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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 α-harmonic maps which for α > 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 α-harmonic maps when α decreases to 1. In general, there are bubbles (harmonic spheres) preventing the smooth convergence of α-harmonic maps. Fortunately, a nonpositive curva- ture 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 ψ 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 ,φ denotes the s-derivative of φ, and F is a nonlinear perturbation satisfying some growth and decay conditions with respect to ψ.
In this paper, we shall systematically study critical point theory according to Palais-Smale for α-Dirac harmonic maps for α > 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 α-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 M ⊗ T N → N and a function F : M ⊗ T N → R. A general element of M ⊗ T N is written as (φ, ψ), where φ ∈ N and ψ ∈ M ⊗ φ * T N, and we write F = F(φ, ψ). We shall first prove the Palais-Smale condition for the action functional L α of perturbed α-Dirac-harmonic maps from closed Riemann surfaces: 3) 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 α-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 α-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(φ, ψ) ∈ C 1,1 loc ( M ⊗ T N, R) and its partial derivatives (F1) There exist p ∈ (2, 4) and C > 0 such that for any (φ, ψ) ∈ F α,1/2 (M, N ). (F2) There exist μ > 2 and R 1 > 0 such that 0 < μF(φ, ψ) ≤ F ψ (φ, ψ), ψ for any (φ, ψ) ∈ F α,1/2 (M, N ) with |ψ| ≥ R 1 . (F3) There exist q < 4 and C > 0 such that for any (φ, ψ) ∈ F α,1/2 (M, N ). (F4) For any (φ, ψ) ∈ F , we have F(φ, ψ) ≥ 0.
Here θ is a given homotopy class of maps, m θ is the minimizing α-energy in θ , c θ is defined as: sup L α (γ (Q θ ;R 1 ,R 2 )), (1.4) and (Q θ ;R 1 ,R 2 ) is defined in Definition 4.12, Q θ ;R 1 ,R 2 is defined by (6.2). In particular, when N has nonpositive curvature, our solution is nontrivial. Since α > 1, φ in the theorem above is continuous. It is natural to expect the smoothness of the weak solutions. Due to the perturbation F, F ψ will produce ψ 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 ∞ -norm of ψ first. The same phenomenon happens in the proof of the ε-regularity. The following regularity theorem shows that such a nontrivial solution is actually smooth. Theorem 1.3 Suppose F ∈ C ∞ satisfies (F1) and (F3) for some p ≤ 2 + 2/α and q ≥ 0. Then any weakly perturbed α-Dirac-harmonic map is smooth. Now, by multiplying F(φ, ψ) by constants 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 = |ψ| 4α 3α−2 . Thus, we get a sequence of perturbed α-Dirac-harmonic maps {(φ k , ψ k )}, which are the critical points of L α k . Furthermore, if such sequence converges smoothly, then the limit will be a critical point of the functional (1.2), which is a α-Dirac-harmonic map. To do so, we need some uniform regularity estimates. Since the proof of Theorem 1.3 depends on the oscillation of φ, which may not be uniformly controlled for φ k , we want to prove another estimate which is called the ε-regularity estimate. This kind of estimate was introduced by Sacks and Uhlenbeck for the α-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 α-Dirac-harmonic maps(α → 1). We have to modify here the growth conditions (F1) and (F3) for the derivatives of F; we need and Since the convergence in the theorem above is smooth on M, the proof of Theorem 1.1 implies that the action functional of the α-Dirac-harmonic map (φ, ψ) is still strictly bigger than m θ . Then the convexity of the action functional tells us that (φ, ψ) is a coupled α-Diracharmonic map. Thus, we obtain: Corollary 1.6 Let M be a closed surface and N a compact Rienmannian manifold with nonpositive curvature. For the α 0 given in Theorem 1.4 and each α ∈ (1, α 0 ), if the sequence of perturbed α-Dirac-harmonic maps {(φ k , ψ k )} satisfies the uniform bounded energy condition (1.9), then there exists a coupled α-Dirac-harmonic map from M to N with φ in the given homotopy class θ .

Remark 1.7
To get the existence of a Dirac-harmonic map, it is natural to consider the convergence of a sequence of α-Dirac-harmonic maps as α 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 α-Dirac-harmonic maps. In Sect. 4, we construct a special pseudo-gradient vector field and deform our configuration space by a negative pseudogradient flow generated by it. Besides, we also recall some facts in linking geometry. In Sect. 5, we prove the uniqueness of α-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.

Euler-Lagrange equations
Let (M, g) be a compact surface with a fixed spin structure. On the spinor bundle M, we denote the Hermitian inner product by ·, · M . For any X ∈ (T M) and ξ ∈ ( M), the Clifford multiplication is skew-adjointness: Let ∇ be the Levi-Civita connection on (M, g). There is a connection (also denoted by ∇) on M compatible with ·, · M . Choosing a local orthonormal basis {e β } β=1,2 on M, the usual Dirac operator is defined as / ∂:=e β · ∇ β , where β = 1, 2 (here and in the sequel, we use the Einstein summation convention). One can find more about spin geometry in [14].
Let φ be a smooth map from M to a compact Riemannian manifold (N , h) of dimension n ≥ 2. Let φ * T N be the pull-back bundle of T N by φ and consider the twisted bundle M ⊗ φ * T N. On this bundle there is a metric ·, · M⊗φ * T N induced from the metric on M and φ * T N. Also, we have a connection∇ on this twisted bundle naturally induced from those on M and φ * T N. In local coordinates {y i } i=1,...,n , the section ψ of M ⊗ φ * T N is written as where each ψ i is a usual spinor on M. We also have following local expression of∇ where i jk are the Christoffel symbols of the Levi-Civita connection of N . The Dirac operator along the map φ is defined as which is self-adjoint [11]. Sometimes, we use / D φ to distinguish the Dirac operators defined on different maps. In [4], the authors introduced the functional where τ m (φ) is the m-th component of the tension field [11] of the map φ with respect to the coordinate on N , ∇φ l · ψ j denotes the Clifford multiplication of the vector field ∇φ l with the spinor ψ j , and R m li j 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 (φ, ψ) 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 Diracharmonic 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 ,φ denotes the s-derivative of φ, and F is a nonlinear interaction term satisfying some growth and decay conditions with respect to ψ. The Euler-Lagrange equations are: (2.9) In particular, Isobe proved that there exists a non-trivial solution (φ, ψ) to (2.9) with φ 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(φ) = M |dφ| 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 L α : (2.12)

Configuration space
We will define a configuration space for our functional L α . In fact, we focus on W 1,2αmaps(α > 1) and H 1/2 -spinors. By the Nash embedding theorem [16], we can embed N into Euclidean space R L for some large L. We define the W 1,2α -maps on N as where φ ∈ W 1,2α (M, R L ) means that both φ and its weak derivative ∇φ are in L 2α (M, R L ). By the Sobolev embedding theorem, any φ ∈ W 1,2α (M, N ) is continuous. Therefore, the pull-back bundle φ * T N is well-defined, and we can consider H 1/2 -spinors along φ ∈ W 1,2α (M, N ) defined as where R L = M × R L is the trivial R L -bundle over M and we regard T φ(x) N as a subset of R L for each x ∈ M by the above embedding. Consider the Banach Since F α,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 (·, ·) 2 is the L 2 -inner product on M and D:=/ ∂ ⊗ 1. With respect to this product structure, we can write where Now, we can define a configuration space F α,1/2 (M, N ) as which is a Banach submanifold of F α,1/2 with the tangent space at (φ, ψ) ∈ F α,1/2 (M, N ) being The space W 1,2α (M, N ) is a Banach manifold [22] whose tangent space at φ ∈ W 1,2α (M, N ) is So, to see (2.21), it suffices to show the vertical part, which is the same as in [8].

The Palais-Smale condition
In this section, we prove the Palais-Smale condition for L α for a certain class of nonlinearities F satisfying (F1)-(F3). From now on, we denote F α,1/2 (M, N ) by F for short. Let us recall the Palais-Smale condition: is a Palais-Smale sequence if the following are satisfied: (2) dL α (φ n , ψ n ) T * (φn ,ψn ) F → 0. We say L α satisfies the Palais-Smale condition on F if any Palais-Smale sequence has a convergent subsequence in F . By Sobolev embedding and conditions (F1) (F3), dL α (φ,ψ) is a bounded linear map on F . Thus, Zorn's proposition (see page 30 in [22]) implies L α is C 1 on F . Therefore, if L α satisfies the Palais-Smale condition on 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 L α 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. Proof We first prove that any Palais-Smale sequence is bounded.
Let {(φ n , ψ n )} ⊂ F be a Palais-Smale sequence. By the structure on F α,1/2 , we have 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 Integrating (F2), we know Consider the spectral decomposition with respect to the operator D:=/ ∂ ⊗ 1, where H − 0 , H 0 0 and H + 0 are the closures in H 1/2 (M, M ⊗ R L ) of the spaces spanned by the negative, the null and the positive eigenspinors of D, respectively. Denote by P − 0 : n,0 and ψ + n,0 being P − 0 ψ n , P 0 0 ψ n and P + 0 ψ n , respectively. The square of the Together with (2.2), this implies where we have used the Hölder inequality and the Sobolev embedding H 1/2 ⊂ 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 ≤ 3 4 μ + 1. Now, plugging (3.11) into (3.9), we have Since α ∈ (1, 2], the Hölder inequality and (3.6) imply Plugging (3.13) and (3.14) into (3.12), we get Since the usual Dirac operator has finite dimensional kernel, dim(H 0 0 ) < ∞. Noting that the H 1/2 -norm and the L 2 -norm are equivalent on H 0 0 ( [1]), we have Combining (3.15), (3.16) and (3.17), we obtain Next, we show {ψ n } has a convergent subsequence. As in [8], we write the vertical gradient of L α as where is bounded linear and Because both |ψ∇φ| and F ψ (φ, ψ) belong in L r with r > 4/3, both K V 1 and K V 2 are compact. We write ψ n = ψ − n,0 + ψ 0 where o(1) → 0 as n → ∞. From the boundedness of {(φ n , ψ n )} and the compactness of (L V ) −1 K on H ± 0 , we know ψ ± n,0 has a convergent subsequence. Again, since dimH 0 0 < ∞, ψ 0 n,0 also has a convergent subsequence. Thus, {ψ n } has a convergent subsequence. Last, for the convergence of {φ n }, we consider the α-energy functional Then the second derivative of e α at dφ with respect to the direction dϕ is (3.25) where a:=dφ i and b:=dφ j . Then (3.25) implies We also have Now, we can control dφ i − dφ j 2α as follows.
where the last inequality comes from the Sublemma 3.18 in [22] stated as follows: for any x, y ∈ V .
To estimate the left-hand side of (3.30), we decompose d H L α as By Sobolev embedding, The boundedness of {(φ n , ψ n )} and the compactness of K H imply that J (φ n ) has a convergent subsequence. In particular, for any ε, there exists an integer N ε such that Together this with (3.30) and where we have used the boundedness of {φ n }. Therefore, {dφ n } contains a Cauchy subsequence in L 2α . Hence, by Sobolev embedding, {φ n } has a convergent subsequence.

Negative pseudo-gradient flow and linking geometry
This section consists of two parts. In the first part, we will construct a special pseudogradient 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 α-Dirac-harmonic maps in Sect. 6.

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 ∇ V L α . 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 ≥ 1) Finsler manifold and f : M → R be a C 1 function. A vector X ∈ T p M is called a pseudo-gradient vector for f at p if X satisfies 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 ∈ M|d f p = 0}.
The main result in this section is Theorem 4.3 Suppose F ∈ C 1,1 loc in the fiber direction. Then there exists a locally Lipschitz pseudo-gradient vector field ω for L α onF of the form of ω = X ⊕ a∇ V L α for some vector field X and a constant a ∈ (1, 2).

Proof We divideF into two subsets
For any a ∈ (1, 2), a∇ V L α always satisfies (i) and (ii) in Definition 4.1 for d V L α . Our assumption implies that a∇ V L α is locally Lipschitz. Therefore, it suffices to show that there is a locally Lipschitz vector field that satisfies (i) and (ii) in Definition 4.
Then extend X φ to be a C 1 vector field in a neighborhood of (φ, ψ) (say by making it "constant" with respect to a chart at (φ, ψ)) [17]. Therefore, for each point For A, we just use the C 1 vector field ω = 0 ⊕ a∇ V L α . Then, for any point inF, we have Together with L α ∈ C 1 , this implies that holds in some neighborhood U of (φ, ψ). So, ω also satisfies (ii) in Definition 4.1. Thus, for each point (φ, ψ) ∈ A, there is a neighborhood U of (φ, ψ) such that ω = 0 ⊕ a∇ V L α is a pseudo-gradient vector field for L α 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. 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 η is chosen such that (4.7) and (4.8) have a global unique solution. In particular, N ). See Appendix A in [18]. Note that the solution ψ t to (4.7) belongs to H 1/2 (M, M ⊗ φ * t T N) for each t ≥ 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 ∈ M, we denote Thus, the vertical part of (4.7) is transformed to 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 F α,1/2 (M, N ). Due to their usefulness, we list them here and refer to [8] for the proofs.

Lemma 4.4
Let (φ t , ψ t ) and P t be as above. P t defines a bounded linear map P t = 1 ⊗ P t : which depends coutinuously on t with respect to the operator norm.

defines a continous family of compact operators with respect to the operator norm.
Note that the decomposition of ∇ V L α 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 ∈ [0, T ], we write (4.14) where / D φ T denotes the Dirac operator along the map φ T , P −1 t→T = 1 ⊗ P −1 t→T and P −1 t→T (x) : Then (4.10) can be written as where L T ,V :=P −1 is compact and continuously depends on t and T . So,K is also compact.
Regarding φ t andψ t in η t (ψ t ) andK (T , t;ψ t ) as already known functions, integration of of (4.15) yieldsψ Since t ∈ [0, T ] is arbitrary, we take T = t and obtaiñ is a compact map.
This fact will be used in the section below.

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 (Q θ ;R 1 ,R 2 ) is the suitable deformation class defined in Definition 4.12, Q θ ;R 1 ,R 2 is a subset of the configuration space. To get a nontrivial critical point, we want to show that c θ is strictly bigger than the α-energy minimizer m θ 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 (Q θ ;R 1 ,R 2 ) and S θ ;ρ link with respect to C. Then, the critical value is bigger or equal to the infimum b θ ;ρ defined in (6.5). Therefore, the nontrivialness of the critical point is equivalent to b θ ;ρ > m θ , 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. , (φ, ψ)) be as in (2). For all 0 ≤ t ≤ 1, there holds 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 ⊂ F be a closed subset and Q ⊂ F a submanifold with relative boundary ∂ Q. We say that S and Q link with respect to C if the following holds: For any

Definition 4.9 We define
Now, we can investigate the linking relationship between two explicit subsets.

Lemma 4.11 For
where F = F − ⊕ F 0 ⊕ F + is the spectral decomposition of F with respect to the operator D, i.e., the fiber over φ ∈ W 1,2α (M, N ), Last, we give an explicit deformation class, which is corresponding to the negative pseudogradient flow. Let Q ⊂ F be a submanifold with relative boundary ∂ Q. For such Q, we define (Q), a class of deformations of Q, as follows: Definition 4. 12 We define N ) is piecewise C 1 . Moreover, ψ t is of the following form: where P t is as in Definition 4.9, K : By Lemma 4.11, we have the following intersection property: Lemma 4.13 Let S ⊂ F be a closed subset and Q a submanifold of F with the relative boundary ∂ Q. Assume that S ⊂ F + and S and Q link with respect to C. Then for any γ ∈ (Q), we have γ (Q) ∩ S = ∅.
As we said in the beginning of this subsection, we will apply Lemma 4.11 and Lemma 4.13 to S θ ;ρ and Q θ ;R 1 ,R 2 to get c θ ≥ b θ ;ρ in Sect. 6.

Uniqueness of˛-harmonic maps
In this section, we wil prove a result about the uniqueness of α-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 α-energy function. So we now derive the second variation formula for the α-energy. Let f st (x):= f (x, s, t) be a smooth family of maps from M × (−ε, ε) × (−ε, ε) to N . The α-energy function is defined as In the following, ∇ is the Levi-Civita connection in the pullback bundle f * T N and everything will be evaluated at s = t = 0. Then . Then (5.2) becomes Here, by the Ricci identity, we have where we used the definition of the Riemann curvature tensor R(X , , Since the Euler-Lagrangian equation for a α-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 α-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 α-energy.

Lemma 5.3 Under the assumptions of Theorem 5.1 with f = f t , for α ≥ 1, if N has nonpositive sectional curvature and
then the α-energy function is convex, that is, By such convexity, one can get the following uniqueness of α-harmonic map as in Theorem 9.7.2 [11].

Existence results
In this section, we will prove the existence of perturbed α-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 θ ∈ [M, N ] be a free homotopy class of maps in N . We define F θ = {(φ, ψ) ∈ F α,1/2 (M, N ) : φ ∈ θ }. For each α, it is well-known that there exists an α-energy minimizing map φ 0 ∈ θ in the class θ [19]. From now on, we fix α > 1. Denote Now, we fix such a map φ 0 ∈ θ in the following. For R 1 , R 2 > 0 and 0 < ρ < R 2 , we define where e + ∈ F + θ,φ 0 is such that e + 1/2,2 = 1 and θ are similar to the ones in Lemma 4.11. We also define In view of the linking geometry proved in Sect. 4, giving good estimates for a θ ;R 1 ,R 2 and b θ ;ρ is crucial to both the existence and the nontrivialness of critical points of L α . So, we will prove these estimates in the next two subsections.

Estimate of a Â;R 1 ,R 2
In this subsection, we shall prove the following estimate. Proof Write ψ = ψ − + ψ 0 + re + , we have where C(e + ) = 1 2 M / D φ 0 e + , e + > 0. By (F2), we get On the other hand, Hölder's inequality implies Combining these, we obtain Since the square of the Plugging (6.10) into (6.9) and noting that the L 2 -norm is equivalent to the H 1/2 -norm on We choose R 2 > 0 such that C(e + )r 2 − Cr μ + C ≤ 0 for r ≥ R 2 . This is possible because μ > 2. We then set M 1 = max 0≤r ≤R 2 (C(e + )r 2 − Cr μ + C) > 0 and take R 1 such For such R 1 and R 2 , the lemma holds. Indeed, any point in ∂ Q R 1 ,R 2 is in one of the following three subsets: For (1) and (3), by (6.11), the choices of R 1 and This case analysis completes the proof.
As before, the square root of ψ → M ψ, / Dψ + ψ 2 2 defines a norm on F + φ 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)  as n → ∞. This is a contradiction. So we complete the proof.
We next prove: 2α (M, N ). By the definition of λ + (φ), for any ε > 0, there exists ψ ε ∈ F + φ such that in general, we cannot use ψ ε as a test spinor to estimate λ + (φ). In order to get a suitable test spinor, we need parallel translation. Denote by ι(N ) > 0 the injectivity radius of N . For any y, z ∈ N with d(y, z) < ι (N ) (d(y, z) is the geodesic distance between y and z in N ), P y,z defined as in Lemma 4.5 depends smoothly on y, z. By Lemma 4.5, T φ,φ ψ ε ∈ Fφ. To make it belong to F + φ , we need to modify it.
We also need to investigate the maps far away from the set M α (θ ). Lemma 6.7 Let δ(θ) > 0 be as in Corollary 6.6. There exists ε(θ) such that for . Proof For any fixed α > 1, if such a ε(θ) does not exist, then there exist {φ n } such that dist(φ n , M α (θ )) ≥ δ(θ) (6.25) and Since m θ is the minimizing energy of E α , {φ n } is a minimizing sequence for E α . Therefore, the Palais-Smale condition implies that, after taking a subsequence if necessary, there is a 2α (M, N ).

Remark 6.8
It follows from the proof above that the estimate (6.13) is uniform in k if we replace F by 1 k F instead.

Critical value of LW
sup L α (γ (Q θ ;R 1 ,R 2 )), (6.35) where (Q θ ;R 1 ,R 2 ) is defined in Definition 4.12. By Lemma 4.7, one can prove that (Q θ ;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 . Let R 1 , R 2 and ρ be as in Lemmas 6.1 and 6.2. Then we have m θ < c θ < ∞ and c θ is a critical value of L α in F .
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 α-Dirac-harmonic maps.
Since the smoothness is a local property, we will prove Theorem 7.1 locally. So let we fix a domain ⊂ M, which is mapped into a local chart {y i } i=1,...,n on N . Then a weakly perturbed α-Dirac-harmonic map (φ, ψ) is a weak solution to the following system: As we said in the introduction, we have to control the L ∞ -norm of ψ first. Since (φ, ψ) ∈ F α,1/2 , by the Sobolev embedding, we can prove that ψ actually belongs to W 1,s for any 2 < s < 2α. Lemma 7.2 Let (φ, ψ) ∈ F α,1/2 be a weak solution of (7.1) and (7.2). Suppose F satisfies (F1) for some p ≤ 2 + 2/α.
where we have used (F3). For the second term in the right-hand side of (7.12), we have (1 + |dφ| 2 ) α−1 |∇η| 2 (7.14) By estimates (7.13), (7.14) and (7.12), we have Since φ is continuous, we can choose ρ so small that C N sup (7.16) Now, take ρ still smaller so that sup where C only depends on N and the constant in (F3).
Now we can prove the main theorem in this section. N ). This can be done just by replacing weak derivatives by difference quotients in the proof of Lemma 7.4. Denote

8˛-Dirac-harmonic maps
In this section, we want to approximate the α-Dirac-harmonic maps by the perturbed α-Dirac-harmonic maps. Precisely, by Theorem 6.11 and Theorem 7.1, we have a sequence of perturbed α-Dirac-harmonic maps (φ k , ψ k ), which are the critical points of the functionals where F satisfies the assumptions in Theorem 6.11 and Theorem 7.1. If (φ n , ψ n ) converges to (φ, ψ) smoothly, then we get the existence of nontrivial α-Dirac-harmonic maps. We now come to the ε-regularity estimate which gives a uniform control on the sequence in Sobolev space.
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