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.
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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 (M, g) 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 , 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 (N, h) 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
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
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
Now, 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 follows
where \(R(\phi ,\psi )\) is defined by
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 (N, g). 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 functional
Critical 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 (N, h) ia a compact Riemannian manifold, we define a nonnegative constant:Footnote 1
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
Here \(\psi \in \Gamma (\Sigma M\otimes \phi ^{*}TN)\) should be understood as a K-tuple of spinors \((\psi ^1, \ldots ,\psi ^K)\) satisfying
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
then \((\phi ,\psi )\in C^\infty (B_{\frac{r_0}{2}}(x_0))\), and it satisfies
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 holds
where \(\phi _t(x)=\phi (x+tY(x))\) 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 \(0<r_1\le r_2<dist(x_0,\partial \Omega )\),
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
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 \(L_\lambda \),
and
where \(\phi _t\) 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:
Thus,
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
Set
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, \((\phi ,\psi )\) satisfies
where
and \(B:=(\frac{\partial \lambda }{\partial z^1}, \ldots ,\frac{\partial \lambda }{\partial z^K})\), \(B^\top \) is the tangential part of B along the map \(\phi \), \(P(\cdot ;\cdot )\) is the shape operator, i.e.
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
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 have
On the other hand, we have
where we used the fact that
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
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
we have
Using the Eq. (2.2) and letting \(\epsilon \rightarrow 0\), we get
which yields
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
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
where \(\Lambda _1\) is as defined in (1.6).
Proof
By Lemma 2.4, we know
which implies
By Sobolev’s embedding and Young’s inequality, we have
and
\(\square \)
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
where
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
then \(\psi \in L^p(B_{1/2})\) and satisfies the estimate
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
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
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
Thus
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
with an antisymmetric potential \(\widehat{\Omega }\) satisfying
and with an error term f satisfying
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
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
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,
Therefore,
By Corollary 2.5, for any \(0<r_0<\frac{1}{2}\min \{dist(x_0,\partial \Omega ),1\}\), we have
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
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
Additionally, we assume
for some \(p>\frac{2m}{3}\). Similar to harmonic maps [36], define the energy concentration set \(\Sigma \) as follows
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
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
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,
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
which implies (similar to deriving (3.6))
By Theorem 1.2, we know
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 \(r\le r_1\),
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
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
Letting \(n\rightarrow \infty \),
for a.e. \(r>0\). Suppose \(x_0\notin \mathrm {sing}(\phi )\), then
whenever \(r>0\) is small enough. Then we have
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
exists for \(H^{m-2}\) a.e. \(x\in \Sigma \). If we denote it by \(\theta _\nu (x)\), then
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
for \(s_i\le t_j\). Letting firstly \(i\rightarrow \infty \) and secondly \(j\rightarrow \infty \), we get
which implies that
exists for any \(x\in \Omega \).
Noting that for \(H^{m-2}\) a.e. \(x\in \Omega \),
therefore,
Obviously, from (4.7), we can get
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
Noting that for \(H^{m-2}\) a.e. \(x\in \Sigma \),
and (4.8), we have
and
\(\square \)
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
If there is a Radon measure \(\mu _*\) such that
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
then
where \([\cdot ,\cdot ]\) 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
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,
in the sense of Radon measures (cf. [27]), where
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
and
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
Noting that \(\nu _{*y,r}=\nu _*\) for \(y\in \Sigma _*\), we also have
These imply
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
Then,
Let \(M(f_n)(x')\) be the Hardy–Littlewood maximal function, i.e.
By the weak \(L^1\)-estimate, for any \(\rho >0\), we have
which implies
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
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
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
On one hand, since \((u_n,v_n)\) is smooth near \(x'_n\times B^2_{1}(0)\), we have
On the other hand, for any \(\sigma >0\), when n is big enough, we must have
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
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
which is a contradiction.
If \(x''_n\rightarrow x''_0\in B^2_{1/2}(0)\) and \(x_0''\ne (0,0)\), for any \(0<\sigma <\frac{|x_0''|}{2}\),
This is also a contradiction. \(\square \)
Let \(x_n=(x'_n,x''_n)\) and
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
By (4.17), we get
By Corollary 2.5, for any \(R>0\), we obtain
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:
Computing directly, one has
On the one hand, by Proposition 4.5,
and (2.3), we have
Noting that
integrating by parts and using Young’s inequality, then the right hand side of (4.22) is controlled by
On the other hand, by Hölder’s inequality, one has
Combining these and letting \(n\rightarrow \infty \), we obtain
uniformly in \(B^{m-2}_6(0)\times B^{2}_{R}(0)\) for any fixed \(R>0\).
Thus, for any \(a=(a',a'')=B^{m-2}_6(0)\times B^{2}_{R}(0)\),
Therefore, for any \(R>0\), when n is big enough, we have
Taking \(C_1(m)\ge 2^mC(m)\), similar to deriving (3.5), we have
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 (u, v) 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
and
Furthermore, since
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. \(\square \)
Notes
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.
References
Adams, D.: A note on Riesz potentials. Duke Math. J. 42(4), 765–778 (1975)
Ammann, B.: A variational problem in conformal spin geometry. Habilitation (Hamburg University) (2003)
Bethuel, F.: On the singular set of stationary harmonic maps. Manu. Math. 78(4), 417–443 (1993)
Branding, V.: Some aspects of Dirac-harmonic maps with curvature term. Differ. Geom. Appl. 40, 1–13 (2015)
Branding, V.: Nonlinear dirac equations, monotonicity formulas and liouville theorems, arxiv:1605.03453
Chang, S.-Y.A., Wang, L., Yang, P.C.: Regularity of harmonic maps. Commun. Pure Appl. Math. 52(9), 1099–1111 (1999)
Chen, Q., Jost, J., Wang, G.: Liouville theorems for Dirac-harmonic maps. J. Math. Phys 48(11), 113517 (2007)
Chen, Q., Jost, J., Li, J., Wang, G.: Regularity theorems and energy identities for Dirac-harmonic maps. Mathematische Zeitschrift 251(1), 61–84 (2005)
Chen, Q., Jost, J., Li, J., Wang, G.: Dirac-harmonic maps. Mathematische Zeitschrift 254(2), 409–432 (2006)
Chen, Q., Jost, J., Sun, L., Zhu, M.: Estimates for solutions of Dirac equations and an application to a geometric elliptic–parabolic problem. J. Eur. Math. Soc. (JEMS) 21(3), 665–707 (2019)
Chen, Q., Jost, J., Wang, G., Zhu, M.: The boundary value problem for Dirac-harmonic maps. J. Eur. Math. Soc. (JEMS) 15(3), 997–1031 (2013)
Deligne, P.: Quantum Fields and Strings: A Course for Mathematicians, vol. 2. American Mathematical Society, Providence (1999)
Evans, L.: Partial regularity for stationary harmonic maps into sphere. Arch. Rat. Mech. Anal. 116(2), 101–113 (1991)
Federer, H.: Geometric Measure Theory. Springer, New York (1969)
Giaquinta, M.: Multiple integrals in the calculus of variations and nonlinear elliptic systems, Ann. Math. Studies, 105. Princeton University Press (1983)
Giaquinta, M., Giusti, E.: On the regularity of the minima of variational integrals. Acta Mathematica 148(1), 31–46 (1982)
Giaquinta, M., Giusti, E.: The singular set of the minima of certain quadratic functionals. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4 e srie, tome 11, n o 1 (1984), pp. 45–55
Hélein, F.: Harmonic Maps, Conservation Laws and Moving Frames. Cambridge Tracts in Mathematics, vol. 150. Cambridge University Press, Cambridge (2002)
Hildebrandt, S., Kaul, H., Widman, K.-O.: An existence theorem for harmonic mappings of Riemannian manifolds. Acta Mathematica 138, 1C16 (1977)
Jost, J.: Geometry and Physics. Springer, Berlin (2009)
Jost, J., Liu, L., Zhu, M.: Geometric analysis of the action functional of the nonlinear supersymmetric sigma model, MPI MIS Preprint: 77/2015
Jost, J., Liu, L., Zhu, M.: A global weak solution of the Dirac-harmonic map flow. Ann. Inst. H. Poincare Anal. Non Lineaire 34(7), 1851–1882 (2017)
Jost, J., Liu, L., Zhu, M.: Blow-up analysis for approximate Dirac-harmonic maps in dimension 2 with applications to Dirac-harmonic heat flow. Calc. Var. Partial Diff. Equ. 56(4), (2017), Art. 108
Jost, J., Liu, L., Zhu, M.: Energy identity for a class of approximate Dirac-harmonic maps from surfaces with boundary. Ann. Inst. H. Poincaré Anal. Non Linéaire 36(2), 365–387 (2019)
Jost, J., Kessler, E., Tolksdorf, J., Wu, R., Zhu, M.: Regularity of solutions of the nolinear sigma model with gravitino. Commun. Math. Phys. 358(1), 171–197 (2018)
Lawson, H., Michelsohn, M.: Spin Geometry, vol. 38. Princeton University Press, Princeton (1989)
Lin, F.: Gradient estimates and blow-up analysis for stationary harmonic maps. Ann. Math. 149, 785–829 (1999)
Liu, L.: No neck for Dirac-harmonic maps. Calc. Var. Partial Differ. Equ. 52(12), 1–15 (2015)
Morrey Jr., C.B.: The problem of Plateau on a Riemannian manifold. Ann. Math. (2) 49, 807–851 (1948)
Naber, A., Valtorta, D.: Rectifiable-Reifenberg and the regularity of stationary and minimizing harmonic maps. Ann. Math. (2) 185(1), 131–227 (2017)
Naber, A., Valtorta, D.: Stratification for the singular set of approximate harmonic maps. Math. Z. 290(3–4), 1415–1455 (2018)
Preiss, D.: Geometry of measures on \(\mathbb{R}^{n}\): distribution, rectifibility, and densities. Ann. Math. 125, 537–643 (1987)
Rivière, T.: Conservation laws for conformally invariant variational problems. Invent. Math. 168(1), 1–22 (2007)
Rivière, T.: The Variations of Yang-Mills Lagrangian, arxiv.org/abs/1506.04554
Rivière, T., Struwe, M.: Partial regularity for harmonic maps and related problems. Commun. Pure Appl. Math. 61(4), 451–63 (2008)
Schoen, R.: Analytic aspects of the harmonic map problem, Math. Sci. Res. Inst. PUBL.2, Springer, New York, pp. 321–358 (1984)
Schoen, R., Uhlenbeck, K.: A regularity theory for harmonic maps. J. Differ. Geom. 17, 253–268 (1982)
Sharp, B.: Higher integrability for solutions to a system of critical elliptic PDE. Methods. Appl. Anal. 21(2), 221–240 (2014)
Sharp, B., Zhu, M.: Regularity at the free boundary for Dirac-harmonic maps from surfaces. Calc. Var. Partial Differ. Equ. 55(2), 55–27 (2016)
Simon, L.: Lectures on Geometric Measure Theory. Australian National University Centre for Mathematical Analysis, Canberra (1983)
Wang, C.: A remark on nolinear Dirac equations. Proc. Am. Math. Soc. 138(10), 3753–3758 (2010)
Wang, C., Xu, D.: Regularity of Dirac-harmonic maps. Int. Math. Res. Not. IMRN 20, 3759–3792 (2009)
Xu, D., Chen, Z.: Regularity for Dirac-Harmonic maps with Ricci type spinor potential. Calc. Var. Partial Differ. Equ. 46(3-4), 571–590 (2013)
Zhao, L.: Energy identities for Dirac-harmonic maps. Calc. Var. Partial Differ. Equ. 28(1), 121–138 (2007)
Zhu, M.: Regularity of weakly Dirac-harmonic maps to hypersurfaces. Ann. Glob. Anal. Geom. 35(4), 405–412 (2009)
Zhu, M.: Dirac-harmonic maps from degenerating spin surfaces I: the Neveu–Schwarz case. Calc. Var. Partial Differ. Equ. 35(2), 169–189 (2009)
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Part of the work was carried out when Lei Liu was visiting the School of Mathematical Sciences at Shanghai Jiao Tong University and he would like to thank the institution for hospitality and support. Miaomiao Zhu was supported in part by National Natural Science Foundation of China (No. 11601325).
Appendix
Appendix
In this section, for reader’s convenience, we recall some known results which are used in this paper.
Lemma 5.1
(Lemma 6.1 in [25]) Let \(m\ge 2\) and \(4<p<\infty \). Let \(\psi \in M^{4,2}(B_1,\mathbb {C}^L\otimes {\mathbb {R}}^K)\) be a weak solution of the nonlinear system
where \(A\in M^{2,2}(B_1,gl(L,\mathbb {C})\otimes gl(K,{\mathbb {R}}))\) and \(B\in M^{2,2}(B_1,\mathbb {C}^L\otimes {\mathbb {R}}^K)\). For any \(U\subseteq B_1\), there exists \(\epsilon _0=\epsilon _0(m,p)>0\) and \(C=C(m,p,U)>0\) such that if
then \(\psi \in L^p(U)\) and the following estimate hold:
Theorem 5.2
([35], Theorem 1.2 in [38]) Let \(m\ge 2\) and \(2<p<\infty \). Let \(u\in W^{1,2}(B_1, {\mathbb {R}}^d)\), \(\Omega \in M^{2,2}(B_1, so(d)\otimes \wedge ^1{\mathbb {R}}^m)\) and \(f\in L^p(B_1, {\mathbb {R}}^d)\) with \(\frac{m}{2}<p<m\), satisfy weakly
Then for any \(U\subseteq B_1\), there exist \(\epsilon =\epsilon (m, d, p)>0\) and \(C=C(m, d, p, U)>0\) such that if \(\Vert \Omega \Vert _{M^{2,2}(B_1)}\le \epsilon \), then
Lemma 5.3
(\(W^{k,p}\)-estimates, c.f. [2]) Let (M, g) be an m-dimensional spin Riemannian manifold. Suppose \(\psi \in \Gamma (\Sigma M),\ \psi \in L^4(B_r(x_0))\) is a weak solution of
where \(B_r(x_0)\) is a geodesic ball of M and \(f\in W^{k,p}(B_r(x_0))\) for some \(1<p<\infty \), \(k\ge 1\). Then \(\psi \in W^{k+1,p}(B_{\frac{r}{2}}(x_0))\) and
Lemma 5.4
(Schauder estimates, c.f. [2]) Let (M, g) be a m-dimensional spin Riemannian manifold. Suppose \(\psi \in \Gamma (\Sigma M),\ \psi \in L^4(B_r(x_0))\) is a weak solution of
where \(f\in C^{k,\alpha }(B_r(x_0))\) for some \(0<\alpha <1\) and \(k\ge 1\). Then \(\psi \in C^{k+1,\alpha }(B_{\frac{r}{2}}(x_0))\) and
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Jost, J., Liu, L. & Zhu, M. Regularity of Dirac-harmonic maps with \(\lambda \)-curvature term in higher dimensions. Calc. Var. 58, 187 (2019). https://doi.org/10.1007/s00526-019-1632-y
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DOI: https://doi.org/10.1007/s00526-019-1632-y