))}{r^{m-2}}\le \limsup _{r\rightarrow 0}\frac{H^{m-2}(\Sigma \cap B_r(x))}{r^{m-2}}\le 1 \end{aligned}$$
and (4.8), we have
$$\begin{aligned} \nu \lfloor \Sigma =\theta (x)H^{m-2}\lfloor \Sigma \end{aligned}$$
and
$$\begin{aligned} \frac{\epsilon _0}{2^m}\le \theta _\nu (x)=\theta (x)\le C(m,p,\Omega ,\Lambda _1,\Lambda ,N)\delta _0^{2-m}. \end{aligned}$$
\(\square \)

By modifying Lin’s method in [27] or applying Preiss’s result [32], we have

Corollary 4.3

The set of energy concentration points \(\Sigma \) is \((m-2)\)-rectifiable.

For any \(x\in \Sigma \) and \(\lambda >0\), we define a scaled Radon measure \(\mu _{y,\lambda }\) by
$$\begin{aligned} \mu _{y,\lambda }(A)=\lambda ^{2-m}\mu (y+\lambda A). \end{aligned}$$
If there is a Radon measure \(\mu _*\) such that
$$\begin{aligned} \mu _{y,\lambda }\rightarrow \mu _* \end{aligned}$$
in the sense of Radon measure as \(r\searrow 0\), then we say that \(\mu _*\) is the tangent measure of \(\mu \) at y. (See [14, 40]).

Lemma 4.4

Suppose \(H^{m-2}(\Sigma )>0\), then there exists a nonconstant harmonic sphere \(S^2\) into N.

Before we prove this lemma, let us state a basic proposition for the Dirac operator.

Proposition 4.5

Suppose \(\phi \in C^2(M,N)\), \(\psi \in C^2(M,\Sigma M\otimes \phi ^\star TN)\). Let \(\{e_\alpha \}_{\alpha =1}^m\) be a unit normal basis of TM and \(X\in \Gamma (TM)\) a section satisfying
$$\begin{aligned} {[}X,e_\alpha ]=0, \quad \alpha =1, \ldots , m, \end{aligned}$$
thenwhere \([\cdot ,\cdot ]\) is the Lie bracket, \(\phi _X=d\phi (X)\) and \(\psi _X=\widetilde{\nabla }_{X}\psi \).

Proof

The proof is similar to the Proposition 2.2 in [23] (see also the computations of Proposition 3.3 in [9]), where the case of a two dimensional domain was considered. \(\square \)

Proof of Lemma 4.4

Since \(\Sigma \) is \((m-2)\)-rectifiable, we can find a point \(x_0\in \Sigma \), such that \(\nu \) has a tangent measure at \(x_0\) and
$$\begin{aligned} \nu _*=\theta (x_0)H^{m-2}\lfloor \Sigma _* \end{aligned}$$
where \(\Sigma _*\subset {\mathbb {R}}^m\) is a \((m-2)\) linear subspace which is usually called the tangent space of \(\Sigma \) at \(x_0\). Without loss of generality, we assume \(x_0=0\) and \(\Sigma _*={\mathbb {R}}^{m-2}\times \{(0,0)\}\).
In fact, by a diagonal argument, we can find a sequence \(r_n\rightarrow 0\), such that,
$$\begin{aligned} |\nabla u_n|^2dx\rightarrow \nu _* \end{aligned}$$
in the sense of Radon measures (cf. [27]), where
$$\begin{aligned} (u_n(x),v_n(x)):=(\phi _n(x_0+r_nx),\sqrt{r_n}\psi _n(x_0+r_nx)). \end{aligned}$$
It is easy to see that \((u_n,v_n)\) is also a stationary Dirac-harmonic map with \(\lambda \)-curvature term. By Lemma 2.4, we have
$$\begin{aligned}&\int _{r_1}^{r_2}\int _{\partial B_r(x_0)}|x-x_0|^{2-m}\left( 2\left| \frac{\partial u_n}{\partial r}\right| ^2+Re\left\langle v_n,\partial _r\cdot \widetilde{\nabla }_{\partial r} v_n\right\rangle \right) dH^{n-1}dr\nonumber \\&=r_2^{2-m}\int _{B_{r_2}(x_0)}\left( |\nabla u_n|^2+\frac{\lambda }{6}R_{ikjl}\langle v_n^i,v_n^j\rangle \langle v_n^k,v_n^l\rangle \right) dx\nonumber \\&\quad -\,r_1^{2-m}\int _{B_{r_1}(x_0)}\left( |\nabla u_n|^2+\frac{\lambda }{6}R_{ikjl}\langle v_n^i,v_n^j\rangle \langle v_n^k,v_n^l\rangle \right) dx. \end{aligned}$$
(4.10)
By (2.9) and (2.10), we have
$$\begin{aligned} \int _{B_r(x_0)}|x-x_0|^{2-m}|v_n||\nabla v_n|dx&\le C(m)\Vert v_n\Vert _{L^{\frac{mp}{m-p}}(B_r(x_0))}\Vert \nabla v_n\Vert _{L^p(B_r(x_0))} r^{3-\frac{2m}{p}}\nonumber \\&= C(m)\Vert \psi _n\Vert _{L^{\frac{mp}{m-p}}}\Vert \nabla \psi _n\Vert _{L^p} (rr_n)^{3-\frac{2m}{p}}\nonumber \\&\le C(m,p,\Lambda ,\Omega )(rr_n)^{3-\frac{2m}{p}} \end{aligned}$$
(4.11)
and
$$\begin{aligned} r_1^{2-m}\int _{B_{r_1}(x_0)}|\psi |^4dx+r_2^{2-m}\int _{B_{r_2}(x_0)}|\psi |^4dx\le C(m,p,\Lambda ,\Omega ) (r_2r_n)^{6-\frac{4m}{p}}. \end{aligned}$$
(4.12)
Since \(r_2^{2-m}\nu _*(B_{r_2}(0))=r_1^{2-m}\nu _*(B_{r_1}(0))\), letting \(n\rightarrow \infty \) in (4.10), we get
$$\begin{aligned} \lim _{n\rightarrow \infty }\int _{B_2(0)}|\frac{\partial u_n}{\partial |x|}|^2dx=0. \end{aligned}$$
(4.13)
Noting that \(\nu _{*y,r}=\nu _*\) for \(y\in \Sigma _*\), we also have
$$\begin{aligned} \lim _{n\rightarrow \infty }\int _{B_2(0)}\left| \frac{\partial u_n}{\partial |x-y|}\right| ^2dx=0,\ for\ y\in \Sigma _*\cap B_2. \end{aligned}$$
(4.14)
These imply
$$\begin{aligned} \lim _{n\rightarrow \infty }\sum _{k=1}^{m-2}\int _{B_2(0)}\left| \frac{\partial u_n}{\partial x^k}\right| ^2dx=0. \end{aligned}$$
(4.15)
Let \(x'=(x_1, \ldots ,x_{m-2}),x''=(x_{m-1},x_m)\), define \(f_n:B^{m-2}_{1}\rightarrow {\mathbb {R}}\) by
$$\begin{aligned} f_n(x'):=\sum _{k=1}^{m-2}\int _{B^2_{1}(0)}\left| \frac{\partial u_n}{\partial x_k}\right| ^2(x',x'')dx''. \end{aligned}$$
Then,
$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert f_n(x')\Vert _{L^1(B^{m-2}_{1}(0))}=0. \end{aligned}$$
Let \(M(f_n)(x')\) be the Hardy–Littlewood maximal function, i.e.
$$\begin{aligned} M(f_n)(x)=\sup _{0<r<\frac{1}{2}}r^{2-m}\int _{B^{m-2}_r(x)}f_n(x')dx',\ x\in B^{m-2}_{1/2}(0). \end{aligned}$$
By the weak \(L^1\)-estimate, for any \(\rho >0\), we have
$$\begin{aligned} \left| \left\{ x\in B^{m-2}_{1/2}(0)|M(f_n)>\rho \right\} \right| \le \frac{C(m)}{\rho }\Vert f_n\Vert _{L^1(B^{m-2}_{1/2}(0))}, \end{aligned}$$
which implies
$$\begin{aligned} \left| \left\{ x\in B^{m-2}_{1/2}(0)|\limsup _{n\rightarrow \infty }M(f_n)>0\right\} \right| =0. \end{aligned}$$
Combining this with Theorem 1.4, there exists a sequence of points \(\{x'_n\in B^{m-2}_{1/2}(0)\}\), such that, passing to a subsequence, \((u_n,v_n)\) is smooth near \((x_n',x'')\) for all \(x''\in B^2_{1}(0)\) and
$$\begin{aligned} \lim _{n\rightarrow \infty }M(f_n)(x'_n)=0. \end{aligned}$$
(4.16)
By the blow-up argument in [27], there exist sequences \(\{\sigma _n\}\) and \(\{x''_n\}\subset B^2_{1/2}(0)\) such that \(\sigma _n\rightarrow 0\), \(x''_n\rightarrow (0,0)\) and
$$\begin{aligned} \max _{x''\in B^2_{1/2}(0)}\sigma _n^{2-m}\int _{B^{m-2}_{\sigma _n}(x'_n)\times B^2_{\sigma _n}(x'')}|\nabla u_n|^2dx=\frac{\epsilon _0^2}{C_1(m)}, \end{aligned}$$
(4.17)
where the maximum is achieved at the point \(x''_n\) and \(C_1(m)\ge 4^m\) is a positive constant to be determined later.
In fact, denote
$$\begin{aligned} g_n(\sigma ):=\max _{x''\in B^2_{1/2}(0)}\sigma ^{2-m}\int _{B^{m-2}_{\sigma }(x'_n)\times B^2_{\sigma }(x'')}|\nabla u_n|^2dx. \end{aligned}$$
On one hand, since \((u_n,v_n)\) is smooth near \(x'_n\times B^2_{1}(0)\), we have
$$\begin{aligned} \lim _{\sigma \rightarrow 0}g_n(\sigma )=0. \end{aligned}$$
On the other hand, for any \(\sigma >0\), when n is big enough, we must have
$$\begin{aligned} g_n(\sigma )\ge \frac{\epsilon _0^2}{2^m}. \end{aligned}$$
Otherwise, by Theorem 1.2 (similar to deriving (3.6)), \(u_n\) will converge strongly in \(W^{1,2}\) which is contradict to \(|\nabla u_n|^2dx\rightarrow \nu _*\). Thus, there exists \(\sigma _n\), such that
$$\begin{aligned} g_n(\sigma _n)=\frac{\epsilon _0^2}{C_1(m)} \end{aligned}$$
and we may assume the maximum is achieved at \(x''_n\). Next, we show \(\sigma _n\rightarrow 0\) and \(x''_n\rightarrow (0,0)\).
If \(\sigma _n\ge \delta >0\), by Corollary 2.5, we have
$$\begin{aligned} \frac{\epsilon _0^2}{C_1(m)}=\limsup _{n\rightarrow \infty }g_n(\sigma _n)\ge \limsup _{n\rightarrow \infty }\left( g_n(\delta )-C(m,p,\Lambda _1,\Lambda ,\Omega ,N)(r_n\sigma _n)^{3-\frac{2m}{p}}\right) \ge \frac{\epsilon _0}{2^m}, \end{aligned}$$
which is a contradiction.
If \(x''_n\rightarrow x''_0\in B^2_{1/2}(0)\) and \(x_0''\ne (0,0)\), for any \(0<\sigma <\frac{|x_0''|}{2}\),
$$\begin{aligned} \frac{\epsilon _0}{2^m}\le \limsup _{n\rightarrow \infty }g_n(\sigma )\le \sigma ^{2-m}\nu _*(B^{m-2}_{1}(0)\times B^2_{2\sigma }(x_0''))=0. \end{aligned}$$
This is also a contradiction. \(\square \)
Let \(x_n=(x'_n,x''_n)\) and
$$\begin{aligned} (\widetilde{u}_n(x),\widetilde{v}_n(x)):=(u_n(x_n+\sigma _nx),\sqrt{\sigma _n}v_n(x_n+\sigma _nx)). \end{aligned}$$
Then \((\widetilde{u}_n(x),\widetilde{v}_n(x))\) is a stationary Dirac-harmonic map with \(\lambda \)-curvature term defined on \(B^{m-2}_{R_n}(0)\times B^2_{R_n}(0)\), where \(R_n=\frac{1}{4\sigma _n}\) which tends to infinity as \(n\rightarrow \infty \).
By (4.16), we have
$$\begin{aligned}&\lim _{n\rightarrow \infty }\sup _{0<R<R_n}R^{2-m}\int _{B^{m-2}_{R}(0)\times B^2_{R_n}(0)}\sum _{k=1}^{m-2}\left| \frac{\partial \widetilde{u}_n}{\partial x_k }\right| ^2dx\nonumber \\&\quad =\lim _{n\rightarrow \infty }\sup _{0<R<R_n}(\sigma _nR)^{2-m}\int _{B^{m-2}_{\sigma _nR}(x'_n)\times B^2_{\sigma _n R_n}(x_n'')}\sum _{k=1}^{m-2}\left| \frac{\partial u_n}{\partial x_k }\right| ^2dx\nonumber \\&\quad \le \lim _{n\rightarrow \infty }M(f_n)(x_n')=0. \end{aligned}$$
(4.18)
By (4.17), we get
$$\begin{aligned} \frac{\epsilon _0^2}{C_1(m)}=\int _{B^{m-2}_{1}(0)\times B^2_{1}(0)}|\nabla \widetilde{u}_n|^2dx=\max _{x''\in B^2_{R_n-1}(0)}\int _{B^{m-2}_{1}(0)\times B^2_{1}(x'')}|\nabla \widetilde{u}_n|^2dx. \end{aligned}$$
(4.19)
By Corollary 2.5, for any \(R>0\), we obtain
$$\begin{aligned} \int _{B^{m-2}_{R}(0)\times B^2_{R}(0)}|\nabla \widetilde{u}_n|^2dx&=(\sigma _n)^{2-m}\int _{B^{m-2}_{\sigma _nR}(x'_n)\times B^2_{\sigma _n R}(x_n'')}|\nabla u_n |^2dx\nonumber \\&\le C(m,p,\delta _0,\Lambda _1,\Lambda ,\Omega ,N)R^{m-2}, \end{aligned}$$
(4.20)
when n is big enough.
Let \(\zeta \in C^\infty _0(B^{m-2}_1(0))\) be a cut-off function such that \(0\le \zeta \le 1\) and \(\zeta |_{B^{m-2}_{1/2}(0)}\equiv 1\). Let \(\eta \in C^\infty _0(B^{2}_1(0))\) also be a cut-off function such that \(0\le \zeta \le 1\) and \(\eta |_{B^{2}_{1/2}(0)}\equiv 1\). Similarly to [27], for any \(R>0\), we define \(F_n(a):B^{m-2}_6(0)\times B^{2}_{R}(0)\rightarrow {\mathbb {R}}\) as follows:
$$\begin{aligned} F_n(a)=\int _{B^{m-2}_1(0)\times B^{2}_1(0)}|\nabla \widetilde{u}_n|^2(a+x)\zeta (x')\eta (x'')dx. \end{aligned}$$
Computing directly, one has
$$\begin{aligned} \frac{\partial F_n(a)}{\partial a_k}&=\int _{B^{m-2}_1(0)\times B^{2}_1(0)}\frac{\partial }{\partial x_k}|\nabla \widetilde{u}_n|^2(a+x)\zeta (x')\eta (x'')dx\\&=2\int _{B^{m-2}_1(0)\times B^{2}_1(0)}\left\langle \frac{\partial \widetilde{u}_n}{\partial x_l}, \frac{\partial ^2 \widetilde{u}_n}{\partial x_l\partial x_k}\right\rangle (a+x)\zeta (x')\eta (x'')dx\\&=-2\int _{B^{m-2}_1(0)\times B^{2}_1(0)}\left\langle \Delta \widetilde{u}_n, \frac{\partial \widetilde{u}_n}{\partial x_k}\right\rangle (a+x)\zeta (x')\eta (x'')dx\\ {}&\quad -2\int _{B^{m-2}_1(0)\times B^{2}_1(0)}\left\langle \frac{\partial \widetilde{u}_n}{\partial x_l}, \frac{\partial \widetilde{u}_n}{\partial x_k}\right\rangle (a+x)\frac{\partial }{\partial x_l}(\zeta (x')\eta (x''))dx. \end{aligned}$$
On the one hand, by Proposition 4.5,and (2.3), we haveNoting that
$$\begin{aligned} \left\langle G(\widetilde{v}_n), \frac{\partial \widetilde{u}_n}{\partial x_k}\right\rangle =\frac{1}{12}\frac{\partial \left( \lambda (\widetilde{u}_n)R_{ijkl}(\widetilde{u}_n)\right) }{{\partial x^k}}\langle v_n^i, v_n^j \rangle \langle v_n^k, v_n^l \rangle , \end{aligned}$$
integrating by parts and using Young’s inequality, then the right hand side of (4.22) is controlled by
$$\begin{aligned}&C(\Lambda _1,N)\int _{B^{m-2}_1(0)\times B^{2}_1(0)}\left( |\nabla \widetilde{v}_n||\widetilde{v}_n|^3+|\widetilde{v}_n|^4+|\nabla \widetilde{v}_n||\widetilde{v}_n|\right) (a+x)dx\\ {}&\le C(m,p,\Lambda _1,\Lambda ,\Omega , N)(\sigma _nr_n)^{3-\frac{2m}{p}}. \end{aligned}$$
On the other hand, by Hölder’s inequality, one has
$$\begin{aligned}&-2\int _{B^{m-2}_1(0)\times B^{2}_1(0)}\left\langle \frac{\partial \widetilde{u}_n}{\partial x_l}, \frac{\partial \widetilde{u}_n}{\partial x_k}\right\rangle (a+x)\frac{\partial }{\partial x_l}(\zeta (x')\eta (x''))dx\\&\quad \le C\left( \int _{B^{m-2}_{R+1}(0)\times B^{2}_{R+1}(0)}|\nabla \widetilde{u}_n|^2dx\right) ^{1/2}\left( \int _{B^{m-2}_{R+1}(0)\times B^{2}_{R+1}(0)}|\frac{\partial \widetilde{u}_n}{\partial x_k}|^2dx\right) ^{1/2}. \end{aligned}$$
Combining these and letting \(n\rightarrow \infty \), we obtain
$$\begin{aligned} \frac{\partial F_n(a)}{\partial a_k}\rightarrow 0,\ k=1, \ldots , m-2, \end{aligned}$$
uniformly in \(B^{m-2}_6(0)\times B^{2}_{R}(0)\) for any fixed \(R>0\).
Thus, for any \(a=(a',a'')=B^{m-2}_6(0)\times B^{2}_{R}(0)\),
$$\begin{aligned} \int _{B^{m-2}_{1/2}(a')\times B^{2}_{1/2}(a'')}|\nabla \widetilde{u}_n|^2dx&\le F_n(a)\\&\le F_n((0,a''))+C(m)\sum _{k=1}^{m-2}\left| \frac{\partial F_n(a)}{\partial a_k}\right| \\&\le \int _{B^{m-2}_{1}(0)\times B^{2}_{1}(a'')}|\nabla \widetilde{u}_n|^2dx+C(m)\sum _{k=1}^{m-2}\left| \frac{\partial F_n(a)}{\partial a_k}\right| \\&\le \frac{\epsilon _0^2}{C_1(m)}+C(m)\sum _{k=1}^{m-2}\left| \frac{\partial F_n(a)}{\partial a_k}\right| . \end{aligned}$$
Therefore, for any \(R>0\), when n is big enough, we have
$$\begin{aligned} 6^{2-m}\int _{B^{m-2}_6(0)\times B^2_6(0)}|\nabla \widetilde{u}_n|^2(x',x''+b)dx\le \frac{C(m)\epsilon _0^2}{C_1(m)} \ for \ all \ b\in B^2_{R-6}. \end{aligned}$$
(4.23)
Taking \(C_1(m)\ge 2^mC(m)\), similar to deriving (3.5), we have
$$\begin{aligned}&\sup _{x_0\in B_3(0),0<r\le 3}r^{2-m}\int _{B_r(x_0)}(|\nabla \widetilde{u}_n|^2+|\lambda |^2|\widetilde{v}_n|^4)(x',x''+b)dx\\&\quad \le \frac{\epsilon _0^2}{4}+C(m,p,\Lambda _1,\Lambda ,\Omega ,N)(r_n\sigma _n)^{3-\frac{2m}{p}} \le \frac{\epsilon _0^2}{2}, \end{aligned}$$
whenever n is large enough.
By Theorem 1.2, we know \((\widetilde{u}_n,\widetilde{v}_n)\) sub-converges to a Dirac-harmonic map with \(\lambda \)-curvature term (uv) in \(C^1_{loc}(B^{m-2}_{3/2}(0)\times {\mathbb {R}}^2)\). Moreover, by (4.18)–(4.20), for any \(R>0\), we have
$$\begin{aligned} \int _{B_R(0)}\sum _{k=1}^{m-2}\left| \frac{\partial u}{\partial x_k}\right| ^2dx=0, \end{aligned}$$
and
$$\begin{aligned} \int _{B_1(0)}|\nabla u|^2dx=\frac{\epsilon _0^2}{C_1(m)},\quad \int _{B_R(0)}|\nabla u|^2dx\le C(m,p,\delta _0,\Lambda _1,\Lambda ,\Omega ,N)R^{m-2}. \end{aligned}$$
Furthermore, since
$$\begin{aligned} \int _{B_R(0)}|v|^4dx=\lim _{n\rightarrow \infty }\int _{B_R(0)}|\widetilde{v}_n|^4dx\le \lim _{n\rightarrow \infty } C(m,p,\delta _0,\Lambda _1,\Lambda ,\Omega ,N)(r_n\sigma _nR)^{3-\frac{2m}{p}}=0, \end{aligned}$$
we know \(v\equiv 0\) and \(u:{\mathbb {R}}^2\rightarrow N\) is a nonconstant harmonic map with finite energy. By the conformal invariance of harmonic maps in dimension two, u can be extended to a nonconstant harmonic sphere.

Proof of Theorem 1.5

The conclusion of Theorem 1.5 follows from Lemma 4.4 and the Federer dimension reduction argument which is similar to [37] for minimizing harmonic maps. We omit the details here. This completes the proof. \(\square \)

Footnotes

  1. 1.

    Here, the constant is the \(C^1\)-norm of \(\lambda \) which can be replaced by the \(W^{1,q}\)-norm for some big constant \(q>1\) in the whole paper, we leave it to interested readers.

Notes

Acknowledgements

Open access funding provided by Max Planck Society.

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Authors and Affiliations

  1. 1.School of Mathematical SciencesShanghai Jiao Tong UniversityShanghaiChina
  2. 2.Max Planck Institute for Mathematics in the SciencesLeipzigGermany

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