Abstract
\(\alpha \)-Dirac-harmonic maps are variations of Dirac-harmonic maps, analogous to \(\alpha \)-harmonic maps that were introduced by Sacks–Uhlenbeck to attack the existence problem for harmonic maps from closed surfaces. For \(\alpha >1\), the latter are known to satisfy a Palais–Smale condition, and so, the technique of Sacks–Uhlenbeck consists in constructing \(\alpha \)-harmonic maps for \(\alpha >1\) and then letting \(\alpha \rightarrow 1\). The extension of this scheme to Dirac-harmonic maps meets with several difficulties, and in this paper, we start attacking those. We first prove the existence of nontrivial perturbed \(\alpha \)-Dirac-harmonic maps when the target manifold has nonpositive curvature. The regularity theorem then shows that they are actually smooth if the perturbation function is smooth. By \(\varepsilon \)-regularity and suitable perturbations, we can then show that such a sequence of perturbed \(\alpha \)-Dirac-harmonic maps converges to a smooth coupled \(\alpha \)-Dirac-harmonic map.
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1 Introduction
Harmonic maps from closed Riemann surfaces and their variants are important both in mathematics as tools to probe the geometry of a Riemannian manifold and in physics as ground states of the nonlinear sigma model of quantum field theory. They represent a borderline case for the Palais–Smale condition and therefore cannot be directly obtained by standard tools. In [19], Sacks–Uhlenbeck introduced the notion of \(\alpha \)-harmonic maps which for \(\alpha >1\) (which we shall always assume in this paper) makes the problem subcritical for the Palais–Smale condition. To get the existence of harmonic maps, they consider the convergence of \(\alpha \)-harmonic maps when \(\alpha \) decreases to 1. In general, there are bubbles (harmonic spheres) preventing the smooth convergence of \(\alpha \)-harmonic maps. Fortunately, a nonpositive curvature condition on the target manifold can exclude such bubbles, and they can therefore obtain the existence of harmonic maps into such manifolds in any given homotopy class.
There is another harmonic map type problem which is not only more difficult and subtle, but also has a profound geometric significance. Motivated by the supersymmetric nonlinear sigma model from quantum field theory [6, 10], Dirac-harmonic maps from spin Riemann surfaces into Riemannian manifolds were introduced in [4]. Mathematically, they are generalizations of the classical harmonic maps and harmonic spinors. From the variational point of view, they are critical points of a conformal invariant action functional whose Euler–Lagrange equations constitute a coupled elliptic system consisting of a second order equation and a Dirac equation.
In fact, the existence of Dirac-harmonic maps from closed surfaces is a tough problem. Different from the Dirichlet problem (see [12] and the references given there), even if there is no bubble, the limit may still be trivial, in the sense that the spinor part \(\psi \) vanishes identically. So far, there are only a few results about the Dirac-harmonic maps from closed surfaces, see [2] and [5] for some existence results of uncoupled Dirac-harmonic maps (here uncoupled means that the map part is harmonic) based on index theory and the Riemann-Roch theorem, respectively. There are also some other interesting approaches, such as the heat flow method [13, 23], the variational method [8] and the homology theory approach [9]. In [8] and [9], the authors discussed the existence of nonlinear Dirac-geodesics, which are the critical points of the following action functional:
where s is the angular coordinate on \(S^1\), \({{\dot{\phi }}}\) denotes the s-derivative of \(\phi \), and F is a nonlinear perturbation satisfying some growth and decay conditions with respect to \(\psi \).
In this paper, we shall systematically study critical point theory according to Palais–Smale for \(\alpha \)-Dirac harmonic maps for \(\alpha >1\). Since the natural space on which the variational integral is defined is not a Hilbert space, but only a Banach space, we need to develop appropriate Banach space tools, like pseudo-gradient flow. Moreover, since our functionals are not bounded from below, we have to look for critical points other than minima, and therefore, we shall need to carefully investigate the Palais–Smale condition. Unfortunately, it follows from [13] that the following action functional does not satisfy the Palais–Smale condition:
This is the functional we are interested in, and its critical points are called \(\alpha \)-Dirac-harmonic maps. Since variational schemes produce only weak solutions, it is also necessary to deal with the issue of their regularity.
In more precise terms, we consider a vector bundle \(\Sigma M\otimes TN\rightarrow N\) and a function \(F: \Sigma M\otimes TN\rightarrow {\mathbb {R}}\). A general element of \(\Sigma M\otimes TN\) is written as \((\phi ,\psi )\), where \(\phi \in N\) and \(\psi \in \Sigma M\otimes \phi ^*TN\), and we write \(F=F(\phi ,\psi )\). We shall first prove the Palais–Smale condition for the action functional \(\mathcal {L}^\alpha \) of perturbed \(\alpha \)-Dirac-harmonic maps from closed Riemann surfaces:
The difficulty is to prove the convergence of the map parts, which will be solved by combining the ideas in [22] and [8]. Next, we obtain a special negative pseudo-gradient flow to deform the configuration space. The existence of a pseudo-gradient vector field is well-known. However, it is generally inexplicit. Our pseudo-gradient flow is special because the spinor part of the pseudo-gradient vector field is explicit. With such a special pseudo-gradient flow and the Palais–Smale condition in hand, by the deformation lemma, we can prove the existence of perturbed \(\alpha \)-Dirac-harmonic maps. In fact, we get a critical value defined in an explicit way. Last, as for the nontrivialness, it suffices to show that the critical value is strictly bigger than the \(\alpha \)-energy minimizer defined in (6.1), where we use the linking geometry theory to give a lower bound of the critical value.
As usual in the calculus of variations, we shall need the following standard growth conditions (F1)-(F5) on the nonlinearity \(F(\phi ,\psi )\in C^{1,1}_{loc}(\Sigma M\otimes TN,{\mathbb {R}})\) and its partial derivatives \(F_\phi (\phi ,\psi )=\frac{\partial F}{\partial \phi }(\phi ,\psi )\in (TN)^*\cong TN\), \(F_\psi (\phi ,\psi )=\frac{\partial F}{\partial \psi }(\phi ,\psi )\in (\Sigma M\otimes TN)^*\cong \Sigma M\otimes TN\) in \(dF=F_\phi d\phi +F_\psi d\psi \).
-
(F1)
There exist \(p\in (2,4)\) and \(C>0\) such that
$$\begin{aligned} |F_\psi (\phi ,\psi )|\le C(1+|\psi |^{p-1}) \end{aligned}$$for any \((\phi ,\psi )\in \mathcal {F}^{\alpha ,1/2}(M,N)\).
-
(F2)
There exist \(\mu >2\) and \(R_1>0\) such that
$$\begin{aligned} 0<\mu {F}(\phi ,\psi )\le \langle F_{\psi }(\phi ,\psi ),\psi \rangle \end{aligned}$$for any \((\phi ,\psi )\in \mathcal {F}^{\alpha ,1/2}(M,N)\) with \(|\psi |\ge R_1\).
-
(F3)
There exist \(q<4\) and \(C>0\) such that
$$\begin{aligned} F_\phi (\phi ,\psi )\le C(1+|\psi |^q) \end{aligned}$$for any \((\phi ,\psi )\in \mathcal {F}^{\alpha ,1/2}(M,N)\).
-
(F4)
For any \((\phi ,\psi )\in \mathcal {F}\), we have
$$\begin{aligned} F(\phi ,\psi )\ge 0. \end{aligned}$$ -
(F5)
As \(|\psi |\rightarrow 0\), we have
$$\begin{aligned} F(\phi ,\psi )=o(|\psi |^2) \text { uniformly in }\phi \in N. \end{aligned}$$
Theorem 1.1
Let M be a closed surface and N a compact manifold. Suppose F satisfies \((\mathrm F1)\)-\((\mathrm F5)\) with \(\frac{4\alpha }{3\alpha -2}\le \mu \le p\le \frac{3}{4}\mu +1\) for \(\alpha \in (1,2]\). Let \(R_1,R_2\) and \(\rho \) be as in Lemmas 6.1 and 6.2. Then we have \(m_\theta<c_\theta <\infty \) and \(c_\theta \) is a critical value of \(\mathcal {L}^\alpha \) in \(\mathcal {F}\).
Here \(\theta \) is a given homotopy class of maps, \(m_\theta \) is the minimizing \(\alpha \)-energy in \(\theta \), \(c_\theta \) is defined as:
and \(\Gamma (Q_{\theta ;R_1,R_2})\) is defined in Definition 4.12, \(Q_{\theta ;R_1,R_2}\) is defined by (6.2). In particular, when N has nonpositive curvature, our solution is nontrivial.
Corollary 1.2
Let M be a closed surface and N a compact Rienmannian manifold with non-positive curvature. Suppose F satisfies \((\mathrm F1)\)-\((\mathrm F5)\) with \(\frac{4\alpha }{3\alpha -2}\le \mu \le p\le \frac{3}{4}\mu +1\) for \(\alpha \in (1,2]\). Then for any homotopy class \(\theta \in [M,N]\), there exists a non-trivial solution \((\phi ,\psi )\in W^{1,2\alpha }(M,N)\times H^{1/2}(M,\Sigma M\otimes \phi ^*TN)\) to the perturbed \(\alpha \)-Dirac-harmonic map Eq. (2.11) and (2.12) with \(\phi \in \theta \).
Since \(\alpha >1\), \(\phi \) in the theorem above is continuous. It is natural to expect the smoothness of the weak solutions. Due to the perturbation F, \(F_\psi \) will produce \(\Vert \psi \Vert ^3_{L^4}\) according to the proof in [3]. Therefore, the proof there can not apply to our situation directly. To overcome it, we need to control the \(L^\infty \)-norm of \(\psi \) first. The same phenomenon happens in the proof of the \(\varepsilon \)-regularity. The following regularity theorem shows that such a nontrivial solution is actually smooth.
Theorem 1.3
Suppose \(F\in C^\infty \) satisfies \((\mathrm F1)\) and \((\mathrm F3)\) for some \(p\le 2+2/\alpha \) and \(q\ge 0\). Then any weakly perturbed \(\alpha \)-Dirac-harmonic map is smooth.
Now, by multiplying \(F(\phi ,\psi )\) by constants \(\frac{1}{k}\), we get a sequence of functionals:
where F satisfies the assumptions in Corollary 1.2 and Theorem 1.3. For example, one can just take \(F=|\psi |^{\frac{4\alpha }{3\alpha -2}}\). Thus, we get a sequence of perturbed \(\alpha \)-Dirac-harmonic maps \(\{(\phi _k,\psi _k)\}\), which are the critical points of \(\mathcal {L}_{k}^{\alpha }\). Furthermore, if such sequence converges smoothly, then the limit will be a critical point of the functional (1.2), which is a \(\alpha \)-Dirac-harmonic map. To do so, we need some uniform regularity estimates. Since the proof of Theorem 1.3 depends on the oscillation of \(\phi \), which may not be uniformly controlled for \(\phi _k\), we want to prove another estimate which is called the \(\varepsilon \)-regularity estimate. This kind of estimate was introduced by Sacks and Uhlenbeck for the \(\alpha \)-harmonic maps in [19]. When coupled with the Dirac equation, this was handled [3] for Dirac-harmonic maps and in [12, 12] for a sequence of \(\alpha \)-Dirac-harmonic maps(\(\alpha \rightarrow 1\)). We have to modify here the growth conditions (F1) and (F3) for the derivatives of F; we need
- \((\mathrm F6)\):
-
\(|F_\psi (\phi ,\psi )|\le C|\psi |^{r-1} \ \text {for} \ 3<r\le 2+2/\alpha ,\)
- \((\mathrm F7)\):
-
\(|F_\phi (\phi ,\psi )|\le C|\psi |^{q} \ \text {for} \ q>2.\)
Theorem 1.4
Suppose F satisfies (F6) and (F7). There is \(\varepsilon _0>0\) and \(\alpha _0>0\) such that if \((\phi ,\psi ):(D,g_{\beta \gamma })\rightarrow (N,g_{ij})\) is a smooth perturbed \(\alpha \)-Dirac-harmonic map satisfying
for \(1\le \alpha <\alpha _0\), then we have
and
forany \({\tilde{D}}\subset D\) and \(p>1\), where \(C(D,N,\Lambda ,p)\) denotes a constant depending on \(D,N,\Lambda \) and p.
With this \(\varepsilon \)-regularity, one can easily prove
Theorem 1.5
For the \(\alpha _0\) given in Theorem 1.4and each \(\alpha \in (1,\alpha _0)\), let \((\phi _k,\psi _k)\) be the smooth critical points of the functional \(L^\alpha _k\) in (1.5) with uniformly bounded energy:
Suppose F satisfies \((\mathrm F6)\) and \((\mathrm F7)\). Then there exist a subsequence, still denoted by \(\{(\phi _k,\psi _k)\}\), and a smooth \(\alpha \)-Dirac-harmonic map \((\phi ,\psi )\) such that
Since the convergence in the theorem above is smooth on M, the proof of Theorem 1.1 implies that the action functional of the \(\alpha \)-Dirac-harmonic map \((\phi ,\psi )\) is still strictly bigger than \(m_\theta \). Then the convexity of the action functional tells us that \((\phi ,\psi )\) is a coupled \(\alpha \)-Dirac-harmonic map. Thus, we obtain:
Corollary 1.6
Let M be a closed surface and N a compact Rienmannian manifold with non-positive curvature. For the \(\alpha _0\) given in Theorem 1.4and each \(\alpha \in (1,\alpha _0)\), if the sequence of perturbed \(\alpha \)-Dirac-harmonic maps \(\{(\phi _k,\psi _k)\}\) satisfies the uniform bounded energy condition (1.9), then there exists a coupled \(\alpha \)-Dirac-harmonic map from M to N with \(\phi \) in the given homotopy class \(\theta \).
Remark 1.7
To get the existence of a Dirac-harmonic map, it is natural to consider the convergence of a sequence of \(\alpha \)-Dirac-harmonic maps as \(\alpha \) decreases to 1. By the joint work of the first author with Lei Liu and Miaomiao Zhu in [12], under the uniform bounded energy condition, there exists a Dirac-harmonic map with the map part in the given homotopy class. However, in general, we cannot guarantee that the limit is nontrivial.
The rest of the paper is organized as follows: In Sect. 2, we derive the Euler–Lagrange equations and define the configuration space. In Sect. 3, we prove the Palais–Smale condition for the action functional of perturbed \(\alpha \)-Dirac-harmonic maps. In Sect. 4, we construct a special pseudo-gradient vector field and deform our configuration space by a negative pseudo-gradient flow generated by it. Besides, we also recall some facts in linking geometry. In Sect. 5, we prove the uniqueness of \(\alpha \)-harmonic maps into nonpositive curved manifolds. In Sect. 6, we give the proof of Theorem 1.1 and Theorem 1.2. In Sect. 7, we prove Theorem 1.3. In the last section, we prove Theorem 1.4 and Theorem 1.5.
2 Euler–Lagrange equations and configuration space
2.1 Euler–Lagrange equations
Let (M, g) be a compact surface with a fixed spin structure. On the spinor bundle \(\Sigma M\), we denote the Hermitian inner product by \(\langle \cdot , \cdot \rangle _{\Sigma M}\). For any \(X\in \Gamma (TM)\) and \(\xi \in \Gamma (\Sigma M)\), the Clifford multiplication is skew-adjointness:
Let \(\nabla \) be the Levi-Civita connection on (M, g). There is a connection (also denoted by \(\nabla \)) on \(\Sigma M\) compatible with \(\langle \cdot , \cdot \rangle _{\Sigma M}\). Choosing a local orthonormal basis \(\{e_{\beta }\}_{\beta =1,2}\) on M, the usual Dirac operator is defined as 
, where \(\beta =1,2\) (here and in the sequel, we use the Einstein summation convention). One can find more about spin geometry in [14].
Let \(\phi \) be a smooth map from M to a compact Riemannian manifold (N, h) of dimension \(n\ge 2\). Let \(\phi ^*TN\) be the pull-back bundle of TN by \(\phi \) and consider the twisted bundle \(\Sigma M\otimes \phi ^*TN\). On this bundle there is a metric \(\langle \cdot ,\cdot \rangle _{\Sigma M\otimes \phi ^*TN}\) induced from the metric on \(\Sigma M\) and \(\phi ^*TN\). Also, we have a connection \({\tilde{\nabla }}\) on this twisted bundle naturally induced from those on \(\Sigma M\) and \(\phi ^*TN\). In local coordinates \(\{y^i\}_{i=1,\dots ,n}\), the section \(\psi \) of \(\Sigma M\otimes \phi ^*TN\) is written as
where each \(\psi ^i\) is a usual spinor on M. We also have following local expression of \({\tilde{\nabla }}\)
where \(\Gamma ^i_{jk}\) are the Christoffel symbols of the Levi-Civita connection of N. The Dirac operator along the map \(\phi \) is defined as
which is self-adjoint [11]. Sometimes, we use 
to distinguish the Dirac operators defined on different maps. In [4], the authors introduced the functional
with Euler–Lagrange equations
where \(\tau ^m(\phi )\) is the m-th component of the tension field [11] of the map \(\phi \) with respect to the coordinate on N, \(\nabla \phi ^l\cdot \psi ^j\) denotes the Clifford multiplication of the vector field \(\nabla \phi ^l\) with the spinor \(\psi ^j\), and \(R^m_{lij}\) stands for the component of the Riemann curvature tensor of the target manifold N. By defining
we can write (2.4) and (2.5) in the global form
Solutions \((\phi ,\psi )\) of (2.7) are called Dirac-harmonic maps from M to N.
In this paper, we use the critical point theory to prove the existence problem of Dirac-harmonic maps from closed surfaces. For the one-dimensional case, Takeshi Isobe [8] proves the existence of nontrivial nonlinear Dirac-geodesics on flat tori, which are critical point of
where s is the angular coordinate on \(S^1\), \({{\dot{\phi }}}\) denotes the s-derivative of \(\phi \), and F is a nonlinear interaction term satisfying some growth and decay conditions with respect to \(\psi \). The Euler–Lagrange equations are:
In particular, Isobe proved that there exists a non-trivial solution \((\phi , \psi )\) to (2.9) with \(\phi \) in any given free homotopy class of loops on a flat torus. In [9], Isobe reconsidered this problem through homology theory. By constructing and computing a Morse-Floer type homology, he obtains several existence results for perturbed Dirac geodesics. Some of them do not need the curvature restriction on the target manifold.
To generalize Isobe’s result to closed surfaces, we need to overcome two obstacles. One is to prove the Palais–Smale condition in the two dimensional setting, the other is to construct a nice pseudo-gradient vector field. Since the energy functional \(E(\phi )=\int _M|d\phi |^2\) in general does not satisfy the Palais–Smale condition in two dimension, we consider
Similar to the computations in [8], one can get the Euler–Lagrange equations for \(\mathcal {L}^{\alpha }\):
2.2 Configuration space
We will define a configuration space for our functional \(\mathcal {L}^{\alpha }\). In fact, we focus on \(W^{1,2\alpha }\)-maps\((\alpha >1)\) and \(H^{1/2}\)-spinors. By the Nash embedding theorem [16], we can embed N into Euclidean space \({\mathbb {R}}^L\) for some large L. We define the \(W^{1,2\alpha }\)-maps on N as
where \(\phi \in W^{1,2\alpha }(M,{\mathbb {R}}^L)\) means that both \(\phi \) and its weak derivative \(\nabla \phi \) are in \(L^{2\alpha }(M,{\mathbb {R}}^L)\). By the Sobolev embedding theorem, any \(\phi \in W^{1,2\alpha }(M,N)\) is continuous. Therefore, the pull-back bundle \(\phi ^*TN\) is well-defined, and we can consider \(H^{1/2}\)-spinors along \(\phi \in W^{1,2\alpha }(M,N)\) defined as
where \(\underline{{\mathbb {R}}}^L=M\times {\mathbb {R}}^L\) is the trivial \({\mathbb {R}}^L\)-bundle over M and we regard \(T_{\phi (x)}N\) as a subset of \({\mathbb {R}}^L\) for each \(x\in M\) by the above embedding.
Consider the Banach space \(\mathcal {F}^{\alpha ,1/2}{:}{=}W^{1,2\alpha }(M,{\mathbb {R}}^L)\times H^{1/2}(M,\Sigma M\otimes \underline{{\mathbb {R}}}^L)\). Since \(\mathcal {F}^{\alpha ,1/2}\) is a product space, the norm can be induced from the norms on the two subspaces. We view the map component as the horizontal part and the spinor component as the vertical part. The horizontal part is a Banach space with the usual norm. The vertical part is actually a Hilbert space with the following inner product [1, 7, 9]:
where \((\cdot ,\cdot )_2\) is the \(L^2\)-inner product on M and 
. With respect to this product structure, we can write
where
and
Now, we can define a configuration space \(\mathcal {F}^{\alpha ,1/2}(M,N)\) as
which is a Banach submanifold of \(\mathcal {F}^{\alpha ,1/2}\) with the tangent space at \((\phi ,\psi )\in \mathcal {F}^{\alpha ,1/2}(M,N)\) being
The space \(W^{1,2\alpha }(M,N)\) is a Banach manifold [22] whose tangent space at \(\phi \in W^{1,2\alpha }(M,N)\) is
So, to see (2.21), it suffices to show the vertical part, which is the same as in [8].
Let us go back to our Lagrangian \(\mathcal {L}^{\alpha }\). On \(\mathcal {F}^{\alpha ,1/2}(M,N)\), since \(H^{1/2}\) can be continuously embedded into \(L^4\) [21], \(\mathcal {L}^{\alpha }\) is well-defined if for example \(F(\phi ,\psi )\) grows at most like \(|\psi |^4\) as \(|\psi |\rightarrow \infty \).
3 The Palais–Smale condition
In this section, we prove the Palais–Smale condition for \(\mathcal {L}^\alpha \) for a certain class of nonlinearities F satisfying (F1)-(F3). From now on, we denote \(\mathcal {F}^{\alpha ,1/2}(M,N)\) by \(\mathcal {F}\) for short. Let us recall the Palais–Smale condition:
Definition 3.1
A sequence \(\{(\phi _n,\psi _n)\}\subset \mathcal {F}\) is a Palais–Smale sequence if the following are satisfied:
-
(1)
\(\sup \limits _{n\ge 1}|\mathcal {L}^{\alpha }(\phi _n,\psi _n)|<\infty \),
-
(2)
\(\Vert d\mathcal {L}^{\alpha }(\phi _n,\psi _n)\Vert _{T^*_{(\phi _n,\psi _n)}\mathcal {F}}\rightarrow 0\).
We say \(\mathcal {L}^\alpha \) satisfies the Palais–Smale condition on \(\mathcal {F}\) if any Palais–Smale sequence has a convergent subsequence in \(\mathcal {F}\). By Sobolev embedding and conditions (F1) (F3), \(d\mathcal {L}^\alpha _{(\phi ,\psi )}\) is a bounded linear map on \(\mathcal {F}\). Thus, Zorn’s proposition (see page 30 in [22]) implies \(\mathcal {L}^\alpha \) is \(C^1\) on \(\mathcal {F}\). Therefore, if \(\mathcal {L}^\alpha \) satisfies the Palais–Smale condition on \(\mathcal {F}\), and so, any Palais–Smale sequence has a subsequence converging to a critical point.
The following is the main theorem in this section. The proof for the vertical part follows from [8]. For the horizontal part, the method in [8] is no longer valid, because \(d^H\mathcal {L}^\alpha \) cannot be written as a combination of a linear and a compact operator. Although the proof in [22] can not be applied to our case either, one estimate there inspired us. For completeness, we also give the proof for the spinor part.
Theorem 3.2
If F satisfies conditions \((\mathrm F1), (F2), (F3)\) with \(\frac{4\alpha }{3\alpha -2}\le \mu \le p\le \frac{3}{4}\mu +1\) for \(\alpha \in (1,2]\), then \(\mathcal {L}^\alpha \) satisfies the Palais–Smale condition on \(\mathcal {F}\).
Proof
We first prove that any Palais–Smale sequence is bounded.
Let \(\{(\phi _n,\psi _n)\}\subset \mathcal {F}\) be a Palais–Smale sequence. By the structure on \(\mathcal {F}^{\alpha ,1/2}\), we have
where we have used \(\Vert d^V\mathcal {L}^{\alpha }(\phi _n,\psi _n)\Vert \rightarrow 0\) as \(n\rightarrow \infty \). This implies
For simplicity, we denote all positive constants that are independent of n by C.
On the other hand, by (F2), there exists a constant \(C>0\) such that
for any \((\phi ,\psi )\in \mathcal {F}^{\alpha ,1/2}(M,N)\). Plugging this into (3.2), we get
Integrating (F2), we know
for any \((\phi ,\psi )\in \mathcal {F}^{\alpha ,1/2}(M,N)\). Plugging this into (3.4) yields
Consider the spectral decomposition
with respect to the operator 
, where \(\mathbf{H}_0^-\), \(\mathbf{H}_0^0\) and \(\mathbf{H}_0^+\) are the closures in \(H^{1/2}(M,\Sigma M\otimes \underline{{\mathbb {R}}}^L)\) of the spaces spanned by the negative, the null and the positive eigenspinors of \(\mathbf{D}\), respectively. Denote by \(\mathrm{P}_0^-: H^{1/2}(M,\Sigma M\otimes \underline{{\mathbb {R}}}^L)\rightarrow \mathbf{H}_0^-\), \(\mathrm{P}_0^0: H^{1/2}(M,\Sigma M\otimes \underline{{\mathbb {R}}}^L)\rightarrow \mathbf{H}_0^0\) and \(\mathrm{P}_0^+: H^{1/2}(M,\Sigma M\otimes \underline{{\mathbb {R}}}^L)\rightarrow \mathbf{H}_0^+\) the corresponding spectral projections. By this decomposition, we write \(\phi _n=\psi _{n,0}^-+\psi _{n,0}^0+\psi _{n,0}^+\) with \(\psi _{n,0}^-\), \(\psi _{n,0}^0\) and \(\psi _{n,0}^+\) being \(\mathrm{P}_0^-\psi _n\), \(\mathrm{P}_0^0\psi _n\) and \(\mathrm{P}_0^+\psi _n\), respectively. The square of the \(H^{1/2}\)- norm of \(\psi _{n,0}^+\in \mathbf{H}_0^+\) is equivalent to ( [1])
Together with (2.2), this implies
where we have used the Hölder inequality and the Sobolev embedding \(H^{1/2}\subset L^4\) for surfaces [21].
Again, as in (3.1), we have
This and (F1) give us
where we have used the Hölder inequality and assumption \(p\le \frac{3}{4}\mu +1\).
Now, plugging (3.11) into (3.9), we have
Since \(\alpha \in (1,2]\), the Hölder inequality and (3.6) imply
and
Plugging (3.13) and (3.14) into (3.12), we get
Similarly,
Since the usual Dirac operator has finite dimensional kernel, \(\mathrm{dim}(\mathbf{H}_0^0)<\infty \). Noting that the \(H^{1/2}\)-norm and the \(L^2\)-norm are equivalent on \(\mathbf{H}_0^0\) ( [1]), we have
Combining (3.15), (3.16) and (3.17), we obtain
Since \(\frac{4\alpha }{3\alpha -2}\le \mu \le p\le \frac{3}{4}\mu +1<\mu +1\) and \(\alpha \in (1,2]\), (3.18) tells us that \(\{\psi _n\}\) is uniformly bounded in \(H^{1/2}(M,\Sigma M\otimes \underline{{\mathbb {R}}}^L)\). Therefore, from (3.14), \(\{\phi _n\}\) is also bounded. Thus, \(\{\phi _n,\psi _n\}\) is a bounded sequence.
Next, we show \(\{\psi _n\}\) has a convergent subsequence. As in [8], we write the vertical gradient of \(\mathcal {L}^\alpha \) as
where
is bounded linear and
Because both \(|\psi \nabla \phi |\) and \(F_\psi (\phi ,\psi )\) belong in \(L^r\) with \(r>4/3\), both \(K^V_1\) and \(K^V_2\) are compact. We write \(\psi _n=\psi _{n,0}^-+\psi _{n,0}^0+\psi _{n,0}^+\). Since \(L^V\) is invertible on \(\mathbf{H}_0^{\pm }\) and \( \nabla ^V\mathcal {L}^\alpha (\phi _n,\psi _n)\rightarrow 0\), we have
where \(o(1)\rightarrow 0\) as \(n\rightarrow \infty \). From the boundedness of \(\{(\phi _n,\psi _n)\}\) and the compactness of \((L^V)^{-1}K\) on \(\mathbf{H}_0^{\pm }\), we know \(\psi _{n,0}^\pm \) has a convergent subsequence. Again, since \(\mathrm{dim}{} \mathbf{H}_0^0<\infty \), \(\psi _{n,0}^0\) also has a convergent subsequence. Thus, \(\{\psi _n\}\) has a convergent subsequence.
Last, for the convergence of \(\{\phi _n\}\), we consider the \(\alpha \)-energy functional
for any \(\phi \in W^{1,2\alpha }(M,{\mathbb {R}}^L)\). Then the second derivative of \(e^\alpha \) at \(d\phi \) with respect to the direction \(d\varphi \) is
For any \(\phi _i,\phi _j\in W^{1,2\alpha }(M,{\mathbb {R}}^L)\), we define
where \(a{:}{=}d\phi _i\) and \(b{:}{=}d\phi _j\). Then (3.25) implies
We also have
and
Now, we can control \(\Vert d\phi _i-d\phi _j\Vert _{2\alpha }\) as follows.
where the last inequality comes from the Sublemma 3.18 in [22] stated as follows:\(\square \)
Lemma 3.3
Let V be a finite dimensional real vector space with a norm \(|\cdot |\). Let m be a positive integer. Then there exists a positive constant C such that
for any \(x,y\in V\).
To estimate the left-hand side of (3.30), we decompose \(d^H\mathcal {L}^\alpha \) as
where
and \(J: W^{1,2\alpha }(M,{\mathbb {R}}^L)\rightarrow (W^{1,2\alpha }(M,{\mathbb {R}}^L))^*=W^{-1,2\alpha /(2\alpha -1)}(M,{\mathbb {R}}^L)\)
By Sobolev embedding, \(K^H=K^H_1+K^H_2+K^H_3:\mathcal {F}\rightarrow (W^{1,2\alpha }(M,{\mathbb {R}}^L))^*\) is compact. Again, we can write
The boundedness of \(\{(\phi _n,\psi _n)\}\) and the compactness of \(K^H\) imply that \(J(\phi _n)\) has a convergent subsequence. In particular, for any \(\varepsilon \), there exists an integer \(N_\varepsilon \) such that
for \(i,j>N_\varepsilon \). Together this with (3.30) and
we get
where we have used the boundedness of \(\{\phi _n\}\). Therefore, \(\{d\phi _n\}\) contains a Cauchy subsequence in \(L^{2\alpha }\). Hence, by Sobolev embedding, \(\{\phi _n\}\) has a convergent subsequence. \(\square \)
4 Negative pseudo-gradient flow and linking geometry
This section consists of two parts. In the first part, we will construct a special pseudo-gradient vector field to deform the configuration space. In the second part, we will recall some important results in linking geometry, which will be used to prove the nontrivialness of the perturbed \(\alpha \)-Dirac-harmonic maps in Sect. 6.
4.1 Negative pseudo-gradient flow
In this section, we want to find a pseudo-gradient vector field with the vertical part being parallel to the vertical gradient \(\nabla ^V\mathcal {L}^\alpha \). Before doing that, let us recall the definition of a pseudo-gradient vector.
Definition 4.1
[17] Let M be a \(C^{r+1}\)(\(r\ge 1\)) Finsler manifold and \(f: M\rightarrow {\mathbb {R}}\) be a \(C^1\) function. A vector \(X\in T_pM\) is called a pseudo-gradient vector for f at p if X satisfies
-
(i)
\(\Vert X\Vert \le 2\Vert df_p\Vert \),
-
(ii)
\(df_p(X)\ge \Vert df\Vert ^2\).
A vector field is called a pseudo-gradient vector field for f if at each point of its domain it is a pseudo-gradient vector for f. It is well-known that
Lemma 4.2
[17] There exists a locally Lipschitz pseudo-gradient vector field for f on \(M^*{:}{=}M-K\), where \(K{:}{=}\{p\in M|df_p=0\}\).
On \(\mathcal {F}^{\alpha ,1/2}(M, N)\), we denote the set of regular points of \(\mathcal {L}^\alpha \) by
The main result in this section is
Theorem 4.3
Suppose \(F\in C^{1,1}_{loc}\) in the fiber direction. Then there exists a locally Lipschitz pseudo-gradient vector field \(\omega \) for \(\mathcal {L}^\alpha \) on \(\tilde{{\mathcal {F}}}\) of the form of \( \omega =X\oplus a\nabla ^V\mathcal {L}^\alpha \) for some vector field X and a constant \(a\in (1,2)\).
Proof
We divide \(\tilde{{\mathcal {F}}}\) into two subsets
For any \(a\in (1,2)\), \(a\nabla ^V\mathcal {L}^\alpha \) always satisfies (i) and (ii) in Definition 4.1 for \(d^V\mathcal {L}^\alpha \). Our assumption implies that \(a\nabla ^V\mathcal {L}^\alpha \) is locally Lipschitz. Therefore, it suffices to show that there is a locally Lipschitz vector field that satisfies (i) and (ii) in Definition 4.1 for \(d^H\mathcal {L}^\alpha \).
Over B, since \(d^H\mathcal {L}\ne 0\), we have a vector \({\tilde{X}}_\phi \) for each \((\phi ,\psi )\in B\) such that \(\Vert {\tilde{X}}_\phi \Vert =1\) and \(d^H\mathcal {L}^\alpha _{(\phi ,\psi )}({\tilde{X}}_\phi )>\frac{2}{3}\Vert d^H\mathcal {L}^\alpha _{(\phi ,\psi )}\Vert \). Then, set \(X_\phi =\frac{3}{2}\Vert d^H\mathcal {L}^\alpha _{(\phi ,\psi )}\Vert {\tilde{X}}_\phi \). So \(X_\phi \) satisfies Definition 4.1 for \(d^H\mathcal {L}^\alpha \) with strict inequalities in (i) and (ii). Then extend \(X_\phi \) to be a \(C^1\) vector field in a neighborhood of \((\phi ,\psi )\) (say by making it “constant” with respect to a chart at \((\phi ,\psi )\)) [17]. Therefore, for each point \((\phi ,\psi )\in B\), there is a \(C^1\) pseudo-gradient vector field \(\omega =X\oplus a\nabla ^V\mathcal {L}^\alpha \) for \(\mathcal {L}^\alpha \) defined in some neighborhood of \((\phi ,\psi )\).
For A, we just use the \(C^1\) vector field \(\omega =0\oplus a\nabla ^V\mathcal {L}^\alpha \). Then, for any point in \(\tilde{{\mathcal {F}}}\), we have
and
So, \(\omega \) satisfies (i) in Definition 4.1. To check (ii), we take any point \((\phi ,\psi )\in A\). Then
Together with \(\mathcal {L}^\alpha \in C^1\), this implies that
holds in some neighborhood U of \((\phi ,\psi )\). So, \(\omega \) also satisfies (ii) in Definition 4.1. Thus, for each point \((\phi ,\psi )\in A\), there is a neighborhood U of \((\phi ,\psi )\) such that \(\omega =0\oplus a\nabla ^V\mathcal {L}^\alpha \) is a pseudo-gradient vector field for \(\mathcal {L}^\alpha \) in U. Finally, to get the pseudo-gradient vector field in the theorem, we can patch these local pseudo-gradient vector fields together by a partition of unity [17, 18]. Thus, we complete the proof.
\(\square \)
Theorem 4.3 gives us a nice pseudo-gradient vector field, but it is only locally Lipschitz. Therefore, its integral curve may not exist globally. To remedy this, it suffices to integrate a truncated pseudo-gradient vector field. The argument can be found in [18]. Different from our case, Isobe [8] used it directly on the gradient of the action functional. Now, we deform our configuration space by integrating the following ODE:
The function \(\eta \) is chosen such that (4.7) and (4.8) have a global unique solution. In particular, \(\eta \) satisfies \(\eta (\phi ,\psi )\Vert \omega (\phi ,\psi )\Vert \le 1\) for all \((\phi ,\psi )\in \mathcal {F}^{\alpha ,1/2}(M,N)\). See Appendix A in [18].
Note that the solution \(\psi _t\) to (4.7) belongs to \(H^{1/2}(M,\Sigma M\otimes \phi _t^*TN)\) for each \(t\ge 0\) and the space depends on t. So we translate it into a flow on a function space which does not depend on t. To do so, we consider the parallel transport. For each \(x\in M\), we denote by \(\mathbf{P}_t(x): T_{\phi (x)}N\rightarrow T_{\phi _t(x)}N\) the parallel transport along the curve \(t\mapsto \phi _t(x)\). We put \({\tilde{\psi }}_t(x){:}{=}(\mathbf{1}\otimes {\mathbf{P}_t(x)}^{-1})\psi _t(x)\), that is, for \(\psi _t(x)=\psi _t^i(x)\otimes \frac{\partial }{\partial y^i}(\phi _t(x))\), we define
Thus, the vertical part of (4.7) is transformed to
where \(\eta _t({\tilde{\psi }}_t)=\eta (\phi _t,\psi _t)\) and a is the constant in Theorem 4.3.
Since the proofs of Lemma 7.2-Lemma 7.6 in [8] only rely on the Sobolev embedding and the vertical part of the gradient vector field, which is kept up to a constant in our case, these nice properties (Lemma 7.2-Lemma 7.6 in [8]) generalize to our configuration space \(\mathcal {F}^{\alpha ,1/2}(M,N)\). Due to their usefulness, we list them here and refer to [8] for the proofs.
Lemma 4.4
Let \((\phi _t,\psi _t)\) and \(\mathbf{P}_t\) be as above. \(\mathbf{P}_t\) defines a bounded linear map \(\mathbf{P}_t=\mathbf{1}\otimes \mathbf{P}_t: H^{1/2}(M,\Sigma M\otimes \phi ^*TN)\rightarrow H^{1/2}(M,\Sigma M\otimes \phi _t^*TN)\subset H^{1/2}(M,\Sigma M\otimes \underline{{\mathbb {R}}}^L)\) which depends coutinuously on t with respect to the operator norm.
Lemma 4.5
Let \(\phi ,{\tilde{\phi }}\in W^{1,2\alpha }(M,N)\) be such that \(\Vert \phi -{\tilde{\phi }}\Vert _\infty <\iota (N)\). We define \(T_{\phi ,{\tilde{\phi }}}(x):\Sigma M\otimes T_{\phi (x)}N\rightarrow \Sigma M\otimes T_{{\tilde{\phi }}(x)}N\) by \(T_{\phi ,{\tilde{\phi }}}(x)=\mathbf{1}\otimes \mathbf{P}_{\phi (x),{\tilde{\phi }}(x)}\), where \(\mathbf{P}_{\phi (x),{\tilde{\phi }}(x)}:T_{\phi (x)}N\rightarrow T_{{\tilde{\phi }}(x)}N\) is the parallel transport along the unique length minimizing geodesic between \(\phi (x)\) and \({\tilde{\phi }}(x)\). Then the map \(T_{\phi ,{\tilde{\phi }}}\) defined by \((T_{\phi ,{\tilde{\phi }}}\psi )(x)=T_{\phi ,{\tilde{\phi }}}(x)\psi (x)\) for a.e. \(x\in M\) is a bounded linear map \(T_{\phi ,{\tilde{\phi }}}:H^{1/2}(M,\Sigma M\otimes \phi ^*TN)\rightarrow H^{1/2}(M,\Sigma M\otimes {\tilde{\phi }}^*TN)\). Moreover, its operator norm satisfies
for some constant \(C(\Vert \phi \Vert _{1,2\alpha },\Vert {\tilde{\phi }}\Vert _{1,2\alpha })\) depending only on \(\Vert \phi \Vert _{1,2\alpha }\) and \(\Vert {\tilde{\phi }}\Vert _{1,2\alpha }\).
Consider \(T_{\phi ,{\tilde{\phi }}}\) as \(T_{\phi ,{\tilde{\phi }}}:H^{1/2}(M,\Sigma M\otimes \phi ^*TN)\rightarrow H^{1/2}(M,\Sigma M\otimes \underline{{\mathbb {R}}}^L)\). For any \(\psi \in H^{1/2}(M,\Sigma M\otimes \phi ^*TN)\subset H^{1/2}(M,\Sigma M\otimes \underline{{\mathbb {R}}}^L)\) we have
for some constant \(C(\Vert \phi \Vert _{1,2\alpha },\Vert {\tilde{\phi }}\Vert _{1,2\alpha })\) depending only on \(\Vert \phi \Vert _{1,2\alpha }\) and \(\Vert {\tilde{\phi }}\Vert _{1,2\alpha }\).
Lemma 4.6
Let \(\phi ,{\tilde{\phi }}\in W^{1,2\alpha }(M,N)\). We denote by \(P^+(\phi ): \mathcal {F}_{\phi }\rightarrow \mathcal {F}_{\phi }^+\) the spectral projection onto the positive eigenspace of 
. For \(\psi \in \mathcal {F}_{\phi }{:}{=}H^{1/2}(M,\Sigma M\otimes \phi ^*TN)\), we have
where the \(H^{1/2}\)-norm on the left side is taken in \(H^{1/2}(M,\Sigma M\otimes \underline{{\mathbb {R}}}^L)\), \(T_{\phi ,{\tilde{\phi }}}\) is defined as in Lemma 4.5and \(C(\Vert \phi \Vert _{1,2},\Vert {\tilde{\phi }}\Vert _{1,2\alpha })\) depending only on \(\Vert \phi \Vert _{1,2\alpha }\) and \(\Vert {\tilde{\phi }}\Vert _{1,2\alpha }\).
Lemma 4.7
Let \(\phi :[0,1]\ni t\mapsto \phi _t\in W^{1,2\alpha }(M,N)\) be piecewise \(C^1\) and \(\mathbf{P}_{t}(x):T_{\phi _0(x)}N\rightarrow T_{\phi _t(x)}N\) the parallel transport along the curve \([0,1]\ni s\mapsto \phi _s(x)\in N\). Set 
. Then 
defines a continous family of compact operators with respect to the operator norm.
Note that the decomposition of \(\nabla ^V\mathcal {L}^\alpha \) is not unique. For our purposes, we need to use a decomposition that is different from those presented in (3.19). Let \(T>0\) be arbitrary. For \(t\in [0,T]\), we write
where 
denotes the Dirac operator along the map \(\phi _T\), \(\mathbf{P}^{-1}_{t\rightarrow T}=\mathbf{1}\otimes \mathbf{P}^{-1}_{t\rightarrow T}\) and \(\mathbf{P}^{-1}_{t\rightarrow T}(x): T_{\phi _t(x)}N\rightarrow T_{\phi _T(x)}N\) is the parallel transport along the curve \([t,T]\ni s\mapsto \phi _s(x)\in N\). Then (4.10) can be written as
where 
. As in Lemma 4.7, \(\mathbf{P}_tL_{T,V}-L^V\mathbf{P}_t:H^{1/2}(M,\Sigma M\otimes \phi ^*TN)\rightarrow H^{1/2}(M,\Sigma M\otimes \underline{{\mathbb {R}}}^L)\) is compact and continuously depends on t and T. So, \({\tilde{K}}\) is also compact.
Regarding \(\phi _t\) and \({\tilde{\psi }}_t\) in \(\eta _t({\tilde{\psi }}_t)\) and \(\tilde{K}(T,t;\tilde{\psi }_t)\) as already known functions, integration of of (4.15) yields
where
is a compact map from \([0,1]\times [0,1]\times H^{1/2}(M,\Sigma M\otimes \phi ^*TN)\) to \(H^{1/2}(M,\Sigma M\otimes \phi ^*TN)\) (see Lemma 7.3 in [8]).
Since \(t\in [0,T]\) is arbitrary, we take \(T=t\) and obtain
where \(K(t,\psi ){:}{=}{\hat{K}}(t,t;\psi ):[0,1]\times H^{1/2}(M,\Sigma M\otimes \phi ^*TN)\rightarrow H^{1/2}(M,\Sigma M\otimes \phi ^*TN)\) is a compact map.
Remark 4.8
The advantage of the form of \({\tilde{\psi }}_t\) in (4.18) is that the bounded linear operator \(\exp \Bigg (-\int _0^t\eta _s({\tilde{\psi }}_s)dsL_{t,V}\Bigg )\) commutes with the spectral projection of the operator 
. This fact will be used in the section below.
4.2 Linking geometry
Before introducing specific definitions and lemmas in linking geometry, let us first explain how we use this theory. As we said in the introduction, since our configuration space is no longer a Hilbert manifold, we have to use the pseudo-gradient flow, which we view as a deformation of the configuration space. Therefore, it is natural to associate with the classical deformation lemma. To use it, it is necessary to define a suitable deformation class according to the special pseudo-gradient flow constructed in the previous subseciton. And since our functionals are not bounded from below, we need to look for critical points, which are saddle points other than minima. Actually, we will use the deformation lemma to prove the following
is a critical value, where \(\Gamma (Q_{\theta ;R_1,R_2})\) is the suitable deformation class defined in Definition 4.12, \(Q_{\theta ;R_1,R_2}\) is a subset of the configuration space. To get a nontrivial critical point, we want to show that \(c_\theta \) is strictly bigger than the \(\alpha \)-energy minimizer \(m_\theta \) defined in (6.1), where we need the linking geometry. Roughly speaking, in the linking geometry, we say two subsets link if the deforming subsets always intersect a fixed subset provided the boundaries of the deforming subsets never intersect that fixed subset. In the following, we will see \(\Gamma (Q_{\theta ;R_1,R_2})\) and \(S_{\theta ;\rho }\) link with respect to \(\mathcal {C}\). Then, the critical value is bigger or equal to the infimum \(b_{\theta ;\rho }\) defined in (6.5). Therefore, the nontrivialness of the critical point is equivalent to \(b_{\theta ;\rho }>m_\theta \), which will be proved in Lemma 6.2.
Now, let us see a linking argument within our framework. Since the proofs only rely on those properties generalized in the section above, we again refer to [8] for details. First, let us define a general deformation class.
Definition 4.9
We define
- (1):
-
\(\Theta (0,(\phi ,\psi ))=(\phi ,\psi )\) for \((\phi ,\psi )\in \mathcal {F}\).
- (2):
-
For \(0\le t\le 1\) and \((\phi ,\psi )\in \mathcal {F}\), writing \(\Theta (t,(\phi ,\psi )){=}{:}(\phi _t,\psi _t)\), we have \(\phi _t\in W^{1,2\alpha }(M,N)\) and \(\psi _t\in H^{1/2}(M,\Sigma M\otimes \phi _t^*TN)\). Moreover, \(\phi :[0,1]\ni t\mapsto \phi _t\in W^{1,2\alpha }(M,N)\) is piecewise \(C^1\).
- (3):
-
Let \((\phi _t,\psi _t)=\Theta (t,(\phi ,\psi ))\) be as in (2). For all \(0\le t\le 1\), there holds
$$\begin{aligned} (P_t^-\oplus P_t^0)\psi _t=P_t^-(\mathbf{P}_t\psi )+P_t^0(\mathbf{P}_t\psi )+K(t,\psi ), \end{aligned}$$
where \(\mathbf{P}_t\psi (x)=(\mathbf{1}\otimes \mathbf{P}_t(x)\psi (x))\) and \(\mathbf{P}_t(x):T_{\phi (x)}N\rightarrow T_{\phi _t(x)}N\) is the parallel transport along the curve \([0,1]\ni s\mapsto \phi _s(x)\in N\), \(P_t^-\) and \(P_t^0\) are the spectral projection of \(H^{1/2}(M,\Sigma M\otimes \phi _t^*TN)\) onto the negative and the null eigenspaces of 
, respectively. \(K:[0,1]\times H^{1/2}(M,\Sigma M\otimes \phi ^*TN)\rightarrow ^{1/2}(M,\Sigma M\otimes \underline{{\mathbb {R}}}^L)\) is a compact map.
Based on this deformation class, we can talk about the linking relationship between two subsets. As we stated in the beginning of this section, this relationship keeps those two subsets intersecting along the deformation.
Definition 4.10
Let \(S\subset \mathcal {F}\) be a closed subset and \(Q\subset \mathcal {F}\) a submanifold with relative boundary \(\partial Q\). We say that S and Q link with respect to \({\mathcal {C}}\) if the following holds: For any \(\Theta \in {\mathcal {C}}\) which satisfies \(\Theta (t,\partial Q)\cap S=\emptyset \) for all \(t\in [0,1]\), we have \(\Theta (t,Q)\cap S\ne \emptyset \) for all \(t\in [0,1]\).
Now, we can investigate the linking relationship between two explicit subsets.
Lemma 4.11
For \(\phi _0\in W^{1,2\alpha }(M,N)\) and \(R_1,R_2,\rho >0\) with \(0<\rho <R_2\), we define
and
where \(\mathcal {F}=\mathcal {F}^-\oplus \mathcal {F}^0\oplus \mathcal {F}^+\) is the spectral decomposition of \(\mathcal {F}\) with respect to the operator \(\mathbf{D}\), i.e., the fiber over \(\phi \in W^{1,2\alpha }(M,N)\), \(\mathcal {F}_\phi {:}{=}H^{1/2}(M,\Sigma M\otimes \phi ^*TN)\) is decomposed as \(\mathcal {F}_\phi =\mathcal {F}_\phi ^-\oplus \mathcal {F}_\phi ^0\oplus \mathcal {F}_\phi ^+\), where \(\mathcal {F}_\phi ^-\), \(\mathcal {F}_\phi ^0\) and \(\mathcal {F}_\phi ^+\) are, respectively, the positive, the null, and the negative spaces with respect to the spectral decomposition of the operator 
and \(e^+\in \mathcal {F}_{\phi _0}^+\) is such that \(\Vert e^+\Vert _{1/2,2}=1\). Then \(S_\rho \) and \(Q_{R_1,R_2}\) link with respect to \({\mathcal {C}}\).
Last, we give an explicit deformation class, which is corresponding to the negative pseudo-gradient flow. Let \(Q\subset \mathcal {F}\) be a submanifold with relative boundary \(\partial Q\). For such Q, we define \(\Gamma (Q)\), a class of deformations of Q, as follows:
Definition 4.12
We define
- (1):
-
\(\gamma (0,(\phi ,\psi ))=(\phi ,\psi )\) for any\((\phi ,\psi )\in \mathcal {F}\).
- (2):
-
\(\gamma (t,(\phi ,\psi ))=(\phi ,\psi )\) for \(0\le t\le 1\) and \((\phi ,\psi )\in \partial Q\).
- (3):
-
\(\gamma (t,(\phi ,\psi ))\) is written as \(\gamma (t,(\phi ,\psi ))=(\phi _t,\psi _t)\), where \(\phi _t\in W^{1,2\alpha }(M,N)\), \(\psi \in H^{1/2}(M,\Sigma M\otimes \phi _t^*TN)\) and \(\phi :[0,1]\ni t\mapsto \phi _t\in W^{1,2\alpha }(M,N)\) is piecewise \(C^1\). Moreover, \(\psi _t\) is of the following form:
$$\begin{aligned} \psi _t=\mathbf{P}_t(L_{\phi _t}(\psi )+K(t,\psi )), \end{aligned}$$where \(\mathbf{P}_t\) is as in Definition 4.9, \(K:[0,1]\times H^{1/2}(M,\Sigma M\otimes \phi ^*TN)\rightarrow H^{1/2}(M,\Sigma M\otimes \phi ^*TN)\) is a compact map and \(L_{\phi _t}(\psi )=\exp (-\sigma (t;\phi ,\psi )L_{t,V})\psi \). Here
and \(\sigma :[0,1]\times \mathcal {F}\rightarrow {\mathbb {R}}\) is a continuous bounded function.
and By Lemma 4.11, we have the following intersection property:
Lemma 4.13
Let \(S\subset \mathcal {F}\) be a closed subset and Q a submanifold of \({\mathcal {F}}\) with the relative boundary \(\partial Q\). Assume that \(S\subset \mathcal {F}^+\) and S and Q link with respect to \({\mathcal {C}}\). Then for any \(\gamma \in \Gamma (Q)\), we have \(\gamma (Q)\cap S\ne \emptyset \).
As we said in the beginning of this subsection, we will apply Lemma 4.11 and Lemma 4.13 to \(S_{\theta ;\rho }\) and \(Q_{\theta ;R_1,R_2}\) to get \(c_\theta \ge b_{\theta ;\rho }\) in Sect. 6.
5 Uniqueness of \(\alpha \)-harmonic maps
In this section, we wil prove a result about the uniqueness of \(\alpha \)-harmonic maps. An analogous theorem can be found in [11] (see Theorem 9.7.2) for harmonic maps. The key is still the convexity of the \(\alpha \)-energy function. So we now derive the second variation formula for the \(\alpha \)-energy. Let \(f_{st}(x){:}{=}f(x,s,t)\) be a smooth family of maps from \(M\times (-\varepsilon , \varepsilon )\times (-\varepsilon , \varepsilon )\) to N. The \(\alpha \)-energy function is defined as
where \(df=\frac{\partial f}{\partial x^\alpha }dx^\alpha =\frac{\partial f^i}{\partial x^\alpha }dx^\alpha \otimes \frac{\partial }{\partial f^i}\). In the following, \(\nabla \) is the Levi-Civita connection in the pullback bundle \(f^*TN\) and everything will be evaluated at \(s=t=0\). Then
Let us set \(V{:}{=}\left. \frac{\partial f}{\partial s}\right| _{s=t=0}\) and \(W{:}{=}\left. \frac{\partial f}{\partial t}\right| _{s=t=0}\). Then (5.2) becomes
Here, by the Ricci identity, we have
where we used the definition of the Riemann curvature tensor \(R(X,Y)Z=\nabla _X\nabla _YZ-\nabla _Y\nabla _XZ-\nabla _{[X,Y]}Z\). Plugging (5.4) into (5.3), we obtain the following second variation formula.
Theorem 5.1
For a smooth family \(f=f_{st}\) of finite \(\alpha \)-energy maps between manifolds M and N with \(f_{st}(x)=f_{00}(x)\) for any point \(x\in \partial M\) in the case of \(\partial M\ne \emptyset \), the second variational formula with \(V{:}{=}\left. \frac{\partial f}{\partial s}\right| _{s=t=0}, W{:}{=}\left. \frac{\partial f}{\partial t}\right| _{s=t=0}\) is
Since the Euler-Lagrangian equation for a \(\alpha \)-harmonic map is
under the assumptions of Theorem 5.1, we can deal with the last term in (5.5) as follows
Therefore, plugging (5.7) into (5.5), we have
Corollary 5.2
All \(\alpha \)-harmonic maps to non-positive curved manifolds are stable.
When the target manifold has non-positive curvature, any two homotopic maps can be connected by geodesics. In this case, we also have the convexity of the \(\alpha \)-energy.
Lemma 5.3
Under the assumptions of Theorem 5.1with \(f=f_t\), for \(\alpha \ge 1\), if N has non-positive sectional curvature and
then the \(\alpha \)-energy function is convex, that is,
By such convexity, one can get the following uniqueness of \(\alpha \)-harmonic map as in Theorem 9.7.2 [11].
Theorem 5.4
Let M be a compact manifold and N a complete Rienmannian manifold. Assume that N has non-positive curvature. Let \(f_0, f_1: M\rightarrow N\) be homotopic \(\alpha \)-harmonic maps (\(\alpha \ge 1\)). Then there exists a family \(f_t: M\rightarrow N\) (\(t\in [0,1]\)) of \(\alpha \)-harmonic maps connecting them, for which the \(\alpha \)-energy \(E^\alpha (f_t)\) is independent of t, and for which every curve \(\gamma _x(t){:}{=}f_t(x)\) is geodesic, and \(\Vert \frac{\partial }{\partial t}\gamma _x(t)\Vert \) is independent of x and t. If N has negative curvature, then \(f_0\) and \(f_1\) either are both constant maps, or they both map M onto the same closed geodesic or they coincide. If M has boundary \(\partial M\) and \(f_0|_{\partial M}=f_1|_{\partial M}\), then \(f_0=f_1\).
6 Existence results
In this section, we will prove the existence of perturbed \(\alpha \)-Dirac-harmonic maps. To this end, in addition to the results showed in the previous sections, we still need some more preparations.
Let N be a compact Riemannian manifold and \(\theta \in [M,N]\) be a free homotopy class of maps in N. We define \(\mathcal {F}_\theta =\{(\phi ,\psi )\in \mathcal {F}^{\alpha ,1/2}(M,N):\phi \in \theta \}\). For each \(\alpha \), it is well-known that there exists an \(\alpha \)-energy minimizing map \(\phi _0\in \theta \) in the class \(\theta \) [19]. From now on, we fix \(\alpha >1\). Denote
Now, we fix such a map \(\phi _0\in \theta \) in the following. For \(R_1,R_2>0\) and \(0<\rho <R_2\), we define
and
where \(e^+\in \mathcal {F}^+_{\theta ,\phi _0}\) is such that \(\Vert e^+\Vert _{1/2,2}=1\) and \(\mathcal {F}_{\theta ,\phi }=\mathcal {F}^-_{\phi }=H^{1/2}(M,\Sigma M\otimes \phi ^*TN)\) is the fiber over \(\phi \) of the fibration \(\mathcal {F}_{\theta }\rightarrow W^{1,2\alpha }(M,N)\cap \theta \). \(\mathcal {F}_{\theta }^\pm ,\mathcal {F}_{\theta }^0\) are similar to the ones in Lemma 4.11. We also define
and
In view of the linking geometry proved in Sect. 4, giving good estimates for \(a_{\theta ;R_1,R_2}\) and \(b_{\theta ;\rho }\) is crucial to both the existence and the nontrivialness of critical points of \(\mathcal {L}^\alpha \). So, we will prove these estimates in the next two subsections.
6.1 Estimate of \(a_{\theta ;R_1,R_2}\)
In this subsection, we shall prove the following estimate.
Lemma 6.1
There exists \(R_1,R_2>0\) such that \(a_{\theta ;R_1,R_2}\le m_\theta \) provided F satisfies \((\mathrm F2)\) and
\((\mathrm F4)\)\(F(\phi ,\psi )\ge 0\) for any \((\phi ,\psi )\in \mathcal {F}\).
Proof
Write \(\psi =\psi ^-+\psi ^0+re^+\), we have
where 
.
By (F2), we get
On the other hand, Hölder’s inequality implies
Combining these, we obtain
Since the square of the \(H^{1/2}\)-norm of \(\psi ^-\in \mathcal {F}_{\phi _0}^-\) is equivalent to 
, there exists \(C(\phi _0)>0\) such that
Plugging (6.10) into (6.9) and noting that the \(L^2\)-norm is equivalent to the \(H^{1/2}\)-norm on \(\mathcal {F}^0_{\theta ,\phi _0}\), one gets
We choose \(R_2>0\) such that \(C(e^+)r^2-Cr^\mu +C\le 0\) for \(r\ge R_2\). This is possible because \(\mu >2\). We then set \(M_1=\max \limits _{0\le r\le R_2}(C(e^+)r^2-Cr^\mu +C)>0\) and take \(R_1\) such that \(C(\phi _0)\Vert \psi ^-\Vert ^2_{1/2,2}+C\Vert \psi ^0\Vert ^\mu _{1/2,2}\ge M_1\) whenever \(\Vert \psi ^-+\psi ^0\Vert ^2_{1/2,2}=\Vert \psi ^-\Vert ^2_{1/2,2}+\Vert \psi ^0\Vert ^2_{1/2,2}\ge R_1^2\).
For such \(R_1\) and \(R_2\), the lemma holds. Indeed, any point in \(\partial Q_{R_1,R_2}\) is in one of the following three subsets:
-
(1)
\(\Vert \psi ^-+\psi ^0\Vert _{1/2,2}=R_1\) and \(0\le r\le R_2\);
-
(2)
\(\Vert \psi ^-+\psi ^0\Vert _{1/2,2}\le R_1\) and \(r=0\);
-
(3)
\(\Vert \psi ^-+\psi ^0\Vert _{1/2,2}\le R_1\) and \(r=R_2\).
For (1) and (3), by (6.11), the choices of \(R_1\) and \(R_2\) imply \(\mathcal {L}^\alpha (\phi _0,\psi )\le m_\theta \).
For (2), by (F4), we have
This case analysis completes the proof.
\(\square \)
6.2 Estimate of \(b_{\theta ,\rho }\)
In this subsection, we will prove the following:
Lemma 6.2
For \(R_2\) in Lemma 6.1, there exists \(0<\rho <R_2\) such that
provided F satisfies \((\mathrm F1),(\mathrm F4)\) and
\((\mathrm F5)\)\(F(\phi ,\psi )=o(|\psi |^2)\) uniformly in \(\phi \in N\) as \(|\psi |\rightarrow 0\).
The proof requires some preparations. For \((\phi ,\psi )\in S_{\theta ,\rho }\), we have
We first estimate the term 
. For this, we define
Thus,
for \((\phi ,\psi )\in S_{\theta ,\rho }\).
First of all, we investigate some properties of \(\lambda ^+(\phi )\).
Lemma 6.3
For any \(\phi \in \theta \cap W^{1,2\alpha }(M,N)\), we have \(\lambda ^+(\phi )>0\).
Proof
By definition of \(\lambda ^+(\phi )\), \(\lambda ^+(\phi )\ge 0\) for any \(\phi \in \theta \cap W^{1,2\alpha }(M,N)\). To prove the lemma, we assume on the contrary that \(\lambda ^+(\phi )=0\) for some \(\phi \in \theta \cap W^{1,2\alpha }(M,N)\). In other words, there exist \(\psi _n\in \mathcal {F}^+_{\phi }\) such that
as \(n\rightarrow \infty \).
We assume that (taking a subsequence if necessary) there exists \(\psi _\infty \in \mathcal {F}^+_\phi \) such that \(\psi _n\rightharpoonup \psi _\infty \) weakly in \(H^{1/2}\) and \(\psi _n\rightharpoonup \psi _\infty \) strongly in \(L^2\) as \(n\rightarrow \infty \).
As before, the square root of 
defines a norm on \(\mathcal {F}^+_\phi \) which is equivalent to the \(H^{1/2}\)-norm. By the weak lower semi-continuity of the norm, we get
By (6.17) and (6.18) and 
(since \(\psi _\infty \in \mathcal {F}^+_\phi \)), we obtain 
. This implies \(\psi _\infty =0\). Then there is a constant C such that
as \(n\rightarrow \infty \). This is a contradiction. So we complete the proof.
\(\square \)
We next prove:
Lemma 6.4
The map \(W^{1,2\alpha }(M,N)\ni \phi \mapsto \lambda ^+(\phi )\in {\mathbb {R}}\) is continuous.
Proof
Let \(\phi \in W^{1,2\alpha }(M,N)\). By the definition of \(\lambda ^+(\phi )\), for any \(\varepsilon >0\), there exists \(\psi _\varepsilon \in \mathcal {F}^+_\phi \) such that
Let \({\tilde{\phi }}\in W^{1,2\alpha }\cap L^\infty (M,N)\) be another map such that \(\Vert \phi -{\tilde{\phi }}\Vert _{W^{1,2\alpha }\cap L^\infty }\) is small. Because \(\psi _\varepsilon \notin \mathcal {F}^+_{{\tilde{\phi }}}\) in general, we cannot use \(\psi _\varepsilon \) as a test spinor to estimate \(\lambda ^+({\tilde{\phi }})\). In order to get a suitable test spinor, we need parallel translation. Denote by \(\iota (N)>0\) the injectivity radius of N. For any \(y,z\in N\) with \(d(y,z)<\iota (N)\) (d(y, z) is the geodesic distance between y and z in N), \(\mathbf{P}_{y,z}\) defined as in Lemma 4.5 depends smoothly on y, z. By Lemma 4.5, \(T_{\phi ,{\tilde{\phi }}}\psi _\varepsilon \in \mathcal {F}_{{\tilde{\phi }}}\). To make it belong to \(\mathcal {F}^+_{{\tilde{\phi }}}\), we need to modify it.
We define \({\tilde{\psi }}_\varepsilon {:}{=}P^+({\tilde{\phi }})T_{\phi ,{\tilde{\phi }}}\psi _\varepsilon \in \mathcal {F}^+_{{\tilde{\phi }}}\). By Lemma 4.6, we have
This and (6.20) give us
where \(\langle \cdot ,\cdot \rangle \) is the metric on \(\Sigma M\otimes \underline{{\mathbb {R}}}^L\).
Since \(\varepsilon >0\) is arbitrary, we obtain
Reversing the roles of \(\phi \) and \({\tilde{\phi }}\) in (6.23), we arrive at
This completes the proof.
\(\square \)
Let \(\mathcal {M}^\alpha (\theta )=\{\phi \in \theta \cap W^{1,2\alpha }(M,N):E^\alpha (\phi )=m_\theta \}\) be the set of energy minimizing maps in the class \(\theta \), where \(E^\alpha \) is defined in (3.24). Because the \(\alpha \)-energy of \(\phi \) satisfies the Palais–Smale condition for \(\alpha >1\) [22], we know
Lemma 6.5
\(\mathcal {M}^\alpha (\theta )\) is compact for \(\alpha >1\).
As a corollary of Lemmas 6.3, 6.4 and 6.5, we have:
Corollary 6.6
There exist \(\delta (\theta )\) and \(\lambda ^+(\theta )>0\) such that for any \(\phi \in \theta \cap W^{1,2\alpha }(M,N)\) with \(\mathrm{dist}(\phi ,\mathcal {M}^\alpha (\theta )){:}{=}\inf \{\Vert \phi -\varphi \Vert _{W^{1,2\alpha }}:\varphi \in \mathcal {M}^\alpha (\theta )\}<\delta (\theta )\), there holds \(\lambda ^+(\phi )\ge \lambda ^+(\theta )\).
Proof
Suppose the corollary is not true, then there exist \(\phi _n\in \theta \cap W^{1,2\alpha }(M,N)\) such that \(\lambda ^+(\phi _n)\rightarrow 0\) and \(\mathrm{dist}(\phi _n,\mathcal {M}^\alpha (\theta ))\rightarrow 0\). By the compactness of \(\mathcal {M}(\theta )\), after taking a subsequence if necessary, we assume that there is \(\phi _\infty \in \mathcal {M}(\theta )\) such that \(\phi \rightarrow \phi _\infty \) in \(W^{1,2\alpha }(M,N)\). By Lemmas 6.3 and 6.4, we have \(\lambda ^+(\phi _n)\rightarrow \lambda ^+(\phi _\infty )>0\), which is a contradiction.
\(\square \)
We also need to investigate the maps far away from the set \(\mathcal {M}^\alpha (\theta )\).
Lemma 6.7
Let \(\delta (\theta )>0\) be as in Corollary 6.6. There exists \(\varepsilon (\theta )\) such that for any \(\phi \in \theta \cap W^{1,2\alpha }(M,N)\) with \(\mathrm{dist}(\phi ,\mathcal {M}^\alpha (\theta ))\ge \delta (\theta )\), we have \(\frac{1}{2}\int _{M}(1+|d\phi |^2)^\alpha \ge m_\theta +\varepsilon (\theta )\).
Proof
For any fixed \(\alpha >1\), if such a \(\varepsilon (\theta )\) does not exist, then there exist \(\{\phi _n\}\) such that
and
Since \(m_\theta \) is the minimizing energy of \(E^\alpha \), \(\{\phi _n\}\) is a minimizing sequence for \(E^\alpha \). Therefore, the Palais–Smale condition implies that, after taking a subsequence if necessary, there is a critical point \(\phi _\infty \in \theta \cap W^{1,2\alpha }(M,N)\) of \(E^\alpha \) such that \(\phi _n\rightarrow \phi _\infty \) strongly in \(W^{1,2\alpha }(M,N)\). In fact, \(\phi _\infty \in \mathcal {M}^\alpha (\theta )\). This contradicts (6.25).
\(\square \)
Now, we can prove Lemma 6.2.
Proof of Lemma 6.2
Let \((\phi ,\psi )\in S_{\theta ,\rho }\) be arbitrary. If \(\phi \) satisfies \(\mathrm{dist}(\phi ,\mathcal {M}^\alpha (\theta ))<\delta (\theta )\), by Corollary 6.6, we have
If \(\phi \) satisfies \(\mathrm{dist}(\phi ,\mathcal {M}^\alpha (\theta ))\ge \delta (\theta )\), Lemma 6.7 implies
Therefore, we have
On the other hand, by \((\mathrm F1),(F4),(F5)\), for any \(\epsilon >0\) there exist \(C_\epsilon \) such that
for any \((\phi ,\psi )\in \mathcal {F}\).
Now take \(\epsilon =\frac{\lambda ^+(\theta )}{4}\) and write \(C=C_\epsilon \). By (6.29), (6.30) and Sobolev embedding, we obtain
There exists \(0<\rho _0<R_2\) such that for \(0<\rho <\rho _0\), we have
and
Thus, for any \(\rho \in (0,\rho _0)\), we have \(\mathcal {L}^\alpha (\phi ,\psi )\ge m_\theta +\frac{1}{2}\rho ^2\varepsilon (\theta )\). This implies
Hence, we complete the proof.
\(\square \)
Remark 6.8
It follows from the proof above that the estimate (6.13) is uniform in k if we replace F by \(\frac{1}{k}F\) instead.
6.3 Critical value of \(\mathcal {L}^\alpha \)
We define
where \(\Gamma (Q_{\theta ;R_1,R_2})\) is defined in Definition 4.12. By Lemma 4.7, one can prove that \(\Gamma (Q_{\theta ;R_1,R_2})\) is closed under composition as in [8]. Once having this property, we obtain the following main existence result in this paper:
Theorem 6.9
Let M be a closed surface and N a compact manifold. Suppose F satisfies \((\mathrm F1)-(F5)\) with \(\frac{4\alpha }{3\alpha -2}\le \mu \le p\le \frac{3}{4}\mu +1\) for \(\alpha \in (1,2]\). Let \(R_1,R_2\) and \(\rho \) be as in Lemmas 6.1 and 6.2. Then we have \(m_\theta<c_\theta <\infty \) and \(c_\theta \) is a critical value of \(\mathcal {L}^\alpha \) in \(\mathcal {F}\).
Proof
By Lemmas 4.11 and 4.13, we know \(\gamma (Q_{\theta ;R_1,R_2})\cap S_{\theta ;\rho }\ne \emptyset \) for any \(\gamma \in \Gamma (Q_{\theta ;R_1,R_2})\). Therefore, Lemma 6.2 implies
On the other hand, since the identity map belongs to \(\Gamma (Q_{\theta ;R_1,R_2})\) and \(Q_{\theta ;R_1,R_2}\) is bounded, by Sobolev embedding, we get
It remains to show that \(c_\theta \) is a critical value of \(\mathcal {F}^\alpha \). Suppose this is not the case, we set \({\bar{\varepsilon }}=\frac{b_{\theta ;\rho }-m_{\theta }}{2}>0\). By integrating (4.7)(see Deformation Lemma in [18, 20]), we can find \(0<\varepsilon <{\bar{\varepsilon }}\) and \(\Phi \in \Gamma (Q_{\theta ;R_1,R_2})\) such that
where \(\mathcal {L}^\alpha _{c}=\{(\phi ,\psi )\in \mathcal {F}_\theta :\mathcal {L}^\alpha (\phi ,\psi )<c\}\). Here, note that our negative pseudo-gradient flow belongs to \(\Gamma (Q_{\theta ;R_1,R_2})\) by (4.18). Also, observing that \(a_{\theta ;R_1,R_2}\le m_{\theta }<c_{\theta }-{\bar{\varepsilon }}\), we have \(\Phi (1,(\phi ,\psi ))=(\phi ,\psi )\) for any \((\phi ,\psi )\in \partial Q_{\theta ;R_1,R_2}\).
Choose \(\gamma \in \Gamma (Q_{\theta ;R_1,R_2})\) such that \(\mathcal {L}^\alpha (\gamma (1,Q_{\theta ;R_1,R_2}))<c_{\theta }+\varepsilon \). Since \(\Phi \circ \gamma \in \Gamma (Q_{\theta ;R_1,R_2})\) as we said before the theorem, we have the following contradiction
Thus we complete the proof.
\(\square \)
Theorem 6.9 gives us a solution to the perturbed \(\alpha \)-Dirac-harmonic map Eqs. (2.11) and (2.12). Let us denote this solution by \((\phi _\theta ,\psi _\theta )\). From (F5), we know \(F(\phi ,0)=0\) and \(F_\psi (\phi ,0)=0\) for any \(\phi \in N\). Therefore, we cannot exclude the possibility that the solution in Theorem 6.9 is a trivial solution, that is, \(\psi _\theta =0\). If \(\psi _\theta =0\), then \(F_\phi (\phi _\theta ,0)=0\) by (F5), which tells us that \(\phi _\theta \) is an \(\alpha \)-harmonic map. On the other hand, \(\mathcal {L}^\alpha (\phi _\theta ,\psi _\theta )=c_\theta >m_\theta \). Therefore, the \(\alpha \)-energy of \(\phi _\theta \) is \(\frac{1}{2}\int _M(1+|d\phi _\theta |^2)^\alpha =c_\theta >m_\theta \). So, we get a corollary:
Corollary 6.10
Under the assumption of Theorem 6.9, we get a critical point \((\phi _\theta ,\psi _\theta )\) with value \(c_\theta \). Then one of the following holds:
(1) There exists a non-trivial perturbed \(\alpha \)-Dirac-harmonic map \((\phi _\theta ,\psi _\theta )\) on N; or
(2) There exists an energy non-minimizing \(\alpha \)-harmonic map \(\phi _\theta \) in \(\theta \).
In particular, when N has non-positive curvature, Theorem 5.4 excludes the possibility (2) in the corollary above. Finally, we obtain an existence result about nontrivial perturbed \(\alpha \)-Dirac-harmonic maps.
Theorem 6.11
Let M be a closed surface and N a compact Rienmannian manifold with non-positive curvature. Suppose F satisfies \((\mathrm F1)-(F5)\) with \(\frac{4\alpha }{3\alpha -2}\le \mu \le p\le \frac{3}{4}\mu +1\) for \(\alpha \in (1,2]\). Then for any homotopy class \(\theta \in [M,N]\), there exists a non-trivial solution \((\phi ,\psi )\in W^{1,2\alpha }(M,N)\times H^{1/2}(M,\Sigma M\otimes \phi ^*TN)\) to the perturbed \(\alpha \)-Dirac-harmonic map equations (2.11) and (2.12) with \(\phi \in \theta \).
7 Regularity theorem
In this section, we are going to prove a regularity theorem for weakly perturbed \(\alpha \)-Dirac-harmonic maps with \(\alpha \in (1,2]\). Precisely, we want to prove:
Theorem 7.1
Suppose \(F\in C^\infty \) satisfies \((\mathrm F1)\) and \((\mathrm F3)\) for some \(p\le 2+2/\alpha \) and \(q\ge 0\). Then any weakly perturbed \(\alpha \)-Dirac-harmonic map is smooth.
Since the smoothness is a local property, we will prove Theorem 7.1 locally. So let we fix a domain \(\Omega \subset M\), which is mapped into a local chart \(\{y^i\}_{i=1,\dots ,n}\) on N. Then a weakly perturbed \(\alpha \)-Dirac-harmonic map \((\phi ,\psi )\) is a weak solution to the following system:
As we said in the introduction, we have to control the \(L^\infty \)-norm of \(\psi \) first. Since \((\phi ,\psi )\in \mathcal {F}^{\alpha ,1/2}\), by the Sobolev embedding, we can prove that \(\psi \) actually belongs to \(W^{1,s}\) for any \(2<s<2\alpha \).
Lemma 7.2
Let \((\phi ,\psi )\in \mathcal {F}^{\alpha ,1/2}\) be a weak solution of (7.1) and (7.2). Suppose F satisfies \((\mathrm F1)\) for some \(p\le 2+2/\alpha \). Then \(\psi \in C^{0,\gamma }(M,\Sigma M\otimes \phi ^*TN)\) for any \(0<\gamma <1-\frac{1}{\alpha }\).
Proof
Since M is closed, it is sufficient to prove the interior estimate. That is, we can apply the equation (7.2) and the elliptic estimate for the first order equation (see [4]) to the spinor multiplied by a cut-off function. For simplicity, we still use \(\psi \) instead. The Hölder inequality implies
for any \(a,s_0>1\). We set
Then \(4/3<s_0=\frac{4\alpha }{\alpha +2}\le 2\). By (F1), to make \(F_\psi \in L^{s_0}\), we need \(p\le 2+2/\alpha \). If \(s_0=2\), that is, \(\alpha =2\) and \(p\le 3\), then the elliptic estimate for the first order equation (see [4]) implies \(\psi \in W^{1,2}\cap L^r\) for any \(r>1\). If \(s_0<2\), then, by Sobolev embedding and (7.2), we have \(\psi \in L^{r_0}\), where \(r_0=\frac{2s_0}{2-s_0}=\frac{4\alpha }{2-\alpha }>4\). Again, by setting
we have \(s_1=\frac{2\alpha r_0}{2\alpha +r_0}\). To make \(F_\psi \in L^{s_1}\), it is sufficient to set \(\frac{\alpha +2}{\alpha }s_1\le r_0\), that is, \(s_1\le \frac{\alpha r_0}{\alpha +2}\). Since \(r_0>4\), \(\frac{2\alpha r_0}{2\alpha +r_0}<\frac{\alpha r_0}{\alpha +2}\). Now, we take \(s_1=\frac{2\alpha r_0}{2\alpha +r_0}\). Then \(s_1>s_0\) and
If \(r_0=\frac{4\alpha }{2-\alpha }\ge \frac{2\alpha }{\alpha -1}\), that is, \(\alpha \ge 4/3\), we have \(s_1\ge 2\). Then \(\psi \in W^{1,2}\cap L^r\) for any \(r>1\). Otherwise, for \(\alpha <4/3\), we get \(s_1<2\). By Sobolev embedding, we have
Again, if \(r_1\ge \frac{2\alpha }{\alpha -1}\), that is, \(\alpha \ge 6/5\), we have \(s_2=\frac{2\alpha r_1}{2\alpha +r_1}\ge 2\). Then \(\psi \in W^{1,2}\cap L^r\) for any \(r>1\). Otherwise, we get \(r_2=\frac{2s_2}{2-s_2}\). Repeating this procedure, for each \(\alpha \), after finitely many steps, we obtain \(r_i\ge \frac{2\alpha }{\alpha -1}\). Then \(\psi \in W^{1,2}\cap L^r\) for any \(r>1\). By Hölder’s inequality, \(\psi \in W^{1,s}\) for any \(2<s<2\alpha \). Then \(\psi \in C^{0,\gamma }\) for any \(0<\gamma<1-\frac{2}{s}<1-\frac{1}{\alpha }\). Thus, we complete the proof.
\(\square \)
Now, following the idea in [3], we need two lemmas. One is
Lemma 7.3
Let \((\phi ,\psi )\) be a weak solution of (7.1) and (7.2). Suppose F satisfies \((\mathrm F3)\) for some \(q\ge 0\). For any \(\varepsilon >0\), there is a \(\rho >0\) such that
where \(B(x_1,\rho )\subset \Omega \), \(\eta \in W_0^{1,2\alpha }(B(x_1,\rho ,{\mathbb {R}})\) and C only depends on N and the constant in \((\mathrm F3)\).
Proof
Denote \(G=(G^1,\cdots ,G^n)\), where
Then the weak form of (7.1) is
for any \(\zeta \in W^{1,2\alpha }(\Omega ,{\mathbb {R}}^L)\). Now, we choose \(\zeta (x)=(\phi (x)-\phi (x_1))\eta ^2\), then
Plugging this into (7.10), one gets
Now, we estimate the left-hand side of (7.12).
where we have used (F3). For the second term in the right-hand side of (7.12), we have
By estimates (7.13), (7.14) and (7.12), we have
Since \(\phi \) is continuous, we can choose \(\rho \) so small that \(C_{N}\sup \limits _{B(x_1,\rho )}|\phi (x)-\phi (x_1)|<\frac{1}{4}\). Then
Now, take \(\rho \) still smaller so that \(\sup \limits _{B(x_1,\rho )}|\phi (x)-\phi (x_1)|\max \{8,\Vert \phi \Vert _{1,2}\}\le \varepsilon \). Then
where C only depends on N and the constant in (F3).
\(\square \)
The other lemma is
Lemma 7.4
Let \(\phi \in W^{1,2\alpha +2}\cap W^{3,2\alpha }(B(x_0,R),N)\) and \((\phi ,\psi )\) be a weak solution of (7.1) and (7.2). Suppose \(F\in C^\infty \) satisfies \((\mathrm{F}1)\) and \((\mathrm{F}3)\) for some \(p\le 2+2/\alpha \) and \(q\ge 0\). Then for R small enough, we have
where \(C_1>0\) is a constant depending on \(|\phi |_{C^0(M,N)}\), R, \(\Vert \psi \Vert _{L^\infty (M,\Sigma M\otimes \phi ^*TN)}\) and the constants in \((\mathrm{F}1)\) and \((\mathrm{F}3)\).
Proof
Choose a local coordinate \(\{y^i\}\) in a neighborhood U of \(\phi (x_0)\) such that \(\Gamma ^i_{jk}(\phi (x_0))=0\). Let R be so small that \(B{:}{=}B(x_0,R)\) is mapped into U by \(\phi \). Since \(\phi \) is continuous, we can choose R small enough such that \(|\phi (x)-\phi (x_0)|<\delta \) for a given \(\delta \) to be determined later. Then (7.2) implies
Noting that 
, we get
where we have replaced the constant C in (F1) by \(C_1\), and \(C(C_1,\Vert \psi \Vert _{L^\infty })\) denotes a constant depending on \(C_1\) and \(\Vert \psi \Vert _{L^\infty (B)}\).
By the elliptic estimate for the first order equation (see [4]), we have
Now, choose R and \(C_0\) so small that \(C_RC_N\delta \Vert d\phi \Vert _{L^2(B)}<\frac{1}{2}\). We obtain from (7.20) and (7.21) that
which implies
For \(\zeta \in W_0^{1,2\alpha }(B,{\mathbb {R}}^L)\), we have
Choosing \(\zeta =g^{\sigma \tau }\nabla _\sigma (\xi ^2\nabla _\tau \phi )\), where \(\xi \in C_0^\infty (B,{\mathbb {R}})\) will be determined later, we get
Note that
We control the last three terms as follows:
Plugging these estimates into (7.26), we have
For the integrands in the left-hand side of (7.25), we have the estimates
and
where we have replaced the constant C in (F3) by \(C_3\) and take \(C_N>1\).
By (7.30), (7.31) and (7.32), (7.25) becomes
For small \(\varepsilon _1>0\), we have
and
Choosing \(\eta =|d\phi |\xi \) in (7.23), we obtain
Now, we control the remaining terms in the right-hand side of (7.33) as follows:
where we denote \((C(C_1,\Vert \psi \Vert _{L^\infty }))^2\) by \(C^2(C_1,\Vert \psi \Vert _{L^\infty })\). By
we have
and
where we have used \(0\le \xi \le 1\). Plugging these estimates into (7.33) and choosing \(\varepsilon _1\) and R so small that
we get
Therefore, we obtain
Now, for \(\varepsilon >0\), let \(\rho >0\) be as in Lemma 7.3, and choose a cut-off function \(\xi \in C^\infty _0(B(x_1,\rho ))\subset B(x_0,R)\) such that
Denoting \(B_\rho {:}{=}B(x_1,\rho )\) for simplicity, we get
It follows from (7.36) and (7.38) that
Therefore,
where the constants C are different between lines as before. Substituting (7.48) into (7.45), we have
hence,
Covering \(B(x_0,\frac{R}{2})\) with \(B(x_1,\frac{\rho }{2})\) and using (7.50) we obtain (7.18).
\(\square \)
Now we can prove the main theorem in this section.
Proof of Theorem 7.1
First we show that \(\phi \in W^{2,2}\cap W^{1,4}(B(x_0,\frac{R}{2}),N)\). This can be done just by replacing weak derivatives by difference quotients in the proof of Lemma 7.4. Denote
where \((E_1,\cdots ,E_L)\) is an orthonormal basis of \({\mathbb {R}}^L\) and \(h\in {\mathbb {R}}\). \(\Delta ^h=(\delta ^h_1,\cdots ,\delta ^h_L)\). Similar to (7.45), we have
Using
and applying the Lemma 7.3 to the second term in the right-hand side of (7.52), we obtain
Then by Lemma A.2.2 in [11] implies the weak derivative \(\nabla ^2\phi \) exists and (7.18) still holds.
Since \(\phi \in W^{2,2}\), we have \(\phi \in W^{1,p}\) for any \(p>0\). Together with the Lemma 7.2 and the equation (7.1), we know \(\phi \in W^{2,p}\) for any \(p>0\). Thus, \(\phi \in C^{1,\gamma }\). By the elliptic estimates for the Eq. (7.2), we have \(\psi \in C^{1,\gamma }\). Then the standard arguments yield that both \(\phi \) and \(\psi \) are smooth. This completes the proof.
\(\square \)
8 \(\alpha \)-Dirac-harmonic maps
In this section, we want to approximate the \(\alpha \)-Dirac-harmonic maps by the perturbed \(\alpha \)-Dirac-harmonic maps. Precisely, by Theorem 6.11 and Theorem 7.1, we have a sequence of perturbed \(\alpha \)-Dirac-harmonic maps \((\phi _k,\psi _k)\), which are the critical points of the functionals
where F satisfies the assumptions in Theorem 6.11 and Theorem 7.1. If \((\phi _n,\psi _n)\) converges to \((\phi ,\psi )\) smoothly, then we get the existence of nontrivial \(\alpha \)-Dirac-harmonic maps. We now come to the \(\varepsilon \)-regularity estimate which gives a uniform control on the sequence in Sobolev space.
Theorem 8.1
Suppose F satisfies
- \((\mathrm F6)\) :
-
\(|F_\psi (\phi ,\psi )|\le C|\psi |^{r-1} \ \text {for} \ 3<r\le 2+2/\alpha ,\)
- \((\mathrm F7)\) :
-
\(|F_\phi (\phi ,\psi )|\le C|\psi |^{q} \ \text {for} \ q\ge 0.\)
There is \(\varepsilon _0>0\) and \(\alpha _0>0\) such that if \((\phi ,\psi ):(D,g_{\beta \gamma })\rightarrow (N,g_{ij})\) is a smooth perturbed \(\alpha \)-Dirac-harmonic map satisfying
for \(1\le \alpha <\alpha _0\), then we have
and
for any \({\tilde{D}}\subset D\) and \(p>1\), where \(C(D,N,\Lambda ,p)\) denotes a constant depending on \(D,N,\Lambda \) and p.
Note that (8.4) is a consequence of the proof of Lemma 7.2 and (F6). This is different from the proof in [3]. The proof of Theorem 8.1 is based on the following lemma, which is a generalization of Lemma 6.1.2 in [15].
Lemma 8.2
There are \(\varepsilon _0>0\) and \(\alpha _0>0\) such that if \((\phi ,\psi )\) and F satisfy (8.2) and \((\mathrm F6)\), \((\mathrm F7)\), respectively, then
Proof
Choose a cut-off function \(\eta \in [0,1]\) with \(\eta |_{D_1}\equiv 1\) and \(\mathrm{Supp}(\eta )\subset D\). As before, \((\phi ,\psi )\) locally satisfies the system
(8.6) implies
Then
Since
and
we have
Therefore,
Thus, for any \(p>1\), we get
Without loss of generality, we assume \(\int _D\phi =0\) so that \(\Vert \phi \Vert _{D,1,p}\le C\Vert d\phi \Vert _{D,0,p}\). By taking \(p=4/3\) in the inequality above and Hölder’s inequality, we obtain
and
On the other hand, let c(p) be the operator norm of \(\Delta ^{-1}:L^p(D)\rightarrow W^{2,p}(D)\cap W^{1,2}_0(D)\). Then we get
Plugging the above estimates into (8.14), we have
Together with the Sobolev inequality \(\Vert \phi \eta \Vert _{D,1,4}\le C_1\Vert \phi \eta \Vert _{D,2,4/3}\), this implies
Now we choose \(\alpha _0,\varepsilon _0>0\) such that
and
then we get (8.5). Thus we complete the proof.
\(\square \)
Proof of Theorem 8.1
Choose \(\tilde{D}\subset D_2\subset D_1\subset D\) and a cut-off function \(\eta \in [0,1]\) with \(\eta |_{D_2}\equiv 1\) and \(\mathrm{Supp}(\eta )\subset D_1\). Taking \(p=2\) in (8.14) on \(D_1\)(we temporarily assume \(\int _{D_1}\phi =0\)), we have
As in the proof of Lemma 8.2, we get
By Lemma 8.2, we obtain
The Sobolev inequality gives us
for any \(p>1\). This also holds for \(\phi \) without \(\int _{D_1}\phi =0\).
Now, since \(\phi \in W^{1,p}\) and \(\psi \in L^p\) for any \(p>1\), the estimates in the theorem follow from the standard \(L^p\)-estimate for the Dirac operator and the \(W^{2,p}\)-estimate for the Laplace operator immediately.
\(\square \)
With the \(\varepsilon \)-regularity in hand, one can easily prove the following theorem.
Theorem 8.3
Let \((\phi _k,\psi _k)\) be smooth critical points of the functional \(L^\alpha _k\) in (8.1) with uniformly bounded energy:
Suppose F satisfies \((\mathrm F6)\) and \((\mathrm F7)\). Then there are a subsequence, still denoted by \(\{(\phi _k,\psi _k)\}\), and a smooth \(\alpha \)-Dirac-harmonic map \((\phi ,\psi )\) such that
Proof
Define the energy concentration set to be
where \(\varepsilon _0\) is the positive constant in Theorem 8.1 and B(x, r) is the geodesic ball in M with center at x and radius r. Suppose S is not empty, let us say \(p\in S\). By the definition, we have
with \(x_k\rightarrow p\) and \(r_k\rightarrow 0\) as \(k\rightarrow \infty \). Let
Then (8.30) becomes
However,
as \(k\rightarrow \infty \). This contradicts (8.32). Thus, S is empty.
Now, for any point \(x\in M\), there exist \(r>0\) and a subsequence of \(k\rightarrow \infty \) such that
Then by the \(\varepsilon \)-regularity Theorem 8.1 and standard elliptic theory, we have
for any \(l>0\). Since \(\{(\phi _k,\psi _k)\}\) has uniformly bounded energy, up to a subsequence if necessary, \(\{(\phi _k,\psi _k)\}\) has a weak limit \((\phi ,\psi )\). Then the regularity Theorem 7.1 tells us that \((\phi ,\psi )\) is a smooth \(\alpha \)-Dirac-harmonic map, and (8.35) implies (8.28).
\(\square \)
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