Blowup behavior of harmonic maps with finite index

In this paper, we study the blow-up phenomena on the αk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _k$$\end{document}-harmonic map sequences with bounded uniformly αk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _k$$\end{document}-energy, denoted by {uαk:αk>1andαk↘1}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{u_{\alpha _k}: \alpha _k>1 \quad \text{ and } \quad \alpha _k\searrow 1\}$$\end{document}, from a compact Riemann surface into a compact Riemannian manifold. If the Ricci curvature of the target manifold has a positive lower bound and the indices of the αk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _k$$\end{document}-harmonic map sequence with respect to the corresponding αk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _k$$\end{document}-energy are bounded, then we can conclude that, if the blow-up phenomena occurs in the convergence of {uαk}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{u_{\alpha _k}\}$$\end{document} as αk↘1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _k\searrow 1$$\end{document}, the limiting necks of the convergence of the sequence consist of finite length geodesics, hence the energy identity holds true. For a harmonic map sequence uk:(Σ,hk)→N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_k:(\Sigma ,h_k)\rightarrow N$$\end{document}, where the conformal class defined by hk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_k$$\end{document} diverges, we also prove some similar results.


Introduction
Let ( , g) be a compact Riemann surface and (N , h) be an n-dimensional smooth compact Riemannian manifold which is embedded in R K isometrically. Usually, we denote the space of Sobolev maps from into N by W k, p ( , N ), which is defined by W k, p ( , N ) = {u ∈ W k, p ( , R K ) : u(x) ∈ N for a.e. x ∈ }.
For u ∈ W 1,2 ( , N ), we define locally the energy density e(u) of u at x ∈ by e(u)(x) = |∇ g u| 2 The energy of u on , denoted by E(u) or E(u, ), is defined by and the critical points of E are called harmonic maps. We know that the energy functional E does not satisfy the Palais-Smale condition. In order to overcome this difficulty, Sacks and Uhlenbeck [16] introduced the so called α-energy E α of u : → N as the following where α > 1. The critical points of E α in W 1,2α ( , N ) are called as the α-harmonic maps from into N . It is well-known that this α-energy functional E α satisfies the Palais-Smale condition and therefore there always exists an α-harmonic maps in each homotopy class of map from into N . The strategy of Sacks and Uhlenbeck is to employ such a sequence of α k -harmonic maps to approximate a harmonic map as α k tends decreasingly to 1. If the convergence of the sequence of α k -harmonic map is smooth, the limiting map is just a harmonic map from into N .
The energy of a map u from a closed Riemann surface is conformally invariance, it means that, if we let g = e 2ϕ g be another conformal metric of , then |∇ g u| 2 dμ g = |∇ g u| 2 dμ g .
Let C g denote the conformal class induced by a metric g, then, the following definition E(u, C g ) = 1 2 |∇ g u| 2 dμ g does make sense. Moreover, it is well-known that the critical points of E(u, C g ) are some branched minimal immersions (see [16,17]). Hence, in order to get a branched minimal surface, we also need to study the convergence behavior of a sequence of harmonic maps u k : ( , h k ) → N with uniformly bounded energy E(u α k ) < C.
No doubt, it is very important to study the convergence of a sequence of α k -harmonic maps from a fixed Riemann surface ( , g) and a sequence of harmonic maps from ( , h k ) into N , where h k is the metric with constant curvature. In fact, these problems on the convergence of harmonic map or approximate harmonic map sequences have been studied extensively by many mathematicians. Although these sequences converge smoothly to harmonic maps under some suitable geometric and topological conditions, generally one found that the convergence of such two classes of sequences might blow up.
First, let's recall the convergence behavior of α k -harmonic map sequences. Suppose that {u α k } be a sequence of α k -harmonic maps from ( , g) with bounded uniformly α k -energy, i.e. E α k (u α k ) ≤ . By the theory of Sacks-Uhlenbeck, there exists a subsequence of {u α k }, still denoted by {u α k }, and a finite set S ⊂ such that the subsequence converges to a harmonic map u 0 in C ∞ loc ( \S). We know that, at each point p i ∈ S, the energy of the subsequence concentrates and the blow-up phenomena occur. Moreover, there exist point where all v i are non-trivial harmonic maps from S 2 into N , and A i ⊂ R 2 is a finite set. In order to explore and describe the asymptotic behavior of {u α k } at each blow-up point, the following two problems were raised naturally. One is whether or not the energy identity, which states that all the concentrated energy can be accounted for by harmonic bubbles, holds true, i.e., Here, B r 0 ( p i ) is a geodesic ball in which contains only one blow-up point p i . The other is whether or not the limiting necks connecting bubbles are some geodesics in N of finite length?
For a harmonic map sequence {u k } from ( , h k ) into (N , h), one also encountered the same problems as above.
In particular, in [9] it is proven that if energy concentration does occur, then a generalized energy identity holds. Moreover, from the view point of analysis some sufficient and necessary conditions were given such that the energy identity holds true. On the other hand, a relation between the blowup radii and the values of α was discovered to ensure the "no neck property". If necks do occur, however, they must converge to geodesics and an example was given to show that there are even some limiting necks (geodesics) of infinite length.
Generally, the energy identity does not hold true. For the case of harmonic map sequence u k : ( , h k ) → (N , h) one has found a counter-example for the energy identity in [13]. Very recently, in [10] a counter-example for the energy identity was given for the case of α k -harmonic map sequence.
Furthermore, from the study in [9,10] we can see that except for α, the topology and geometry of the target manifold (N , h) also play an important role in the convergence of αharmonic map sequence from a compact surface. From the viewpoint of differential geometry, it is therefore natural and interesting to find some reasonable geometric and topological conditions on the domain or target manifold such that the energy identity holds. In particular, a natural question is whether or not we can exploit some geometric and topological conditions to ensure the limiting necks are some geodesics of finite length, which implies that the energy identity holds true? For this goal, in this paper we obtain the following two theorems: Theorem 1.1 Let ( , g) be a closed Riemann surface and (N , h) be a closed Riemannian manifold with Ric N > λ > 0. Let α k → 1 and {u α k } be a sequence of maps from ( , g) → (N , h) such that each u k is an α k -harmonic map, the indices and energy satisfy respectively If {u α k } blows up, then the limiting necks consist of some finite length geodesics.  [1] or [9], the fact the limiting necks are of finite length implies that the energy identity is true. We should also mention that, when each u α k in {u α k } is the minimizer of the corresponding E α k in a fixed homotopy class, Chen and Tian [1] have proved that the necks are just some geodesics of finite length in N .

Remark 1.4
The curvature condition in Theorem 1.1 and 1.2 is used to ensure that any geodesic of infinite length lying on N is not stable. In fact, we will prove in this paper that, if the necks contain an unstable geodesic of infinite length, then the indices of the harmonic (or α-harmonic) map sequence can not be bounded from the above.

The Proofs of Theorem 1.1
Our strategy is to show that the indices of the sequence {u α k } in Theorem 1.1 are not bounded if there exists a infinite length geodesic in the set of the limiting necks {u α k }. For this goal, first we need to recall the definition of the index of a α-harmonic map and the second variational formula of α-energy functional.

The index of a α-harmonic map
Obviously, u 0 = u. Then, the formula of the second variation of E α reads For more details we refer to [12]. Let (L) denotes the linear space of the smooth sections of L. Then, the index of u is the maximal dimension of the linear subspaces of (L) on which the (2.1) is negative definite.

The index of the necks
We have known the limiting necks of {u α k } are some geodesics in N , a natural question is there exists some relations between the indices of these geodesics and the indices of the necks of {u α k }. In this subsection we need to analyse the asymptotic behavior of the necks of {u α k } and try to establish the desired relations.
Let α k → 1 and each u α k of the map sequence {u α k : k = 1, 2, . . .} be a α k harmonic map from ( , g) into (N , h). For convenience we always embed (N , h) into R K isometrically and set u k = u α k . Assume that {u k } blows up only at a point p ∈ . Then, for any , we have Choose an isothermal coordinate chart (D; For simplicity, we assume u k has only one blowup point in D. Put Then, we have that x k → 0, r k → 0 and there exists a bubble v, which can be considered as a harmonic map from S 2 into N , such that u k (x k + r k x) converges to v. Without loss of generality, we may assume x k = 0.
By the arguments in [9], we only need to prove Theorem 1.1 for the case there exists one bubble in the convergence of {u k }. So, we always assume that only one bubble appears in the convergence of {u k } in this section. Now, we consider the case that the limiting necks contain a geodesic of infinite length. In fact, the present paper is a follow up of the papers [9] and [2], first of all, we need to recall some results proved in [9]. Lemma 2.1 Let α k → 1 and {u k } be a map sequence such that each u k is an α k -harmonic map from ( , g) into (N , h). If there is a positive constant such that E α k (u k ) < for any α k , then, there exists a positive constant C such that, neglecting a subsequence, there holds For the proof of this Lemma and more details we refer to Remark 1.2 in [9]. Moreover, for the convergence radii and α k we have following relations: Here, r k is defined as before.
where μ max ≥ 1 is a positive constant. Therefore, it follows that there holds Since the limiting neck is of infinite length, from Theorem 1.3 in [9] we known that It follows that Thus we complete the proof.
As a direct corollary of the Proposition 4.3 in [9], we have Suppose that there is a positive constant such that E α k (u k ) < for any α k and there exists only one bubble in the convergence of {u k }. Then, for any t k → t ∈ (0, 1), there exist a vector ξ ∈ R K and a subsequence of {u k } such that where μ is defined by Now we define the approximate curve of u k , denoted by u * k (r ), by Since the target manifold (N , h) is embedded in R K , u * k (r ) is a space curve of R K and we denote the arc-length parametrization of u * k by s such that s(r t 1 k ) = 0, where t 1 ∈ (0, 1).
Thus t k i → t 1 . By Lemma 2.3, after passing to a subsequence, we have Therefore, noting Thus, we get a contradiction. Hence, it follows From the arguments in the above and [9] we conclude that for any fixed θ a] → 0. By the same way, we can prove (2.4).

Lemma 2.6 Suppose that {u k } satisfies the same conditions as in Lemma 2.3. Then, for any fixed R
Proof Assume this is not true. After passing to a subsequence, we can find t k → t ∈ [t 1 , t 2 ], such that 1 However, by Proposition 4.2 in [9], This is a contradiction, Thus we complete the proof of the lemma.

Now, let's recall the definition of stability of a geodesic on a Riemannian manifold (N , h). A geodesic γ is called unstable if and only if the second variation formula of its length satisfies
Here R is the curvature operator of N . We have such that Proof Since the limiting neck of {u k }, denoted by γ : [0, a] → N , is not a stable geodesic, there exists a vector field V 0 on γ with V 0 | γ (0) = 0 and V 0 | γ (a) = 0 such that Let P be projection from T R K to T N. We define where s is the arc-length parametrization of u * . By Corollary 2.5, for any fixed θ , we have Then (2.7) Next, we will show that We compute Firstly, we calculate I:

By Lemma 2.3 we can see easily that
It follows from the fact μ ≥ 1 (see [9]) and the above Hence, we infer from the above and Corollary 2.5 Next, we calculate the term II. By the definition we have where ∂ V k ∂θ is the derivative in R n . This leads to Hence, we have For a given R > 0, set It is easy to see that Then It follows from Lemma 2.6 and the above inequality that there holds Lastly, we consider the term III. It is easy to check that So, there exists a constant C such that This leads to Thus, we obtain the desired estimate and finish the proof.
Since that (N , h) is a complete Riemannian manifold with Ric N ≥ λ > 0, then, the well-known Myers theorem tells us that the diameter of (N , h) satisfies and any geodesic γ lying on (N , h) is unstable if its length l(γ ) satisfies Hence, for any given positive number a such that a ≥ l N + 2 and any geodesic γ lying on (N , h) which is parameterized by arc-length in [0, a], there always exists a vector field V 0 (s), which is smooth on γ , and 0 on γ | [0, ] and γ | [a− , a] , such that the second variation of the length of γ satisfies Proof Since the limiting neck of {u k } is a geodesic of infinite length, then, for given t 1 , the above arguments in Lemma 2.7 tell us that we can always choose a suitable positive constant 1 such that, as k is large enough, the arc length a of u * k (s) on D r t 1 Therefore, there exists a section V 1 By the same method, for t 2 = t 1 + 2 1 , we can also pick 2 > 0 and construct a section V 2 k , which is 0 outside D r t 2 Since the limiting neck is a geodesic of infinite length, then, when k is sufficiently large, there exists i k with i k → ∞ such that we can construct by the same way as above a series of sections {V 3 k , V 4 k , . . . , V i k k } satisfying that for any 1 ≤ i ≤ i k there holds true Obviously, V 1 k , V 2 k , . . ., V i k k are linearly independent. This means that Index(E α k (u k )) ≥ i k .
Thus, we complete the proof of the lemma.
The proof of Theorem 1.1 Obviously, Theorem 1.1 is just a direct corollary of the above lemma.

The Proofs of Theorem 1.2
From the arguments and the Appendix in [2] we know that one only need to consider the convergence behavior of harmonic map sequences from two dimensional flat cylinders, although the original harmonic map sequence is from a sequence of hyperbolic or flat closed Riemann surfaces respectively. First, we recall some fundamental notions such as the index of a harmonic map with respect to the energy functional.
Let T k → ∞ be a series of positive numbers and u : It is well-known that the second variational formula of energy functional E is the following: Let (L) denote the linear space of the smooth sections of L. Then, the index of u is just the maximal dimension of a linear subspace of (L) on which the above is negative definite. Let u k be an harmonic map from (−T k , T k ) × S 1 into (N , h). We assume that, for any t k ∈ (−T k , T k ), Moreover, we assume that u k ((−T k , T k ) × S 1 ) converges to an infinite length geodesic. By the arguments in [2], we can see easily that Theorem 1.2 in this paper can be deduced from the following lemma: . .} be a harmonic map sequence such that for any t k ∈ (−T k , T k ), there holds true |∇u k (θ, t k )| → 0. If Ric N ≥ λ > 0 and u k ((−T k , T k ) × S 1 ) converges to an infinite length geodesic, then the index of u k tends to infinity.
In order to prove Proposition 3.1, we need to recall some known results which were established in [2]. We first recall a useful observation in [19].
Next, we recall some known results proved in [2], which are used in the following arguments. N , k = 1, 2, . . .} be a sequence of harmonic maps such that for any t k ∈ (−T k , T k ), there holds true |∇u k (θ, t k )| → 0. Assume that u k ((−T k , T k ) × S 1 ) converges to an infinite length geodesic. Then, as k → 0, we have As in [2], we introduce the following sequence of curves in R K defined by Obviously, these curves are smooth. Now, for each k, let s be the arc-length parametrization of the the curve u * k (t) with s(0) = 0. By the arguments in [

Here C(a) is a positive constant which depends on a.
Now we turn to the discussions on the asymptotic behavior of the index and the second variation of the energy of u k . Since Ric N ≥ λ > 0, by the well-known Myers theorem we know that, if a ≥ π λ(n − 1) −1 + 2 , then there exists a tangent vector field V 0 (s) on N , which is smooth on γ , and 0 on γ | [0, ] and γ | [a− ,a] , such that the second variational of length of γ satisfies Following the arguments in Sect. 2, we can see easily that the conclusions in Proposition 3.1 are implied by the following Lemma.
Proof Let P be projection from T R K to T N. We define V k (t, θ) = P u k (s,θ ) (V 0 (s)), where s is the arc-length parametrization of u * k (t) with s(0) = 0. Then, V k is a smooth section of u −1 k (T N) which is supported in [0, t a k ] × S 1 . By Corollary 3.7, for any fixed θ , we have Next, we will show On the other hand, we have Hence, it follows In view of Lemma 3.5, we concludes lim k→∞ 1 √ |Re β(u k )| II = 0.
Immediately, it follows Hence, for k large enough, we have the desired inequality Thus, we complete the proof of this lemma.