Bubbling analysis for approximate Lorentzian harmonic maps from Riemann surfaces

For a sequence of approximate harmonic maps (un,vn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(u_n,v_n)$$\end{document} (meaning that they satisfy the harmonic system up to controlled error terms) from a compact Riemann surface with smooth boundary to a standard static Lorentzian manifold with bounded energy, we prove that identities for the Lorentzian energy hold during the blow-up process. In particular, in the special case where the Lorentzian target metric is of the form gN-βdt2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_N -\beta dt^2$$\end{document} for some Riemannian metric gN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_N$$\end{document} and some positive function β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} on N, we prove that such identities also hold for the positive energy (obtained by changing the sign of the negative part of the Lorentzian energy) and there is no neck between the limit map and the bubbles. As an application, we complete the blow-up picture of singularities for a harmonic map flow into a standard static Lorentzian manifold. We prove that the energy identities of the flow hold at both finite and infinite singular times. Moreover, the no neck property of the flow at infinite singular time is true.


Introduction
Harmonic maps constitute one of the fundamental objects in the field of geometric analysis. When the domain is two-dimensional, particularly interesting features arise. The conformal invariance of the energy functional leads to non-compactness of the set of harmonic maps in dimension two, and the blow-up behavior has been studied extensively in [5,13,20,23,24,27] for the interior case and [10,15,16] for the boundary case. Roughly speaking, the energy identities for harmonic maps tell us that, during the weak convergence of a sequence of harmonic maps, the loss of energy is concentrated at finitely many points and can be quantized by a sum of energies of harmonic spheres and harmonic disks. Also for many other elliptic and parabolic nonlinear variational problems arising in geometry and physics, such as Jholomorphic curves or Yang-Mills fields, to understand the convergence properties of a sequence and the emergence of singularities is of special importance.
In physics, harmonic maps arise as a mathematical representation of the nonlinear sigma model and this leads to several generalizations. For example, motivated by the supersymmetric sigma model, Dirac harmonic maps where a map is coupled with a spinor field have been extensively studied. One can refer to [4,14,31] and the references therein. From the perspective of general relativity, it is also natural to generalize the target of a harmonic map to a Lorentzian manifold. Recent work on minimal surfaces in anti-de-Sitter space and their applications in theoretical physics (see e.g. Alday and Maldacena [1]) shows the importance of this extension. Geometrically, the link between harmonic maps into S 4 1 and the conformal Gauss maps of Willmore surfaces in S 3 [3] also naturally leads to such harmonic maps.
Thus, in this paper, we investigate harmonic maps from Riemann surfaces into Lorentzian manifolds. In order to gain some special structure, we consider a Lorentzian manifold N × R that is equipped with a warped product metric of the form where (R, dθ 2 ) is the 1-dimensional Euclidean space, (N , g N ) is an n-dimensional compact Riemannian manifold which by Nash's theorem can be isometrically embedded into some R K , β is a positive C ∞ function on N and ω is a smooth 1-form on N . Since N is compact, β and ω are both bounded on N . We suppose for any p ∈ N , 0 < λ 1 < β(p) < λ 2 , |ω( p)| + |∇ω( p)| + |∇β( p)| ≤ λ 2 .
A Lorentzian manifold with a metric of the form (1.1) is called a standard static manifold. For more details on such manifolds, we refer to [17,22]. Let (M, h) be a compact Riemann surface with smooth boundary ∂ M. For a map (u, v) ∈ C 2 (M, N × R) with fixed boundary data (u, v)| ∂ M = (φ, ψ), we define the functional Zhu [32] has derived the Euler-Lagrange equations for (1.2), u + A(u)(∇u, ∇u) − H = 0 in M, (1.4) div β(u)(∇v + ω i ∇u i ) = 0 in M (1.5) with the boundary data for some α ∈ (0, 1). Here A is the second fundamental form of N in R K , H is the tangential part of H = (H 1 , . . . , H K ) along the map (u, v) with Let us now recall some related results. The existence of geodesics in Lorentzian manifolds was studied in [2]. Variational methods for such harmonic maps were developed in [6,7]. Recently, [8] studied the corresponding heat flow under the assumption that ω ≡ 0 and proved the existence of a Lorentzian harmonic map in any given homotopic class under either some geometric conditions on N or a small energy condition of the initial maps. The regularity theory of Lorentzian harmonic maps was studied in [11,12,19,32].
In [9], the authors proved identities of the Lorentzian energy for a blow-up sequence of Lorentzian harmonic maps when M is a compact Riemann surface without boundary. They showed the tangential Lorentzian energy of the sequence in the neck region has no concentration by comparing the energy with piece-wise linear functions (i.e. geodesics). Then they used the Hopf differentials to control the radial Lorentzian energy.
In any case, the analysis of Lorentzian harmonic maps is more difficult than that of standard (Riemannian) harmonic maps, because one cannot no longer use positivity properties of the target metric. This is a technical reason why we restrict ourselves to standard static Lorentzian manifolds.
In this paper, we shall prove some energy identities of an approximate Lorentzian harmonic map sequence and get the no neck property during a blow-up process when M is a compact Riemann surface with boundary. We work with approximate sequences which means that we allow for error terms in the Lorentzian harmonic maps system. The reason is that this has a direct application in studying the singularities of the parabolic version, the Lorentzian harmonic map flow (see [8]). Moreover, since we assume that the domain M is a manifold with boundary, blow-up analysis on the boundary must be included in our case. Here, we will use the method of integrating by parts (cf. [20] for harmonic maps) to prove a Pohozaev type identity instead of using the Hopf differential. The Pohozaev identity method is more general and powerful than the Hopf differential method. We first prove identities for the Lorentzian energy for a blow-up sequence of approximate Lorentzian harmonic maps. Furthermore, for the special case ω ≡ 0, we show that also such identities for the positive energy and no neck properties hold.
Throughout this paper, we call a map into N × R a Lorentzian map and when we have a map into the Riemannian manifold N , we just call it a map. We first give the definition of an approximate Lorentzian harmonic map.
Now we can present our first main result.
After taking a subsequence, still denoted by {u n , v n }, we can find a finite set S = {p 1 , . . . , p I } and a limit map Here and in the sequel, "finite" includes "possibly empty", that is, singularities need not always arise. Since this is obvious, it will not be explicitly mentioned.
When ω ≡ 0, the equations for Lorentzian harmonic maps become and B is the tangential part of B along the map u. In this case, the blow-up behavior is simpler. We show that the identities for the positive energy hold and there is no neck during the process.
) is a connected set in N .
As an application of Theorem 1.2, we consider a harmonic map heat flow with the boundary-initial data This kind of harmonic map heat flow is a parabolic-elliptic system and was first studied in [8].
We proved the problem (1.14) and (1.15) admits a unique solution (u, v) ∈ V(M T 1 0 ; N × R) (see the notation at the end of this section), where T 1 is the first singular time and some bubbles (nontrivial harmonic spheres) split off at t = T 1 . In this paper, we complete the blow-up picture at the singularities of this flow. First, we have When the flow blows up at finite time, we have (1.14) and (1.15) with T 1 < ∞ as its first singular time. Then there exist finitely many harmonic spheres 20) where (u(T 1 ), v(T 1 )) is the weak limit of (u(t), v(t)) in W 1,2 (M) as t → T 1 .
The paper is organized as follows. In Sect. 2, we derive some basic lemmas including the small energy regularity, a Pohozaev type identity and a removable singularity result. In Sect. 3, we prove the energy identities and no neck property for a sequence of approximate Lorentzian harmonic maps (Theorems 1.1, 1.2). In Sect. 4, we apply these two results to the harmonic map heat flow and prove Theorems 1.3 and 1.4. Throughout this paper, we use C to denote a universal constant and denote D 1 (0) := {(x, y) ∈ R 2 ||x| 2 + |y| 2 ≤ 1},

some basic lemmas
In this section, we will prove some basic lemmas for Lorentzian harmonic maps, such as the small energy regularity, a Pohozaev type identity and a removable singularity result. First, we present two small energy regularity lemmas corresponding to the interior case and the boundary case. For harmonic maps, such results have been obtained in [5,27] for the interior case and in [10,15,16] for the boundary case. We usē to denote the average value of a function u on the domain . Here and in the sequel, we shall view (φ, ψ) as the restriction of some C 2+α (M, N × R) map on ∂ M and for simplicity, we still denote it by (φ, ψ).
Moreover, by the Sobolev embedding W 2, p → C 0 , we have For the boundary case, we have On the boundary we assume that u| ∂ 0 D + = φ(x) and Moreover, by the Sobolev embedding W 2, p → C 0 , we have Since the proof of the interior case is similar to and simpler than that of the boundary case, we only prove Lemma 2.2 and omit the proof of Lemma 2.1.
Proof Without loss of generality, we assumeφ Similarly, First we assume that 1 < p < 2. By standard elliptic estimates and Poincare's inequality, we obtain Thus we have proved the lemma for the case 1 < p < 2.
If p = 2, one can derive the above estimate for p = 4 3 at first. Such an estimate implies that ∇u and ∇v are bounded in L 4 (D + 3/4 ). Then one can apply the W 2,2 −boundary estimate to the equation and get the conclusion of the lemma with p = 2.
For an approximate Lorentzian harmonic map, we can prove the following Pohozaev type identity which is useful in the blow-up analysis. This kind of equality was first introduced in [20] for the interior case of harmonic maps and extended in [10,15,16] for some boundary cases.

Lemma 2.3 Let D ⊂ R 2 be the unit disk and
where (r, θ) are polar coordinates in D centered at 0. Since we use the Euclidean metric, we have that the covariant derivative ∇ r u equals to ∂u ∂r and we denote them with a unified notation ∂u ∂r or just u r for brevity. Proof Multiplying (1.8) by r (v r + ω i u i r ) and integrating by parts, we get By direct computations, noting that Similarly, multiplying (1.7) by ru r and integrating by parts, we get Noting that and combining (2.4) with (2.2) and (2.3), we obtain the conclusion of the lemma.
By Hölder's inequality and integrating (2.1) about ρ from r 0 to 2r 0 , we get For the boundary case, we have where (r, θ) are polar coordinates in D centered at 0.
Proof The proof is similar to the proof of Lemma 2.3.
Multiplying (1.8) by r ( v r + ω i u i r ) and integrating by parts, we get By direct computations, we have Noting that Similarly, multiplying (1.7) by r u r and integrating by parts, we get Combining (2.4) with (2.6) and (2.7), we obtain the conclusion of the lemma.
Proof By Hölder's inequality, it is easy to find that the right hand side of (2.5) is bounded by . Then the conclusion of the corollary follows by an integration about ρ from r 0 to 2r 0 .
Similar as for harmonic maps into a Riemannian manifold, there is also an energy gap for a nontrivial Lorentzian harmonic map.
then (u, v) is a constant map. Here S 2 denotes the unit sphere in R 3 .
Proof One can find the proof of the theorem in [9] for the case (u, v) : S 2 → N × R. By Eqs. (1.4) and (1.5), we have The standard elliptic theory tells us that .
It is easy to get that, if 0 is small enough, (u, v) must be a constant map. If (u, v) : R 2 + → N × R is a smooth Lorentzian harmonic map with constant Dirichlet boundary condition, choosing 0 ≤ 2 where 2 is the positive constant in Lemma 2.2, then by Lemma 2.2 (taking (φ, ψ) = constant, (τ, κ) = 0 and any constant p > 2) and Sobolev embedding, for any R > 0, we have Sending R to infinity yields that (u, v) must be a constant map.
It is necessary for the singularities to be removable during the blow-up process. Removability of singularities for a Lorentzian harmonic map (i.e. τ = κ = 0) is proved in [9]. By assuming additionally that ω ≡ 0, for an approximate Lorentzian harmonic map (i.e. (τ, κ) = 0) with singularities either in the interior or on the boundary, we can also remove them. For an approximate Lorentzian harmonic map then (u, v) can also be extended to D + in W 2,2 (D + ).
Proof We prove the theorem for the boundary case and the interior case can be proved similarly.
On the one hand, it is easy to see that (u, v) is a weak solution of (1.7) and (1.8). By Theorem 1.2 in [30] which is developed from the regularity theory for critical elliptic systems with an anti-symmetric structure in [25,26,28,29,32], we know that v ∈ W 2, p (D + ρ (0)) for some ρ > 0 and any 1 < p < 2. In fact, the anti-symmetric term in the equation for v equals to zero. This implies that ∇v ∈ L 4 (D + ).
On the other hand, since the Eq. (1.7) can be written as an elliptic system with an antisymmetric potential ( [25]) u = · ∇u + f with ∈ L 2 (D + , so(n) ⊗ R 2 ) and f ∈ L 2 (D + ), using Theorem 1.2 in [30] again, we have u ∈ W 2, p (D + ρ (0)) for some ρ > 0 and any 1 < p < 2. Then the higher regularity can be derived by a standard bootstrap argument.

Energy identity and analysis on the neck
In this section, we shall study the behavior at blow-up points both in the interior and on the boundary for an approximate Lorentzian harmonic map sequence {(u n , v n )}. To this end, we first define the blow-up set and show that the blow-up points for such a sequence are finite in number. Throughout this section, we suppose that there exists a constant > 0 such that the sequence satisfies (3.1)   For points on the boundary of M, we can proceed similarly and finally, we get a subsequence {(u n , v n )} which converges strongly to some (u, v) in W 1,2 loc (M \ S), where S = S 1 ∪ S 2 = {p 1 , p 2 , . . . , p I } is a finite set.

Definition 3.1 For an approximate Lorentzian harmonic map sequence
We consider the case that the blow-up points are in the interior first. Since the blow-up set S 1 is finite, we can find small geodesic disks D δ i (by conformal invariance, we can assume that they are flat disks) for each blow-up point p i such that D δ i ∩ D δ j = ∅ for i = j, i, j = 1, 2, . . . , I , and on M \ ∪ I i=1 D δ i , (u n , v n ) converges strongly to a limit map (u, v). Without loss of generality, we discuss the case that there is only one blow-up point 0 ∈ D 1 (0) in S 1 and the sequence {(u n , v n )} satisfies that there is some (u, v) such that Proof According to the standard induction argument in [5], we can assume that there is only one bubble at the singular point 0 ∈ D 1 (0). To prove (3.5) is equivalent to prove that there exists a Lorentzian harmonic sphere (σ, ξ ) such that By the standard argument of blow-up analysis, for any n, there exist sequences x n → 0 and r n → 0 such that E(u n , v n ; D r n /2 (x n )) = sup x∈D δ ,r ≤r n D r (x)⊂D δ E(u n , v n ; D r/2 (x)) = 1 8 .
Without loss of generality, we may assume that x n = 0 and denoteũ n = u n (r n x),ṽ n = v n (r n x). Then we have E(ũ n ,ṽ n ; D 1/2 ) = E(u n , v n ; D r n /2 ) = 1 8 < 1 (3.8) and E(ũ n ,ṽ n ; D R ) = E(u n , v n ; D r n R ) < .
By (3.7), we can apply Lemma 2.1 on D R for {(ũ n ,ṽ n )} and get that {(ũ n ,ṽ n )} converges strongly to some Lorentzian harmonic map (σ, ξ ) in W 1,2 (D R , N × R) for any R ≥ 1. By stereographic projection and the removable singularity theorem [9], we get a nonconstant harmonic sphere (σ, ξ ). Thus we get the first bubble at the blow-up point and to prove (3.6) is equivalent to prove that Since we assume that there is only one bubble, we have that, for any > 0, there holds that as n → ∞, R → ∞ and δ → 0. Otherwise, we will get a second bubble and this is a contradiction to the assumption that L = 1. One can refer to [5,15,31] for details of this kind of arguments. Then by Lemma 2.1 and a standard scaling argument, for any ρ ∈ [r n R, δ 2 ], we have By (3.11), we know that and similarly, Then we get by integrating by parts that Since (u n , v n ) is an approximate harmonic map, we have (3.14) Then we get from (3.12), (3.13) and (3.14) that By Lemma 2.1 and the trace theory, we obtain for the boundary term in (3.15) that Similarly, Combining these, we have Similarly, we can obtain that Without loss of generality, we may assume δ = 2 m n r n R, where m n is a positive integer which tends to ∞ as n → ∞. By Corollary 2.4, for i = 0, 1, . . . , m n − 1, we have Since m n −1 i=0 2 i r n R = 2 m n r n R = δ, (3.17) from which (3.9) follows immediately.
When the 1-form ω ≡ 0, the behavior of the sequence at the blow-up points is clearer. In fact, we can get identities for the positive energy E instead of for the Lorentzian energy E g and there is no neck between the limit map and the bubbles. More precisely, we have Lemma 3.3 Assume that {(u n , v n )} is an approximate Lorentzian harmonic map sequence as in Lemma 3.2 and additionally, we assume that ω ≡ 0 and ∇v n L p ≤ for some p > 2, then we have that σ i : R 2 ∪ {∞} → N is a nontrivial harmonic sphere, ξ i is a constant map and (3.5) becomes is a connected set.
Proof Similar to the proof of Lemma 3. Since ∇v n L p (D) ≤ for some p > 2, we get Since ω ≡ 0, (3.17) implies that where 0 ≤ s 0 ≤ m n and 0 ≤ s ≤ min{s 0 , m n − s 0 }. Integrating by parts, we get By (3.12) and (3.14), we obtain We deduce from Corollary 2.4 that For the boundary term in (3.25), by Hölder's inequality and Poincare's inequality, we have Similarly, we also have Taking and δ sufficiently small, we get from (3.25) that where we can take C sufficiently large such that 1 − 2 p − 1 C > 0. Integrating from 2 to L, we arrive at Let s 0 = i and L = L i := min{i, m n − i}. Noting that Q(L i ) ⊂ D δ \ D r n R , we have where the last inequality follows from the energy identity (3.21). By using Lemma 2.1, now it is easy to deduce (3.24) from (3.22) and the above estimates (3.26) for energy decay.
For the case that the blow-up point is on the boundary of the manifold, the behavior is similar to those in Lemmas 3.2 and 3.3. But the analysis is more complicated. More precisely, we consider an approximate Lorentzian harmonic map sequence for some 0 < α < 1 which satisfies that (∇u n , ∇v n ) L 2 (D + 1 (0)) + (τ n , κ n ) L 2 (D + 1 (0)) ≤ . (3.28) Without loss of generality, we still suppose that there is only one blow-up point 0 ∈ D + 1 (0) and the sequence {(u n , v n )} satisfies that there is some (u, v) such that For such a sequence, we have be a sequence of approximate Lorentzian harmonic maps satisfying (3.27), (3.28) and (3.29). Up to a subsequence which is still denoted by {(u n , v n )}, we can find a positive integer L, points x i n ∈ D + 1 (0) and r i n > 0 satisfying x i n → 0 and r i n → 0, i = 1, . . . , L as n → ∞ and both of the following two cases may appear during the blow-up process.
there is a nontrivial Lorentzian harmonic sphere (σ i , ξ i ) : R 2 ∪{∞} → N ×R which is the weak limit of (u n (x i n +r i n x), v n (x i n +r i n x)) in W 1,2 loc (R 2 ).
Furthermore, for both of the two cases, there holds the energy identity Here, L just stands for a nonnegative integer which may different from the constant in Lemma 3.2.
Proof Similar to what we have done in the proof of Lemma 3.2, for any n, there exist sequences x n → 0 and r n → 0 such that We have that either lim sup n→∞ d n r n < ∞ or lim sup n→∞ d n r n = ∞. We discuss these two cases respectively. Case (a) lim sup n→∞ d n r n < ∞. By taking a subsequence, we may assume that lim n→∞ d n r n = a ≥ 0. Denote We have that as n → ∞, It is easy to get that ( u n , v n ) : B n → N × R is an approximate Lorentzian harmonic map with ( τ n , κ n ) = r 2 n (τ n , κ n ) and ( u n (x), v n (x)) = (ϕ(x n + r n x), ψ(x n + r n x)), if x n + r n x ∈ ∂ 0 D + .

Lemma 2.2 and (3.31) tell us that for any
By a similar argument as in Section 4 of [8], after taking a subsequence of ( u n , v n ) if necessary (still denoted by ( u n , v n )), there is a Lorentzian harmonic map ( u, v) ∈ W 1,2 (R 2 a , N × R) with the constant boundary condition ( u, v)| ∂R 2 a = (φ(0), ψ(0)) such that, for any R > 0, lim In this case, ( u n , v n ) lives in B n which tends to R 2 as n → ∞. Moreover, for any x ∈ R 2 , when n is sufficiently large, by (3.31), we have E( u n , v n ; D 1 (x)) ≤ 1 8 .
According to Lemma 2.1, there exist a subsequence of ( u n , v n ) which is still denoted by ( u n , v n )) and a Lorentzian harmonic By Theorem 2.8,( u,v) can be extended to a Lorentzian harmonic sphere and (3.31) tells us that it is nontrivial. We call the Lorentzian harmonic map ( u, v) obtained in these two cases the first bubble. Without loss of generality, we assume that there is only one bubble at the blow-up point 0 ∈ D + 1 (0). Under this assumption, similar to (3.10), we have that, for any > 0, there exist constants δ > 0 and R > 0 such that when n is large enough. Now to prove the energy identity (3.30) is equivalent to prove We shall prove (3.33) for the two cases respectively. For case (a) lim n→∞ d n r n = a < ∞. For n and R are sufficiently large, we decompose the neck domain D + δ (x n ) \ D + r n R (x n ) into three parts which follows the decomposition in [15,16].
Here x n ∈ ∂ 0 D + is the projection of x n , i.e. d n = |x n − x n |.
Since lim n→∞ d n r n = a, when n and R are large enough, it is easy to get that Moreover, for any ρ ∈ [r n R, δ 2 ], there holds Then we get from (3.32) that E(u n , v n ; 1 ) + E(u n , v n ; 3 ) ≤ 2 (3.34) and E(u n , v n ; D + 2ρ (x n ) \ D + ρ (x n )) ≤ 2 for any ρ ∈ r n R, By Lemma 2.1, we have For case (b) lim n→∞ d n r n = ∞. The result for this case can be derived from case (a) and Lemma 3.2. In fact, in this case, for n sufficiently large, we decompose the neck domain D + δ (x n ) \ D + r n R (x n ) as in [15,16] as follows (3.43) Since lim n→∞ d n = 0 and lim n→∞ d n r n = ∞, when n is large enough, it is easy to get that Moreover, for any ρ ∈ [d n , δ 2 ], there holds Noting that 4 To estimate the energy concentration in 2 , we can use the same arguments as for case (a) to get that E g (u n , v n ; 2 ) ≤ C( + δ). (3.44) Thus we finish the proof of the lemma.
Furthermore, the image is a connected set.
Proof We use the same symbols as in Lemma 3.4. First, let us show that if ω ≡ 0, Case (a) will not happen. In fact, since v satisfies div(β( u)∇ v) = 0 in R 2 a and v| ∂R 2 a ≡ ψ(0), v must be a constant map. Thus, u is a harmonic map from R 2 a with constant boundary data u| ∂R 2 a = φ(0) which implies that u is a constant map [18]. This is a contradiction with E( u, v; R 2 a ) ≥ 1 8 min{ 1 , 2 }. For case (b), when ω ≡ 0, it is clear that v satisfies the equation div(β( u)∇ v) = 0 in S 2 with finite energy ∇ v L 2 (S 2 ) ≤ C which implies that v must be a constant map. Therefore u : S 2 → N is a nontrivial harmonic sphere. Now to prove the energy identities when n, R are large and δ is small, which implies that there is no neck on 1 ∪ 3 . Moreover, for any d n ≤ ρ ≤ δ 2 , there holds when n is big enough, then (3.32) tells us |∇u n | 2 dx ≤ 2 for any ρ ∈ (d n , δ 2 ).
Combining this with Lemma 2.2, we get for any ρ ∈ (d n , δ 2 ). Noting that 4 = D + d n (x n ) \ D + r n R (x n ) = D d n (x n ) \ D r n R (x n ), the proofs of (3.48) and To prove that there is no energy loss on 2 , noting that ∇v n L p (D + ) ≤ for some p > 2, we get D + δ (x n )\D + rn R (x n ) |∇v n | 2 dx ≤ Cδ Integrating from 2 to L, we arrive at The rest proof is the same as the proof in Lemma 3.3. Thus we finish the analysis of energy loss and no neck property on 1 ∪ 3 , 4 and 2 and get (3.48) and (3.49).
We can now prove Theorems 1.1 and 1.2. Then, Theorem 1.2 is a direct conclusion of Lemmas 3.3 and 3.5.

Applications to the Lorentzian harmonic map flow
At the beginning of this section, let us recall a lemma in [8] which is useful in this part.