Asymptotic analysis for Dirac-harmonic maps from degenerating spin surfaces and with bounded index

We study the refined blow-up behaviour of a sequence of Dirac-harmonic maps from degenerating spin surfaces with uniformly bounded energy in the case that the domain surfaces converge to a spin surface with only Neveu–Schwarz type nodes. For Dirac-harmonic necks appearing near the nodes, we show that the limit of the map part of each neck is a geodesic in the target manifold. Moreover, we give a length formula for the limit geodesics appearing near the node in terms of the Pohozaev type constants associated to the sequence. In particular, if the Ricci curvature of the target manifold has a positive lower bound and the Dirac-harmonic sequence has bounded index, then the limit of the map part of the necks consist of geodesics of finite length and the energy identities hold.

izes the theory of harmonic maps and harmonic spinors, which has been widely studied. In dimension two, similarly to harmonic maps, the conformal invariance of the Dirac-harmonic map system leads to non-compactness of the space of Dirac-harmonic maps. In general, bubbling phenomena can occur, due to the possible energy concentration at finitely many points in the domain. When the domain is fixed, the blow-up theory for a sequence of Dirac-harmonic maps with uniformly bounded energy has been systematically studied in [2,19,26] for Diracharmonic maps and in [11] for the case of a more general model. To study the existence problem for Dirac-harmonic maps, a heat flow approach was investigated in [4,5,12,15]; see [13][14][15] for the blow-up analysis of the corresponding approximate Dirac-harmonic maps. Roughly speaking, the results of these works assert that the failure of strong convergence occurs at finitely many energy concentration points. At such points, finitely many bubbles, i.e. non-trivial Dirac-harmonic spheres, can separate, and the sum of the energies of these bubbles accounts for the loss of energy during the process of weak convergence. This is known as the energy identity. Moreover, the image of the map part of the limit Dirac-harmonic map and those of the bubbles are connected in the target manifold. This is called the no neck property.
However, if we allow the domain surface to vary, then the energy identity does not hold in general, see e.g. the example in the case of harmonic maps constructed by Parker [20]. Such a situation was first systematically investigated in [28] where the author proved a generalized energy identity for the sequence when the domain surfaces degenerate to a spin surface with only Neveu-Schwarz type nodes and gave a sufficient and necessary condition that the energy identity holds, extending the case of harmonic maps from degenerating surfaces systematically studied in [29]. In this paper, we shall explore the finer asymptotic behaviour of the Dirac-harmonic necks appearing near the degenerating region of the domain. In the case of harmonic maps, where the spinors are vanishing, such necks converge to geodesics in the target [7,8,10]. In the more general case of Dirac-harmonic maps investigated in the present paper, it is natural to conjecture that the limit of the map part of these Dirac-harmonic necks are curves in the target manifold with specific properties similar to geodesics. Also, it should be possible to calculate the lengths of the limiting curves of the necks appearing near the nodes in terms of some geometric quantities associated to the Dirac-harmonic map sequence, as is achieved in the case of harmonic maps in [7,29].
To state our problem more precisely, let φ be a smooth map from a spin Riemann surface (M, h) with metric h to another compact Riemannian manifold (N , g) with 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 with induced metric ·, · M⊗φ T N and induced connection ∇. A smooth section ψ of the twisted bundle M ⊗ φ T N is called a spinor field along the map φ. Critical points (φ, ψ) of the action functional are called Dirac-harmonic maps from M to N . Here / D is the Dirac operator along the map φ, defined by / Dψ := e α · ∇ e α ψ. See Sect. 2 for more details on the notations and definitions. Now, let (M n , h n , c n , σ n ) be a sequence of closed hyperbolic Riemann surface of genus g > 1, equipped with hyperbolic metrics h n , compatible complex structures c n and spin structures σ n . We consider a sequence of smooth Dirac-harmonic maps We shall often omit the domain M from the notation and simply write E(φ) = E(φ; M), E(ψ) = E(ψ; M) and E(φ, ψ) = E(φ, ψ; M).
In two dimensional geometric variational problems, Pohozaev type identities play an important role in the study of the qualitative behavior for a sequence of solutions. When the domain surface varies and possibly degenerates, one needs to study solutions defined on long cylinders and in such situations, Pohozaev type identities in general no longer hold. Therefore, we need to find some geometric quantities associated to the sequence of solutions and domains to characterize the asymptotic behaviour of the solutions. Such a scheme was first explored in [29] for harmonic maps and then in [28] for Dirac-harmonic maps. Consider a Dirac-harmonic map (φ, ψ) defined on a standard cylinder P = [t 1 , t 2 ] × S 1 with flat metric ds 2 = dt 2 + dθ 2 . Denote (1.1) It is well known that the quadratic differential T dz 2 is holomorphic. See [3] for details. By Lemma 3.4 in [28] (or Lemma 2.1 in the present paper), we know that is a complex number which is independent of t ∈ [t 1 , t 2 ]. The quantity defined in (1.2) can be considered as a Pohozaev type constant associated to the Dirac-harmonic map (φ, ψ) defined on the cylinder P and it measures the extent to which the Pohozaev type identity fails. In the case of a vanishing spinor field ψ ≡ 0, it reduces to the Pohozaev type constant associated to the harmonic map φ introduced in [29] (page 64). Such quantities played an important role in the study of the asymptotic behaviour of harmonic maps and Dirac-harmonic maps from degenerating surfaces [28,29]. Pohozaev type constants appear also in other two dimensional geometric variational problems, like the singular super-Liouville type systems [16]. Now, we assume that (M n , h n , c n ) degenerates to a hyperbolic Riemann surface (M, h, c) by collapsing p (1 ≤ p ≤ 3g − 3) pairwise disjoint simple closed geodesics γ j n of lengths l j n , j = 1, . . . , p. For each j, the geodesics γ j n degenerate into a pair of punctures ( j,1 , j,2 ). Let P j n be the standard cylindrical collar about γ j n . We associate the sequence (φ n , ψ n , M n ) with a sequence of p-tuples (α 1 n , . . . , α p n ), where α j n := α(φ n , ψ n , P j n ) ∈ C are quantities defined by (1.2). Pulling back the hyperbolic metrics h n and the compatible complex structures c n by suitable diffeomorphisms M → M n \ ∪ p j=1 γ j n and passing to a subsequence, we can think of (h n , c n ) as all living on the limit surface M, converging in C ∞ loc to (h, c), and the pull back of σ n is a fixed spin structure σ on M. Note that M has p pairs of punctures. As in [28], we require the following assumptions on the limit surface:

All punctures of the limit spin sur f ace (M, σ ) are o f N eveu−Schwar z t ype.
Then σ extends to some spin structure σ on M, where M is the surface obtained by adding a point at each puncture of M. Thus, (φ n , ψ n ) can be considered as a sequence of Dirac-harmonic maps defined on (M, h n , c n , σ ).
In [28], the following generalized energy identities for Dirac-harmonic maps from degenerating surfaces were proved: Theorem 1.1 Assumptions and notations as above. Then there exist at most finitely many blow-up points {x 1 , . . . , x I } which are away from the punctures {( j,1 , j,2 ), j = 1, 2, . . . , p} and finitely many Dirac-harmonic maps such that after selection of a subsequence, (φ n , ψ n ) converges to (φ, ψ) in C ∞ loc × C ∞ loc on M\{x 1 , . . . , x I } and the following identities hold: When the domain is fixed, it was shown in [19] that the images of the map parts of the weak limit and all the bubbles emerging from any blow-up point are connected, which is known as the no neck property. By slightly modifying the arguments in [19], it is easy to see that the same property holds if the domain surfaces are non-degenerating and stay in a compact region of the moduli space. Therefore, in Theorem 1.1, we know that the image φ(M) ∪ I i=1 ∪ L i l=1 σ i,l is a connected set, as the bubbles (σ i,l , ξ i,l ) occur at blow-up points that are away from the punctures.
In this paper, we shall explore the finer asymptotic behaviour of the Dirac-harmonic necks that appear near the nodes when the domain surfaces degenerate. It turns out that the map part of these Dirac-harmonic necks converge to geodesics in the target manifold (see Theorem 3.1). To achieve this, we need to carry out a refined neck analysis for Dirac-harmonic maps from long cylinders. Our first main result is the following:

Remark 1.3
In the above theorem, the length of each geodesic can be finite or infinite and a geodesic of zero length means that the map part of the corresponding neck is converging to a point in the target.
To prove Theorem 1.2, we need to establish a key lemma about the exponential decay along long cylinders of the tangential energies of both the map part and the spinor part. For the case of ψ n ≡ 0, i.e. when the φ n are harmonic maps, this is achieved in [7] where the authors used the ideas in [9] to derive a differential inequality on a long cylinder. However, this kind of technique requires the structure of a harmonic map type equation and cannot be applied to the Dirac equation. Instead, we shall apply the three circle theorem for Dirac-harmonic type systems developed in [19]. The idea is inspired from the work [21], which used a special case of the three circle theorem due to [25] to show that the tangential energy of the sequence on the long cylinder decays exponentially. To derive the exponential decay of the whole energy of spinor, we shall firstly conformally change the long cylinder to an annulus, then we apply a Hardy type inequality as in [11,15] to obtain some differential inequality on this annulus.
According to Theorem 1.1, it is easy to see that the energy identities hold if and only if the following analytical condition is satisfied A natural question then is whether we can exploit some geometric conditions to ensure that the limiting necks are some geodesics of finite length so that the energy identity follows immediately.
To achieve this, in analogy to the works on minimal hypersurfaces [23] and harmonic maps [18], we shall impose the assumptions that the Ricci curvature of the target manifold has a positive lower bound and the sequence of Dirac-harmonic maps has bounded Morse index (see Sect. 4). In the following, we state our second main result: Theorem 1.4 Under the assumption of Theorem 1.1, suppose the Ricci curvature of the target manifold (N , g) has a positive lower bound, i.e. there exists a positive constant λ 0 > 0 such that Ric N ≥ λ 0 > 0 and assume the sequence (φ n , ψ n ) has bounded index, then the limit of the map part of the necks consists of geodesics of finite length. Moreover, the energy identities hold, i.e.
We remark that a similar but more subtle and complicated analysis for α-Dirac-harmonic maps was carried out in [15], that is, when the functional is perturbed in the sense of [22]. This problem naturally emerges in our context, because our existence scheme for Diracharmonic maps works with such a perturbation to control the asymptotic behavior of our elliptic-parabolic flow.
The rest of the paper is organized as follows. In Sect. 2, we shall review some geometric and analytic aspects of Dirac-harmonic maps. Then we establish the three circle theorem for Dirac-harmonic maps on long cylinders and derive the exponential decay of tangential energies of both the map and the spinor. In Sect. 3, we prove our main Theorem 1.2. In Sect. 4, we calculate the second variation formula for the functional L, provide the notion of index of Dirac-harmonic maps and then prove Theorem 1.4.

Preliminaries and some basic lemmas
In this section, we shall first recall the definition of Dirac-harmonic maps and some basic lemmas like the energy gap theorem, the small energy regularity theorem and so on. Then we will establish the three circle lemma for Dirac-harmonic maps on long cylinder which yields the energy exponential decay in the tangential direction.
Let (M, h, σ ) be a Riemann surface, equipped with a Riemannian metric h and with a fixed spin structure σ , M be the spinor bundle over M and ·, · M be the metric on M induced by the Riemannian metric h. Choosing a local orthonormal basis e α , α = 1, 2 on M, the usual Dirac operator is defined as / ∂ := e α · ∇ e α , where ∇ is the spin connection on M. The usual Dirac operator / ∂ on a surface can be seen as the Cauchy-Riemann operator. Consider R 2 with the Euclidean metric dx 2 + dy 2 . Let e 1 = ∂ ∂ x and e 2 = ∂ ∂ y be the standard orthonormal frame. A spinor field is simply a map ψ : R 2 → 2 = C 2 , and the action of e 1 and e 2 on spinors can be identified with multiplication with matrices

is a spinor field, then the Dirac operator is
For more details on spin geometry and Dirac operators, one can refer to [17]. Let φ be a smooth map from M to another compact Riemannian manifold (N , g) with dimension n ≥ 2. Let φ T N be the pull-back bundle of T N by φ and then we get the twisted bundle M ⊗ φ T N. Naturally, there is a metric ·, · M⊗φ T N on M ⊗ φ T N which is induced from the metrics on M and φ T N. Also we have a natural connection ∇ on M ⊗ φ T N which is induced from the connections on M and φ T N. Let ψ be a section of the bundle M ⊗ φ T N. In local coordinates, it can be written as where each ψ i is a usual spinor on M and ∂ y i is the nature local basis on N . Then ∇ becomes where i jk are the Christoffel symbols of the Levi-Civita connection of N . The Dirac operator along the map φ is defined by / Dψ := e α · ∇ e α ψ.
We consider the following functional The functional L(φ, ψ) is conformally invariant. That is , for any conformal diffeomor- The Euler-Lagrange equations of the functional L are Here R m li j stands for the Riemann curvature tensor of the target manifold (N , g). By the Nash embedding theorem, we embed N into R K . Then, the critical points (φ, ψ) satisfy the Euler-Lagrange equations where / ∂ is the usual Dirac operator, A is the second fundamental form of N in R K , and Here P(ξ ; ·) denotes the shape operator, defined by , and Re(z) denotes the real part of z ∈ C. We refer to [2,6,11,24,27] for more details.
Next, we recall some basic lemmas which will be used in this paper. Then

Lemma 2.2 (Theorem 3.1 in [2]). Let M be a closed spin Riemann surface with a fixed spin structure and N be a compact Riemannian manifold. Then there is a small constant
then φ is a constant map and ψ is a tuple of harmonic spinors. .
can be extended smoothly to the whole disk D.
with L is given and large. Then there exists a positive constant In the end of this section, we derive the exponential decay of the tangential energy on the long cylinder. Set If there is no energy concentration for (φ n , ψ n ), then for any fixed k > 0 and for any t n ∈ [−T n + T + k, T n − T − k], when n and T are sufficiently large, there holds where lim T →∞ lim n→∞ o(n, T ) = 0 and C = C(L) is a positive constant, L is the positive constant in Proposition 2.5.

Proof
The idea is similar to the argument in the proof of Theorem 1.4 in [19]. Since there is no energy concentration for (φ n , ψ n ), by a standard contradiction argument in [10,26], we have This means that for any > 0, there exist two positive integers T and N which are large enough such that when n ≥ N , there holds (2.10) Now, by (2.10) and Lemma 2.3, we have that for any small > 0, when n and T are large enough, there holds by a direct computation as in [19], we can prove that (u, v) satisfy Eqs. (2.7)-(2.8) (see equations (3.17) and (3.18) in [19]). Moreover, using the estimate (2.11), (u, v) satisfy the conditions in Proposition 2.5 (see also [19] for details).
For simplicity of notations, we also set where L > 0 is a constant in Proposition 2.5. Without loss of generality, we may assume where k n is a positive integer which goes to infinity as n → ∞. By (c) of Proposition 2.5, we obtain Then, using the (a) and (b) of Proposition 2.5, by iterating, we have This yields Applying the interior estimates Lemma 2.6, we get and Then it is easy to see that the conclusion of the lemma follows immediately from the above decay estimates. In fact, there exist two positive integers i 1 , i 2 , such that 1 ≤ i 1 ≤ i 2 ≤ k n and According to (2.14) and (2.15), we have Lastly, by (2.11), it is easy to see that ( φ n − φ * n L 2 (P 1 ∪P kn ) + ψ n − ψ * n L 2 (P 1 ∪P kn ) ) = o(n, T ). We finished the proof of the lemma.
As a corollary of the above lemma, we have   . (2.16) We recall the notation and without loss of generality, we assume 2(T n − T ) = k n L where L is the positive constant in Proposition 2.5 and k n is a positive integer which will go to infinity as n → ∞. By Lemma 2.7, we get Similarly, In order to prove by the elliptic estimates for the Dirac operator, we have where we used the Hölder inequality in the last estimate. Then (2.20) follows immediately from (2.19) and (2.10). Thus, we proved the statement (1) of the lemma.
For the statement (2), similar to the argument in (2.16), we may obtain where the last inequality follows from (2.14) and (2.15).

Combining this with Lemma 2.3, we get
Therefore, Then the conclusion of statement (2) follows immediately and we finished the proof of this lemma.

Convergence to geodesics
In this section, we will prove our main Theorem 1.2.
In fact, in [28], it was shown that the energy concentration at the nodes can be reduced to the study of the asymptotic behaviour of a sequence of Dirac-harmonic maps from long cylinders. Therefore, we just need to study the following problem: Let P n = [−T n , T n ]× S 1 with standard metric ds 2 = dt 2 + dθ 2 and T n → ∞ as n → ∞. Given a sequence of Dirac-harmonic maps (φ n , ψ n ) from P n to N with uniformly bounded energy E(φ n , ψ n ) ≤ < ∞, which satisfies and does φ n converges to a geodesic? If so, how to compute the lengths of these curves? where [−T n , T n ] × S 1 is a cylinder with standard flat metric ds 2 = dt 2 + dθ 2 and T n → ∞ as n → ∞. Suppose that
To show Theorem 3.1, we shall first prove some lemmas. Since μ > 0, when n is big enough, there holds Thus, Letting n → ∞ and then T → ∞, we will get the conclusion of the lemma.
From the above proof, it is easy to see that a stronger property holds. Next, we will show that there is no concentration for some stronger energy norm of the spinor part. The proof is based on applying some Hardy-type inequality on R 2 , as was done in [11,15].

Proof
The key of the proof is the Hardy-type inequality on R 2 that for any where the constant 1 is the best possible constant (for a simple proof, see [1]). Firstly, we introduce a new coordinate system. Let (r , θ) be polar coordinates centered at 0. Let F : with the metric g = dt 2 + dθ 2 , which is conformal to the standard Euclidean metric ds 2 on R 2 . In fact, Taking f = η| n | 2 in the inequality (3.5), we get On the one hand, from the equation or spinor and conformal invariance, we know On the other hand, by inequality (2.11), we have Combining these, we get (3.8) Since we can take sufficiently small, we then have Combining this with (2.10) again, we have Combing Lemma 3.4 with the Hardy type inequality (3.5), we shall derive the exponential decay of the energy of spinor part on the neck region. for any t 0 ∈ (1, k n ) and 0 ≤ t ≤ min{t 0 , k n − t 0 }. For any ρ > 0, taking the cut-off function η ∈ C ∞ 0 (D e (t 0 +t)L e −Tn +T +ρ \D e (t 0 −t)L e −Tn +T −ρ ) such that 0 ≤ η ≤ 1 and η ≡ 1 on D e (t 0 +t)L e −Tn +T \D e (t 0 −t)L e −Tn +T and |∇η| ≤ 2 ρ . Taking f = η| n | 2 in the Hardy inequality (3.5) and by (3.8), we get (3.11) Taking > 0 sufficiently small such that C ≤ 1 2 , we have where ρ is small and the last inequality is from Lemma 2.7.
Letting ρ → 0, we get This is Without loss of generality, we may assume t 0 ≤ k n − t 0 . Then, integrating the above differential inequality from 1 to t 0 , we get where the second equality follows from Lemma 3.4 since In the case of k α − t 0 ≤ t 0 , we can apply similar arguments to get Thus, we have proved (3.9). The estimate (3.10) is a consequence of (3.9) and (2.11).
For (3.13), we prove it by a contradiction argument which is similar to the proof of Lemma 2.7 in [7].
If it was false, then there would exist t n ∈ [−λT n , λT n ] and θ n ∈ [0, 2π] such that By (2.21), for any t ∈ [−λT n , λT n ] × S 1 and k > 0, when n is big enough, we have Combining this with (3.15) and (3.17), since λ ∈ (0, 1), when n is sufficiently large, we get (3.18) By small energy regularity Lemma 2.3, we obtain that for any β ∈ (0, 1), there holds Setting u n := 1 √ |Re α(φ n , ψ n )| (φ n (t n + t, θ) − φ n (t n , 0)) , then from (3.19), it is easy to see that k). (3.20) Combining this with the fact that u n (0, 0) = 0, we have Noting that |ψ n | ≤ C|Re α(φ n , ψ n )|, it is easy to prove that u n satisfies the following equation Therefore, we know that after passing to a subsequence, there holds Moreover, by Lemma 3.2, we get Thus, we know that the harmonic function u must be of the form where − → a = (a 1 , . . . , a K ) ∈ T y N . Let lim n→∞ θ n = θ 0 , then it is obvious that However, by (3.16), we have which is a contradiction. So, (3.13) holds and we finished the proof of the lemma.

Lemma 3.7
Under the assumptions of Theorem 3.1, let μ > 0 and λ ∈ (0, 1), then there holds Proof In fact, if (3.21) does not hold, then there exist t n ∈ [−λT n , λT n ] and θ n ∈ [0, 2π] such that By similar argument as in the proof of Lemma 3.6, we may assume where Combining (3.23) with (3.22), we get which is also a contradiction. Thus, (3.21) holds and we finished the proof of this lemma. Now, with the help of the previous lemmas, we can prove Theorem 3.1.

Proof of Theorem 3.1
When μ = 0, then the conclusions of Theorem 3.1 follow immediately from Lemma 2.8. We just need to consider the case that μ > 0. Denote where t ∈ [−λT n , λT n ], λ ∈ (0, 1). By Lemma 2.3, it is easy to see that γ n converges to some curve on N denoted by γ . Next, we will show that γ is just a geodesic on N .
Let s be the arc length parameter of φ * n (t), i.e.
On the other hand, also from Lemmas 3.6 and 3.7, it is easy to see that Therefore, we obviously see that Thus, φ * n (s) will converge locally to a smooth vector valued function from [0, s] into R K , denoted by ω(s), in the sense of C 1 , i.e. γ n | [−λT n ,λT n ] converges locally to the curve γ . Next, we will show that γ is a geodesic.
By Lemmas 3.6 and 3.7, we obtain Therefore, we get that the Dirac-harmonic neck converges to a geodesic γ on the target manifold N .
Finally, we compute the length of the geodesic. We consider the following two cases.

By (2.22), there holds
Similarly, Therefore, we get Combining this with Lemma 3.6, the length of the limit geodesic is Case 2: μ = ∞.
In this case, from the above argument, it is easy to see that the neck contains at least an infinite length geodesic.
In the end of this section, we give the proof of Theorem 1.2. By Theorem 3.1, we know that map parts φ n on I i n are converging to the geodesics on N of length and the spinor parts ψ n on I i n are converging to 0. By applying the neck analysis for Diracharmonic maps from a fixed domain in [19], the maps φ n on J j n are converging to a point, i.e., we get the no neck property. Thus, the maps φ n on the whole cylinder P n converge to some geodesics on N . The sum of the lengths of these geodesics can be calculated as follows: which implies the conclusions of the theorem immediately.

Proof of Theorem 1.4
In this section, we shall calculate the second variation formula for the functional L and define the notion of index of Dirac-harmonic maps. Similar calculation and definition for α-Dirac-harmonic maps were given in [15]. Then, in analogy to [18], we shall give some geometric conditions on the target manifold which ensures that the energy identities hold for Dirac-harmonic maps from degenerating surfaces with bounded index. Let (φ, ψ) : M → N be a Dirac-harmonic map. φ * (T N) is the pull-back bundle over M. Let V be a section of L. Consider the following variation of (φ, ψ): It is well-known that the following second variational formula for the energy of the map holds [22] is the tension field of the map φ. Next, we compute Choosing a local orthonormal basis {e α } on M such that [e α , ∂ ∂τ ] = 0, ∇ e α e β = 0 at a considered point, then we have Noting that Definition 4.2 Let (φ * T N) denote the linear space of smooth sections of φ * T N. The index of (φ, ψ) is defined as the maximal dimension of a linear subspace of (φ * T N) on which the second variation of L with respect to the variations (4.1) is negative, i.e., for any V ∈ ⊂ (φ * T N), there holds Before giving the proof of Theorem 1.4, we first prove the following theorem. where [−T n , T n ] × S 1 is a cylinder with standard flat metric ds 2 = dt 2 + dθ 2 and T n → ∞ as n → ∞. Suppose there is no energy concentration for (φ n , ψ n ), if μ = ∞ and Ric N ≥ λ N > 0, then the index of (φ n , ψ n ) tends to infinity.

Lemma 4.4
Under the assumptions of Theorem 4.3, for sufficiently large n, there exists a section V n of φ * n (T N), which is supported in [0, t a n ], such that δ 2 L(φ n , ψ n )(V n , V n ) < 0.
Proof Let P be projection from T R K to T N. We define V n (s, θ) = P φ n (s,θ ) (V 0 (s)), where s is the arc-length parametrization of φ * n (t) with s(0) = 0. Then, it is easy to see that V n is a smooth section of φ * n (T N) which is supported in [0, t a k ]× S 1 . Since φ n (s, θ) → γ (s) in C 1 ([0, a] × S 1 ), we have V n (φ n (s, θ)) → V 0 (s) in C 1 ([0, a] × S 1 ).
On the other hand, we get For the term I 2 , we have For any given T > 0, we set m n = t a n T + 1, which tends to infinity as n → ∞. By (4.3), there holds |Re α(φ n , ψ n )|m n ≤ C(T ).
Hence, for n large enough, we have the desired inequality δ 2 L(φ n , ψ n )(V n , V n ) < 0.
Thus, we complete the proof of this lemma.