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Bubbling analysis for approximate Lorentzian harmonic maps from Riemann surfaces

  • Xiaoli Han
  • Jürgen JostEmail author
  • Lei Liu
  • Liang Zhao
Open Access
Article

Abstract

For a sequence of approximate harmonic maps \((u_n,v_n)\) (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 \(g_N -\beta dt^2\) for some Riemannian metric \(g_N\) and some positive function \(\beta \) 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.

Mathematics Subject Classification

53C43 58E20 

1 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 J-holomorphic 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 \({\mathbb {S}}_1^4\) and the conformal Gauss maps of Willmore surfaces in \({\mathbb {S}}^3\) [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 \(N\times {\mathbb {R}}\) that is equipped with a warped product metric of the form
$$\begin{aligned} g=g_N-\beta (d\theta +\omega )^2, \end{aligned}$$
(1.1)
where \(({\mathbb {R}}, d\theta ^2)\) 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 \({\mathbb {R}^K}\), \(\beta \) is a positive \(C^{\infty }\) function on N and \(\omega \) is a smooth 1-form on N. Since N is compact, \(\beta \) and \(\omega \) are both bounded on N. We suppose for any \(p\in N\),
$$\begin{aligned} 0<\lambda _1<\beta (p)<\lambda _2,\ |\omega (p)|+|\nabla \omega (p)|+|\nabla \beta (p)|\le \lambda _2. \end{aligned}$$
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 (Mh) be a compact Riemann surface with smooth boundary \(\partial M\). For a map \((u,v)\in C^2(M,N\times {\mathbb {R}})\) with fixed boundary data \((u,v)|_{\partial M}=(\phi ,\psi )\), we define the functional
$$\begin{aligned} E_g(u,v)= \frac{1}{2}\int _{M} \left\{ |\nabla u|^2- \beta (u)|\nabla v+\omega _i(u)\nabla u^i|^2 \right\} dv_h, \end{aligned}$$
(1.2)
which is called the Lorentzian energy of the map (uv) on M. Critical points (uv) in \(C^2(M,N\times {\mathbb {R}})\) of the functional (1.2) are called Lorentzian harmonic maps from (Mh) into the Lorentzian manifold \((N\times {\mathbb {R}},g)\). Besides the Lorentzian energy \(E_g(u,v)\), we also consider
$$\begin{aligned} E(u)= \frac{1}{2}\int _{M} |\nabla u|^2 dv_h\ \ {\text {and}}\ \ E(u,v)= \frac{1}{2}\int _{M} \left\{ |\nabla u|^2+|\nabla v|^2 \right\} dv_h \end{aligned}$$
(1.3)
and call it the positive energy of u and (uv) on M respectively. It is obvious that both the Lorentzian and positive energy functionals are conformally invariant when \(\dim M=2\).
Zhu [32] has derived the Euler–Lagrange equations for (1.2),
$$\begin{aligned} \Delta u +A(u)(\nabla u,\nabla u)-H^{\top }&=0\ \text {in}\ M, \end{aligned}$$
(1.4)
$$\begin{aligned} div\left( \beta (u)(\nabla v+\omega _i\nabla u^i)\right)&=0\ \text {in}\ M \end{aligned}$$
(1.5)
with the boundary data
$$\begin{aligned} (u,v)|_{\partial M}=(\phi ,\psi )\in C^{2+\alpha }(\partial M,N\times {\mathbb {R}}) \end{aligned}$$
(1.6)
for some \(\alpha \in (0,1)\). Here A is the second fundamental form of N in \({\mathbb {R}}^K\), \(H^{\top }\) is the tangential part of \(H=(H^1,\ldots ,H^K)\) along the map (uv) with
$$\begin{aligned} H^j:=\beta (\nabla v+\omega _i\nabla u^i)\cdot \nabla u^k\left( \frac{\partial \omega _j}{\partial y^k}-\frac{\partial \omega _k}{\partial y^j}\right) -\frac{1}{2}\frac{\partial \beta }{\partial y^j}|\nabla v+\omega _i\nabla u^i|^2\ , j=1,\ldots ,K. \end{aligned}$$
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 \(\omega \equiv 0\) 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 \(\omega \equiv 0\), we show that also such identities for the positive energy and no neck properties hold.

Throughout this paper, we call a map into \(N\times \mathbb {R}\) 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.

Definition 1.1

\((u,v)\in W^{2,2}(M,N\times {\mathbb {R}})\) is called an approximate Lorentzian harmonic map with Dirichlet boundary data \((\phi ,\psi )\), if there exist fields \((\tau (u,v),\kappa (u,v))\in L^1(M)\) such that (uv) satisfies
$$\begin{aligned} \Delta u +A(u)(\nabla u,\nabla u)-H^{\top }&=\tau (u,v)\ \text {in}\ M, \end{aligned}$$
(1.7)
$$\begin{aligned} div\left( \beta (u)(\nabla v+\omega _i\nabla u^i)\right)&=\kappa (u,v) \ \text {in}\ M \end{aligned}$$
(1.8)
with the boundary condition \((u,v)|_{\partial M}=(\phi ,\psi )\).

Now we can present our first main result.

Theorem 1.1

Let \((u_n,v_n)\in W^{2,2}(M, N\times \mathbb {R})\) be a sequence of approximate harmonic maps with Dirichlet boundary \((u_n,v_n)|_{\partial M}=(\phi ,\psi )\in C^{2+\alpha }(\partial M,N\times {\mathbb {R}})\) satisfying
$$\begin{aligned} E(u_n,v_n)+\Vert (\tau _n,\kappa _n)\Vert _{L^2(M)}\le \Lambda <\infty , \end{aligned}$$
where \(\Vert (\tau _n,\kappa _n)\Vert _{L^2(M)}^2=\Vert \tau _n\Vert ^2_{L^2(M)}+\Vert \kappa _n\Vert _{L^2(M)}^2\). After taking a subsequence, still denoted by \(\{u_n, v_n\}\), we can find a finite set \(\mathcal {S}=\{p_1,\ldots ,p_I\}\) and a limit map \((u_0,v_0)\in W^{1,2}(M,N\times \mathbb {R})\) with Dirichlet boundary data \((u_0,v_0)|_{\partial M}=(\phi ,\psi )\) such that \(\{(u_n,v_n)\}\) converges weakly in \(W^{2,2}_{loc}(M \setminus \mathcal {S})\) to \((u_0,v_0)\). Moreover, there are finitely many nontrivial Lorentzian harmonic spheres \((\sigma _i^l,\xi _i^l):{\mathbb {S}}^2\rightarrow N\times \mathbb {R}\) and nontrivial Lorentzian harmonic maps \((\sigma _i^k,\xi _i^k):\mathbb {R}^2_+:=\{(x^1,x^2)\in \mathbb {R}^2|x^2\ge 0\}\rightarrow N\times \mathbb {R}\) with constant boundary values, where \(i=1,\ldots ,I\), \(l=1,\ldots ,l_i\) and \(k=1,\ldots ,k_i\) with \(l_i,k_i\ge 0\) and \(l_i+k_i\ge 1\), such that
$$\begin{aligned} \lim _{n\rightarrow \infty }E_g(u_n,v_n)=E_g(u_0,v_0)+\sum _{i=1}^I\sum _{l=1}^{l_i}E_g(\sigma ^l_i,\xi ^l_i) +\sum _{i=1}^I\sum _{k=1}^{k_i}E_g(\sigma ^k_i,\xi ^k_i). \end{aligned}$$
(1.9)

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 \(\omega \equiv 0\), the equations for Lorentzian harmonic maps become
$$\begin{aligned} \Delta u +A(u)(\nabla u,\nabla u)-B^\top (u)|\nabla v|^2&=0\ \text {in}\ M, \end{aligned}$$
(1.10)
$$\begin{aligned} div(\beta (u)\nabla v)&=0\ \text {in}\ M \end{aligned}$$
(1.11)
where \(B(u):=(B^1, B^2, \ldots , B^K)\) with
$$\begin{aligned} B^j:=-\frac{1}{2}\frac{\partial \beta (u)}{\partial y^j} \end{aligned}$$
and \(B^{\top }\) 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.

Theorem 1.2

If we additionally assume \(\omega \equiv 0\) in Theorem 1.1, there cannot emerge any Lorentzian harmonic maps \((\sigma _i^k,\xi _i^k):\mathbb {R}^2_+:=\{(x^1,x^2)\in \mathbb {R}^2|x^2\ge 0\}\rightarrow N\times \mathbb {R}\) during the blow-up process [i.e. \(k_i=0\) in (1.9)]. Moreover, the components \(\xi _i^l\) of the maps \((\sigma ^l_i,\xi _i^l)\) are constant and \(\sigma ^l_i:{\mathbb {S}}^2\rightarrow N\), \(1\le l\le l_i\) are nontrivial harmonic spheres. In this case, (1.9) becomes
$$\begin{aligned} \lim _{n\rightarrow \infty }E(u_n)= & {} E(u_0)+\sum _{i=1}^I\sum _{l=1}^{l_i}E(\sigma ^l_i), \end{aligned}$$
(1.12)
$$\begin{aligned} \lim _{n\rightarrow \infty }E(v_n)= & {} E(v_0). \end{aligned}$$
(1.13)
and the image \(u_0(M)\cup _{i=1}^I\cup _{l=1}^{l_i}(\sigma ^l_i({\mathbb {S}}^2))\) is a connected set in N.
As an application of Theorem 1.2, we consider a harmonic map heat flow
$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u=\Delta u+ A(u)(\nabla u,\nabla u)-B^\top (u)|\nabla v|^2,\ &{}in\ M\times [0,T) \\ -div (\beta (u)\nabla v)=0, \ &{}in\ M\times [0,T) \end{array}\right. } \end{aligned}$$
(1.14)
with the boundary-initial data
$$\begin{aligned} {\left\{ \begin{array}{ll} u(x,t)=\phi _0(x)\ &{}on\ M\times \{t=0\},\\ u(x,t)=\phi (x),\ &{}on\ \partial M\times \{t>0\},\\ v(x,t)=\psi (x),\ &{}on \ \partial M\times \{t>0\},\\ \phi _0(x)=\phi (x)\ &{}on \ \partial M. \end{array}\right. } \end{aligned}$$
(1.15)
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 \((u,v)\in \mathcal {V}(M_0^{T_1};N\times \mathbb {R})\) (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

Theorem 1.3

Suppose the problem (1.14) and (1.15) admits a unique global solution \((u,v)\in \mathcal {V}(M_0^{\infty };N\times \mathbb {R})\) which blows up at infinity, i.e. \(T_1=\infty \). By choosing some \(t_n\rightarrow \infty \), there exist a smooth Lorentzian harmonic map \((u_\infty ,v_\infty ):M\rightarrow N\times \mathbb {R}\) with boundary data \((u_\infty ,v_\infty )|_{\partial M}=(\phi ,\psi )\) and finitely many harmonic spheres \(\{\sigma ^i\}_{i=1}^L:\mathbb {R}^2\cup \{\infty \} \rightarrow N\) such that
$$\begin{aligned} \lim _{n\rightarrow \infty }E(u(t_n);M)&=E(u_\infty ,M)+\sum _{i=1}^LE(\sigma ^i), \end{aligned}$$
(1.16)
$$\begin{aligned} \lim _{t\rightarrow \infty }E_g(u(t),v(t);M)&=E_g(u_\infty , v_\infty ;M)+\sum _{i=1}^LE(\sigma ^i). \end{aligned}$$
(1.17)
Furthermore, there exist sequences \(\{x_n^i\}_{i=1}^L\subset M\) and \(\{r_n^i\}_{i=1}^L\subset \mathbb {R}_+\) such that
$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert u(\cdot ,t_n)-u_\infty (\cdot )-\sum _{i=1}^L\sigma ^i_n(\cdot )\Vert _{L^\infty (M)} =0, \end{aligned}$$
(1.18)
where \(\sigma ^i_n(\cdot )=\sigma ^i\left( \frac{\cdot -x^i_n}{r^i_n}\right) -\sigma _i(\infty )\).

When the flow blows up at finite time, we have

Theorem 1.4

Let \((u,v)\in \mathcal {V}(M_0^{T_1};N\times \mathbb {R})\) be a solution to (1.14) and (1.15) with \(T_1<\infty \) as its first singular time. Then there exist finitely many harmonic spheres \(\{\sigma ^i\}_{i=1}^L:\mathbb {R}^2\cup \{\infty \}\rightarrow N\) such that
$$\begin{aligned} \lim _{t\nearrow T_1}E(u(t);M)&=E(u(T_1),M)+\sum _{i=1}^LE(\sigma ^i), \end{aligned}$$
(1.19)
$$\begin{aligned} \lim _{t\nearrow T_1}E_g(u(t),v(t);M)&=E_g(u(T_1), v(T_1);M)+\sum _{i=1}^LE(\sigma ^i), \end{aligned}$$
(1.20)
where \((u(T_1),v(T_1))\) is the weak limit of (u(t), v(t)) in \(W^{1,2}(M)\) as \(t\rightarrow T_1\).

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 \(D_1(0):=\{(x,y)\in \mathbb {R}^2||x|^2+|y|^2\le 1\}\), \(D_1^+(0):=\{(x,y)\in \mathbb {R}^2||x|^2+|y|^2\le 1, y\ge 0\}\), \(\partial ^0D^+_1(0):=\{(x,y)\in D^+_1(0)|y=0\}\) and \(\partial ^+D^+_1(0):=\{(x,y)\in \partial D^+_1(0)|y>0\}\).

Notation We denote
$$\begin{aligned} \mathcal {V}(M_s^t;N\times \mathbb {R}):=&\,\left\{ (u,v):M\times [s,t)\rightarrow N\times \mathbb {R},\ v\in L^\infty ([s,t);C^{2+\alpha }(M)),\right. \\&v,\nabla v\in \cap _{s<\rho<t} C^{\alpha ,\alpha /2}(M\times [s,\rho ]),\\&\left. u\in \cap _{s<\rho <t} C^{2+\alpha ,1+\alpha /2}(M\times [s,\rho ])\right\} . \end{aligned}$$

2 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 use
$$\begin{aligned} \bar{u}_\Omega =\frac{1}{|\Omega |}\int _\Omega u dx \end{aligned}$$
to denote the average value of a function u on the domain \(\Omega \). Here and in the sequel, we shall view \((\phi ,\psi )\) as the restriction of some \(C^{2+\alpha }(M,N\times \mathbb {R})\) map on \(\partial M\) and for simplicity, we still denote it by \((\phi ,\psi )\).

Lemma 2.1

Let \((u,v)\in W^{2,p}(D,N\times \mathbb {R})\), \(1<p\le 2\) be an approximate Lorentzian harmonic map with \((\tau ,\kappa )\in L^p(D)\). There exist constants \(\epsilon _1=\epsilon _1(p,\lambda _1,\lambda _2,N)>0\) and \(C=C(p,\lambda _1,\lambda _2,N)>0\), such that if \(E(u,v;D)\le \epsilon _1\), we have
$$\begin{aligned}&\Vert u-\bar{u}_{D_{1/2}}\Vert _{W^{2,p}(D_{1/2})} +\Vert v-\bar{v}_{D_{1/2}}\Vert _{W^{2,p}(D_{1/2})}\\&\quad \le C(\Vert \nabla u\Vert _{L^2(D)}+\Vert \nabla v\Vert _{L^2(D)}+\Vert \tau \Vert _{L^p(D)}+\Vert \kappa \Vert _{L^p(D)}). \end{aligned}$$
Moreover, by the Sobolev embedding \(W^{2,p}\hookrightarrow C^0\), we have
$$\begin{aligned} \Vert u\Vert _{osc(D_{1/2})}&=\sup _{x,y\in D_{1/2}}|u(x)-u(y)|\le C(\Vert (\nabla u,\nabla v)\Vert _{L^2(D)}+\Vert (\tau ,\kappa )\Vert _{L^p(D)}). \end{aligned}$$

For the boundary case, we have

Lemma 2.2

Let \((u,v)\in W^{2,p}(D^+,N\times \mathbb {R})\), \(1<p\le 2\) be an approximate Lorentzian harmonic map with \((\tau ,\kappa )\in L^p(D^+)\). On the boundary we assume that \(u|_{\partial ^0D^+}=\phi (x)\) and \(v|_{\partial ^0D^+}=\psi (x)\) where \((\phi ,\psi )\in C^{2+\alpha }(D)\). There exist constants \(\epsilon _2=\epsilon _2(p,\lambda _1,\lambda _2,N)>0\) and \(C=C(p,\lambda _1,\lambda _2,N)>0\), such that if \(E(u,v;D_1^+)\le \epsilon _2\), we have
$$\begin{aligned}&\Vert u-\bar{\phi }_{\partial ^0{D^+}}\Vert _{W^{2,p}(D_{1/2}^+)}+ \Vert v-\bar{\psi }_{\partial ^0{D^+}}\Vert _{W^{2,p}(D_{1/2}^+)}\\&\le C\left( \Vert (\nabla u,\nabla v)\Vert _{L^2(D^+)}+\Vert (\nabla \phi ,\nabla \psi )\Vert _{W^{1,p}(D^+)}+\Vert (\tau ,\kappa )\Vert _{L^p(D^+)}\right) , \end{aligned}$$
where \(\bar{\phi }_{\partial ^0{D^+}}=\frac{1}{2}\int _{\partial ^0 D^+_{1}}\phi dx\) and \(\bar{\psi }_{\partial ^0{D^+}}=\frac{1}{2}\int _{\partial ^0 D^+_{1}}\psi dx\).
Moreover, by the Sobolev embedding \(W^{2,p}\hookrightarrow C^0\), we have
$$\begin{aligned} \Vert u\Vert _{osc(D_{1/2}^+)}&=\sup _{x,y\in D_{1/2}^+}|u(x)-u(y)|\\&\le C(\Vert (\nabla u,\nabla v)\Vert _{L^2(D^+)}+\Vert (\nabla \phi ,\nabla \psi )\Vert _{W^{1,p}(D^+)} +\Vert (\tau ,\kappa )\Vert _{L^p(D^+)}). \end{aligned}$$

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 assume \(\bar{\phi }_{\partial ^0{D^+}}=\bar{\psi }_{\partial ^0{D^+}}=0\). Choosing a cut-off function \(\eta \in C_0^\infty (D^+)\) satisfying \(0\le \eta \le 1,\eta |_{D^+_{3/4}}\equiv 1,|\nabla \eta |+|\nabla ^2\eta |\le C\) and computing directly, we get
$$\begin{aligned} |\Delta (\eta u)|&=|\eta \Delta u+2\nabla \eta \nabla u+u\Delta \eta |\\&\le C\left( |u|+|\nabla u|+(|\nabla u|+|\nabla v|)(|\eta \nabla u|+|\eta \nabla v|)+|\tau |\right) \\&\le C(|\nabla u|+|\nabla v|)(|\nabla (\eta u)|+|\nabla (\eta v)|)\\&\quad + C\left( |u|+(1+|v|)(|\nabla u|+|\nabla v|)+|\tau |\right) . \end{aligned}$$
Similarly,
$$\begin{aligned} |\Delta (\eta v)|&=|\eta \Delta v+2\nabla \eta \nabla v+v\Delta \eta |\\&\le C\left( |v|+|\nabla v|+(|\nabla u|+|\nabla v|)(|\eta \nabla u|+|\eta \nabla v|)+|\tau |+|\kappa |\right) \\&\le C(|\nabla u|+|\nabla v|)(|\nabla (\eta u)|+| \nabla (\eta v)|)\\ {}&\quad + C\left( |v|+(1+|v|)(|\nabla u|+|\nabla v|)+|\tau |+|\kappa |\right) . \end{aligned}$$
First we assume that \(1<p<2\). By standard elliptic estimates and Poincare’s inequality, we obtain
$$\begin{aligned}&\Vert \eta u\Vert _{W^{2,p}(D)}+\Vert \eta v\Vert _{W^{2,p}(D)}\\&\quad \le C\Vert (\nabla u,\nabla v)\Vert _{L^2(D^+)}\Vert (\nabla (\eta u),\nabla (\eta v))\Vert _{L^{\frac{2p}{2-p}}(D^+)}+C\Vert (u,v)\Vert _{W^{1,p}(D^+)}\\&\qquad +C\left( \Vert (\nabla u,\nabla v)\Vert _{L^2(D^+)}\Vert v\Vert _{L^{\frac{2p}{2-p}(D^+)}} +\Vert |(\phi ,\psi )|\Vert _{W^{2,p}(D^+)}+\Vert |(\tau ,\kappa )|\Vert _{L^{p}(D^+)}\right) \\&\quad \le C\epsilon _2\Vert (\nabla (\eta u),\nabla (\eta v))\Vert _{L^{\frac{2p}{2-p}}(D^+)}+C(\Vert (\nabla u,\nabla v)\Vert _{L^2(D^+)}+\Vert |(\nabla \phi ,\nabla \psi )|\Vert _{W^{1,p}(D^+)} \\&\qquad +\Vert |(\tau ,\kappa )|\Vert _{L^{p}(D^+)}), \end{aligned}$$
where we use the Sobolev inequality
$$\begin{aligned} \Vert v\Vert _{L^{\frac{2p}{2-p}(D^+)}}\le C(p)\Vert \nabla v\Vert _{L^2(D^+)}. \end{aligned}$$
Taking \(\epsilon _2>0\) sufficiently small, we have
$$\begin{aligned}&\Vert u\Vert _{W^{2,p}(D^+_{3/4})}+\Vert v\Vert _{W^{2,p}(D^+_{3/4})}\le \Vert \eta u\Vert _{W^{2,p}(D^+)}+\Vert \eta v\Vert _{W^{2,p}(D^+)}\\&\quad \le C\left( \Vert (\nabla u,\nabla v)\Vert _{L^2(D^+)} +\Vert |(\nabla \phi ,\nabla \psi )|\Vert _{W^{1,p}(D^+)}+\Vert |(\tau ,\kappa )|\Vert _{L^{p}(D^+)}\right) . \end{aligned}$$
Thus we have proved the lemma for the case \(1<p<2\).

If \(p=2\), one can derive the above estimate for \(p=\frac{4}{3}\) at first. Such an estimate implies that \(\nabla u\) and \(\nabla v\) 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 \(p=2\). \(\square \)

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 \(D\subset \mathbb {R}^2\) be the unit disk and \((u,v)\in W^{2,2}(D,N\times \mathbb {R})\) be an approximate Lorentzian harmonic map with \((\tau ,\kappa )\in L^2(D)\), then for any \(0<\rho <\frac{1}{2}\), we have
$$\begin{aligned}&\rho \int _{\partial D_{\rho }}\left( |u_r|^2-\beta (u)|v_r+\omega _i u_r^i|^2-\frac{1}{2}|\nabla u|^2+\frac{1}{2}\beta (u)|\nabla v+\omega _i\nabla u^i|^2\right) ds\nonumber \\&=\int _{D_{\rho }}ru_r\tau dx-\int _{D_{\rho }}r(v_r+\omega _iu_r^i)\kappa dx. \end{aligned}$$
(2.1)
where \((r,\theta )\) are polar coordinates in D centered at 0. Since we use the Euclidean metric, we have that the covariant derivative \(\nabla _r u\) equals to \(\frac{\partial u}{\partial r}\) and we denote them with a unified notation \(\frac{\partial u}{\partial r}\) or just \(u_r\) for brevity.

Proof

Multiplying (1.8) by \(r(v_r+\omega _iu_r^i)\) and integrating by parts, we get
$$\begin{aligned}&\int _{D_{\rho }}r(v_r+\omega _iu_r^i)\kappa dx\\&\quad =\int _{D_{\rho }}div(\beta (u)(\nabla v+\omega _i\nabla u^i))\cdot r(v_r+\omega _iu_r^i) dx\\&\quad =\int _{\partial D_{\rho }}r\beta (u)|v_r+\omega _iu_r^i|^2 ds-\int _{D_{\rho }}\beta (u)(\nabla v+\omega _i\nabla u^i)\cdot \nabla \left( r(v_r+\omega _iu_r^i)\right) dx\\&\quad =\int _{\partial D_{\rho }}r\beta (u)|v_r+\omega _iu_r^i|^2 ds-\int _{D_{\rho }}\beta (u)|\nabla v+\omega _i\nabla u^i|^2dx\\&\qquad -\int _{D_{\rho }}\beta (u)(\nabla v+\omega _i\nabla u^i)\cdot r\left( \frac{\partial }{\partial r}(\nabla v+\omega _i\nabla u^i)-\frac{\partial \omega _i}{\partial r}\nabla u^i+\frac{\partial u_i}{\partial r}\nabla \omega ^i \right) dx. \end{aligned}$$
By direct computations, noting that
$$\begin{aligned}&-\int _{D_{\rho }}\beta (u)(\nabla v+\omega _i\nabla u^i)\cdot r\frac{\partial }{\partial r}(\nabla v+\omega _i\nabla u^i)dx\\&\quad =-\frac{1}{2}\int _{D_{\rho }}\beta (u)r\frac{\partial }{\partial r}|\nabla v+\omega _i\nabla u^i|^2dx\\&\quad =-\frac{1}{2}\int _{\partial D_{\rho }}r\beta (u)|\nabla v+\omega _i\nabla u^i|^2 ds+ \int _{D_{\rho }}\beta (u)|\nabla v+\omega _i\nabla u^i|^2dx\\&\,\qquad +\frac{1}{2}\int _{D_{\rho }}r\frac{\partial \beta (u)}{\partial r}|\nabla v+\omega _i\nabla u^i|^2dx, \end{aligned}$$
we have
$$\begin{aligned} \int _{D_{\rho }}r(v_r+\omega _iu_r^i)\kappa dx&= \int _{\partial D_{\rho }}r\beta (u)\left( |v_r+\omega _iu_r^i|^2 -\frac{1}{2}|\nabla v+\omega _i\nabla u^i|^2\right) ds \nonumber \\&\qquad +\int _{D_{\rho }}\beta (u)(\nabla v+\omega _i\nabla u^i)\cdot r\left( \frac{\partial \omega _i}{\partial r}\nabla u^i-\frac{\partial u_i}{\partial r}\nabla \omega ^i \right) dx\nonumber \\&\qquad +\frac{1}{2}\int _{D_{\rho }}r\frac{\partial \beta (u)}{\partial r}|\nabla v+\omega _i\nabla u^i|^2dx. \end{aligned}$$
(2.2)
Similarly, multiplying (1.7) by \(ru_r\) and integrating by parts, we get
$$\begin{aligned} \int _{D_{\rho }}ru_r\tau dx&=\int _{D_{\rho }}(\Delta u-H^{\top }) ru_r dx\nonumber \\&=\int _{\partial D_{\rho }}r|u_r|^2 ds-\int _{D_{\rho }}\nabla u\cdot \nabla (ru_r) dx-\int _{D_{\rho }}H\cdot ru_r dx\nonumber \\&=\int _{\partial D_{\rho }}r\left( |u_r|^2-\frac{1}{2}|\nabla u|^2\right) -\int _{D_{\rho }}H\cdot ru_r dx. \end{aligned}$$
(2.3)
Noting that
$$\begin{aligned} H\cdot ru_r&=rH^ju^j_r\nonumber \\&=r\left( \beta (\nabla v+\omega _i\nabla u^i)\cdot \nabla u^k\left( \frac{\partial \omega _j}{\partial y^k}-\frac{\partial \omega _k}{\partial y^j}\right) -\frac{1}{2}\frac{\partial \beta }{\partial y^j}|\nabla v+\omega _i\nabla u^i|^2\right) u^j_r\nonumber \\&= r\beta (\nabla v+\omega _i\nabla u^i)\left( u^j_r\nabla \omega _j-\frac{\partial \omega _j}{\partial r}\nabla u^j\right) -\frac{1}{2}r\frac{\partial \beta (u)}{\partial r}|\nabla v+\omega _i\nabla u^i|^2 \end{aligned}$$
(2.4)
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 \(\rho \) from \(r_0\) to \(2r_0\), we get

Corollary 2.4

For (uv) in Lemma 2.3, if \(\Vert (\nabla u,\nabla v)\Vert _{L^2(D)}+\Vert (\tau ,\kappa )\Vert _{L^2(D)}\le \Lambda \), then for any \(0<r_0<\frac{1}{4}\), we have
$$\begin{aligned} \int _{D_{2r_0}\setminus D_{r_0}}\left( |u_r|^2-\beta (u)|v_r+\omega _i u_r^i|^2-\frac{1}{2}|\nabla u|^2+\frac{1}{2}\beta (u)|\nabla v+\omega _i\nabla u^i|^2\right) dx\le Cr_0, \end{aligned}$$
where \(C=C(\lambda _1,\lambda _2,\Lambda ,N)>0\) is a constant.

Denote \( \widetilde{u}=u-\phi \ \text {and}\ \widetilde{v}=v-\psi . \) For the boundary case, we have

Lemma 2.5

Let \(D^+\subset \mathbb {R}^2\) be the upper unit disk and \((u,v)\in W^{2,2}(D^+,N\times \mathbb {R})\) be an approximate Lorentzian harmonic map with Dirichlet boundary data \((u,v)|_{\partial ^0D^+}=(\phi ,\psi )\in C^{2+\alpha }(D)\) and \((\tau ,\kappa )\in L^2(D^+)\), then for any \(0<\rho <\frac{1}{2}\), we have
$$\begin{aligned}&\rho \int _{\partial ^+ D_{\rho }^+}(|u_r|^2-\beta (u)|v_r+\omega _i u_r^i|^2-\frac{1}{2}|\nabla u|^2+\frac{1}{2}\beta (u)|\nabla v+\omega _i\nabla u^i|^2)ds\nonumber \\&=\int _{D_{\rho }^+}r\widetilde{u}_r\tau dx-\int _{D_{\rho }^+}r(\widetilde{v}_r+\omega _i\widetilde{u}_r^i)\kappa dx +\int _{\partial ^+ D_{\rho }^+}r\frac{\partial u}{\partial r}\frac{\partial \phi }{\partial r}ds-\int _{D_{\rho }^+}\nabla u\nabla (r\phi _r) dx\nonumber \\&\qquad +\int _{D_{\rho }^+}(A(u)(\nabla u,\nabla u)-H^{\top })\cdot r\phi _r dx-\int _{\partial ^+D_{\rho }^+}r\beta (u)( v_r+\omega _i u_r^i)(\psi _r+\omega _i \phi _r^i)ds\nonumber \\&\qquad +\int _{D_{\rho }^+}\beta (u)(\nabla v+\omega _i\nabla u^i)\cdot (\nabla \psi +\omega _i\nabla \phi ^i)dx\nonumber \\&\qquad +\int _{D_{\rho }^+}r\beta (u)(\nabla v+\omega _i\nabla u^i)\cdot \nabla (\psi _r+\omega _i\phi _r^i)dx \end{aligned}$$
(2.5)
where \((r,\theta )\) are polar coordinates in D centered at 0.

Proof

The proof is similar to the proof of Lemma 2.3.

Multiplying (1.8) by \(r(\widetilde{v}_r+\omega _i\widetilde{u}_r^i)\) and integrating by parts, we get
$$\begin{aligned} \int _{D_{\rho }^+}r(\widetilde{v}_r+\omega _i\widetilde{u}_r^i)\kappa dx&=\int _{D_{\rho }^+}div(\beta (u)(\nabla v+\omega _i\nabla u^i))\cdot r(\widetilde{v}_r+\omega _i\widetilde{u}_r^i) dx\\&=\int _{\partial ^+D_{\rho }^+}\beta (u)( v_r+\omega _i u_r^i)\cdot r(\widetilde{v}_r+\omega _i\widetilde{u}_r^i)ds\\&\quad -\int _{D_{\rho }^+}\beta (u)(\nabla v+\omega _i\nabla u^i)\cdot \nabla ( r(\widetilde{v}_r+\omega _i\widetilde{u}_r^i))dx\\&=\int _{\partial ^+D_{\rho }^+}\beta (u)( v_r+\omega _i u_r^i)\cdot r(\widetilde{v}_r+\omega _i\widetilde{u}_r^i)ds\\&\quad -\int _{D_{\rho }^+}\beta (u)(\nabla v+\omega _i\nabla u^i)\cdot (\nabla \widetilde{v}+\omega _i\nabla \widetilde{u}^i)dx\\&\quad -\int _{D_{\rho }^+}r\beta (u)(\nabla v+\omega _i\nabla u^i)\cdot \nabla (\widetilde{v}_r+\omega _i\widetilde{u}_r^i)dx =\mathbb {I}+\mathbb {II}+\mathbb {III}. \end{aligned}$$
By direct computations, we have
$$\begin{aligned} \mathbb {I}&=\int _{\partial ^+D_{\rho }^+}r\beta (u)| v_r+\omega _i u_r^i|^2ds-\int _{\partial ^+D_{\rho }^+}r\beta (u)( v_r+\omega _i u_r^i)(\psi _r+\omega _i \phi _r^i)ds,\\ {\mathbb {II}}&=-\int _{D_{\rho }^+}\beta (u)|\nabla v+\omega _i\nabla u^i|^2dx+\int _{D_{\rho }^+}\beta (u)(\nabla v+\omega _i\nabla u^i)\cdot (\nabla \psi +\omega _i\nabla \phi ^i)dx, \end{aligned}$$
and
$$\begin{aligned} \mathbb {III}&=-\frac{1}{2}\int _{D_{\rho }^+} r\beta (u)\frac{\partial }{\partial r}|\nabla v+\omega _i\nabla u^i|^2dx +\int _{D_{\rho }^+}r\beta (u)(\nabla v+\omega _i\nabla u^i)\cdot \nabla (\psi _r+\omega _i\phi _r^i)dx\\&\,\quad +\int _{D_{\rho }^+}r\beta (u)(\nabla v+\omega _i\nabla u^i)\cdot \left( \frac{\partial \omega _i}{\partial r}\nabla u^i-\nabla \omega _i u_r^i\right) dx. \end{aligned}$$
Noting that
$$\begin{aligned}&-\frac{1}{2}\int _{D_{\rho }^+}\beta (u)r\frac{\partial }{\partial r}|\nabla v+\omega _i\nabla u^i|^2dx\\&\quad = -\frac{1}{2}\int _{\partial ^+ D^+_{\rho }}r\beta (u)|\nabla v+\omega _i\nabla u^i|^2ds+ \int _{D_{\rho }^+}\beta (u)|\nabla v+\omega _i\nabla u^i|^2dx\\&\,\qquad +\frac{1}{2}\int _{D_{\rho }^+}r\frac{\partial \beta (u)}{\partial r}|\nabla v+\omega _i\nabla u^i|^2dx, \end{aligned}$$
we have
$$\begin{aligned} \int _{D_{\rho }^+}r(\widetilde{v}_r+\omega _i\widetilde{u}_r^i)\kappa dx&= \int _{\partial ^+ D_{\rho }^+}r\beta (u)\left( |v_r+\omega _iu_r^i|^2-\frac{1}{2}|\nabla v+\omega _i\nabla u^i|^2\right) ds\nonumber \\&\quad +\int _{D_{\rho }^+}\beta (u)(\nabla v+\omega _i\nabla u^i)\cdot r\left( \frac{\partial \omega _i}{\partial r}\nabla u^i-\frac{\partial u_i}{\partial r}\nabla \omega ^i \right) dx\nonumber \\&\quad +\frac{1}{2}\int _{D_{\rho }^+}r\frac{\partial \beta (u)}{\partial r}|\nabla v+\omega _i\nabla u^i|^2dx\nonumber \\&\quad -\int _{\partial ^+D_{\rho }^+}r\beta (u)( v_r+\omega _i u_r^i)(\psi _r+\omega _i \phi _r^i)ds\nonumber \\&\quad +\int _{D_{\rho }^+}\beta (u)(\nabla v+\omega _i\nabla u^i)\cdot (\nabla \psi +\omega _i\nabla \phi ^i)dx\nonumber \\&\quad +\int _{D_{\rho }^+}r\beta (u)(\nabla v+\omega _i\nabla u^i)\cdot \nabla (\psi _r+\omega _i\phi _r^i)dx. \end{aligned}$$
(2.6)
Similarly, multiplying (1.7) by \(r\widetilde{u}_r\) and integrating by parts, we get
$$\begin{aligned} \int _{D_{\rho }^+}r\widetilde{u}_r\tau dx&=\int _{D_{\rho }^+}(\Delta u+A(u)(\nabla u,\nabla u)-H^{\top })\cdot r\widetilde{u}_r dx\nonumber \\&= \int _{\partial ^+ D_{\rho }^+}r(|u_r|^2-\frac{1}{2}|\nabla u|^2)ds -\int _{\partial ^+ D_{\rho }^+}r\frac{\partial u}{\partial r}\frac{\partial \phi }{\partial r} +\int _{D_{\rho }^+}\nabla u\nabla (r\phi _r) dx\nonumber \\&\quad -\int _{D_{\rho }^+}(A(u)(\nabla u,\nabla u)-H^{\top })\cdot r\phi _r dx -\int _{D_{\rho }^+}H\cdot ru_r dx. \end{aligned}$$
(2.7)
Combining (2.4) with (2.6) and (2.7), we obtain the conclusion of the lemma. \(\square \)

Corollary 2.6

For (uv) in Lemma 2.5, if \(\Vert (\nabla u,\nabla v)\Vert _{L^2(D^+)}+\Vert (\tau ,\kappa )\Vert _{L^2(D^+)}\le \Lambda \), then for any \(0<r_0<\frac{1}{4}\), we have
$$\begin{aligned} \int _{D^+_{2r_0}\setminus D^+_{r_0}}(|u_r|^2-\beta (u)|v_r+\omega _i u_r^i|^2-\frac{1}{2}|\nabla u|^2+\frac{1}{2}\beta (u)|\nabla v+\omega _i\nabla u^i|^2)dx\le Cr_0, \end{aligned}$$
where \(C=C(\lambda _1,\lambda _2,\Lambda ,N)>0\) is a constant.

Proof

By Hölder’s inequality, it is easy to find that the right hand side of (2.5) is bounded by
$$\begin{aligned}C\rho \int _{\partial ^+D_{\rho }^+}(|\nabla u|+|\nabla v|)+C\rho ,\end{aligned}$$
where \(C=C(\lambda _1,\lambda _2,N,\Vert \phi \Vert _{C^2},\Vert \psi \Vert _{C^2})\). Then the conclusion of the corollary follows by an integration about \(\rho \) from \(r_0\) to \(2r_0\). \(\square \)

Similar as for harmonic maps into a Riemannian manifold, there is also an energy gap for a nontrivial Lorentzian harmonic map.

Theorem 2.7

(Gap phenomenon) Suppose either \((u,v):{\mathbb {S}}^2\rightarrow N\times \mathbb {R}\) is a smooth Lorentzian harmonic map or \((u,v):\mathbb {R}_+^2\rightarrow N\times \mathbb {R}\) is a smooth Lorentzian harmonic map with Dirichlet boundary condition:
$$\begin{aligned} (u,v)|_{\partial \mathbb {R}^2_+}=constant, \end{aligned}$$
then there exists a constant \(\epsilon _0>0\) depending on \((N\times \mathbb {R},g)\), such that if either
$$\begin{aligned} E(u,v;{\mathbb {S}}^2)\le \epsilon _0\ or\ E(u,v;\mathbb {R}_+^2)\le \epsilon _0, \end{aligned}$$
then (uv) is a constant map. Here \({\mathbb {S}}^2\) denotes the unit sphere in \({\mathbb {R}}^3\).

Proof

One can find the proof of the theorem in [9] for the case \((u,v):{\mathbb {S}}^2\rightarrow N\times \mathbb {R}\).

By Eqs. (1.4) and (1.5), we have
$$\begin{aligned} |\Delta u|+|\Delta v|\le C(|\nabla u|^2+|\nabla v|^2). \end{aligned}$$
The standard elliptic theory tells us that
$$\begin{aligned} \Vert \nabla u\Vert _{W^{1,\frac{4}{3}}}+\Vert \nabla v\Vert _{W^{1,\frac{4}{3}}}&\le C(\Vert \Delta u\Vert _{L^{\frac{4}{3}}}+\Vert \Delta v\Vert _{L^{\frac{4}{3}}})\\&\le CE^{\frac{1}{2}}(u,v;{\mathbb {S}}^2)(\Vert \nabla u\Vert _{L^4}+\Vert \nabla v\Vert _{L^4})\\&\le C\sqrt{\epsilon _0}(\Vert \nabla u\Vert _{W^{1,\frac{4}{3}}}+\Vert \nabla v\Vert _{W^{1,\frac{4}{3}}}). \end{aligned}$$
It is easy to get that, if \(\epsilon _0\) is small enough, (uv) must be a constant map.
If \((u,v):\mathbb {R}_+^2\rightarrow N\times \mathbb {R}\) is a smooth Lorentzian harmonic map with constant Dirichlet boundary condition, choosing \(\epsilon _0\le \epsilon _2\) where \(\epsilon _2\) is the positive constant in Lemma 2.2, then by Lemma 2.2 (taking \((\phi ,\psi )=constant\), \((\tau ,\kappa )=0\) and any constant \(p>2\)) and Sobolev embedding, for any \(R>0\), we have
$$\begin{aligned} R\Vert \nabla u\Vert _{L^{\infty }(D^+_{R/2})}+R\Vert \nabla v\Vert _{L^{\infty }(D^+_{R/2})}\le CE^{\frac{1}{2}}(u,v;D^+_{R})\le C\epsilon _0^{\frac{1}{2}}. \end{aligned}$$
Sending R to infinity yields that (uv) must be a constant map. \(\square \)

It is necessary for the singularities to be removable during the blow-up process. Removability of singularities for a Lorentzian harmonic map (i.e. \(\tau =\kappa =0\)) is proved in [9]. By assuming additionally that \(\omega \equiv 0\), for an approximate Lorentzian harmonic map (i.e. \((\tau ,\kappa )\ne 0\)) with singularities either in the interior or on the boundary, we can also remove them.

Theorem 2.8

Suppose \((u,v)\in W_{loc}^{2,2}(D\setminus \{0\})\) is an approximate Lorentzian harmonic map from the punctured disk \(D\setminus \{0\}\) to \((N\times \mathbb {R},g_N-\beta d^2\theta )\). If \(E(u,v;D)<\infty \) and \((\tau ,\kappa )\in L^2(D)\), then (uv) can be extended to the whole disk D in \(W^{2,2}(D)\).

For an approximate Lorentzian harmonic map \((u,v)\in W_{loc}^{2,2}(D^+\setminus \{0\})\) which is from \(D^+\setminus \{0\}\) to \((N\times \mathbb {R},g_N-\beta d^2\theta )\) with boundary data \((u,v)|_{\partial ^0D^+}=(\phi ,\psi )\). If \(E(u,v;D^+)<\infty \) and \((\tau ,\kappa )\in L^2(D^+)\), then (uv) 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 (uv) 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 \(v\in W^{2,p}(D^+_{\rho }(0))\) for some \(\rho >0\) and any \(1<p<2\). In fact, the anti-symmetric term in the equation for v equals to zero. This implies that \(\nabla v\in L^4(D^+)\).

On the other hand, since the Eq. (1.7) can be written as an elliptic system with an anti-symmetric potential ([25])
$$\begin{aligned} \Delta u=\Omega \cdot \nabla u+f \end{aligned}$$
with \(\Omega \in L^2(D^+,so(n)\otimes \mathbb {R}^2)\) and \(f\in L^2(D^+)\), using Theorem 1.2 in [30] again, we have \(u\in W^{2,p}(D^+_\rho (0))\) for some \(\rho >0\) and any \(1<p<2\). Then the higher regularity can be derived by a standard bootstrap argument. \(\square \)

3 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 \(\Lambda >0\) such that the sequence satisfies
$$\begin{aligned} \Vert (\nabla u_n,\nabla v_n)\Vert _{L^{2}(D_1(0))}+\Vert (\tau _n,\kappa _n)\Vert _{L^{2}(D_1(0))}\le \Lambda . \end{aligned}$$
(3.1)

Definition 3.1

For an approximate Lorentzian harmonic map sequence \(\{(u_n, v_n)\}\), define
$$\begin{aligned} \mathcal {S}_1:=\cap _{r>0}\left\{ x\in M|\liminf _{n\rightarrow \infty }\int _{D_r(x)}(|\nabla u_n|^2+|\nabla v_n|^2)dv_h\ge \epsilon _1\right\} , \end{aligned}$$
and
$$\begin{aligned} \mathcal {S}_2:=\cap _{r>0}\left\{ x\in \partial M|\liminf _{n\rightarrow \infty }\int _{D^+_r(x)}(|\nabla u_n|^2+|\nabla v_n|^2)dv_h\ge \epsilon _2\right\} , \end{aligned}$$
where \(\epsilon _1\) and \(\epsilon _2\) are constants in Lemmas 2.1 and 2.2. The blow-up set of \(\{(u_n, v_n)\}\) is defined to be \(\mathcal {S}:=\mathcal {S}_1\cup \mathcal {S}_2\).

Lemma 3.1

For an approximate harmonic map sequence satisfying (3.1), the blow-up set \(\mathcal {S}\) is a finite set.

Proof

By (3.1), we can take a subsequence and still denote it by \(\{(u_n,v_n)\}\), such that \(\{(u_n,v_n)\}\) converges weakly in \(W^{1,2}(M)\) to a limit map \((u,v):M\rightarrow ({N\times {\mathbb {R}}},g)\) . If for any point \(x\in M\),
$$\begin{aligned} \lim _{r\rightarrow 0}\limsup _{n\rightarrow \infty }\int _{D_r(x)}|\nabla u_n|^2+|\nabla v_n|^2<\epsilon _1, \end{aligned}$$
(3.2)
Lemma 2.1 tells that the convergence is strong in \(W^{1,2}(M)\). Obviously in this case, \({\mathcal {S}_1}\) is empty. Otherwise, if there exists a point \(p_1\in M\) such that
$$\begin{aligned} \lim _{r\rightarrow 0}\limsup _{n\rightarrow \infty }\int _{D_r(p_1)}|\nabla u_n|^2+|\nabla v_n|^2\ge \epsilon _1, \end{aligned}$$
(3.3)
By taking a subsequence, we can assume that
$$\begin{aligned} \lim _{r\rightarrow 0}\lim _{n\rightarrow \infty }\int _{D_r(p_1)}|\nabla u_n|^2+|\nabla v_n|^2\ge \epsilon _1. \end{aligned}$$
If (3.2) holds for any point \(x\in M\setminus \{p_1\}\), we get that \({\mathcal {S}_1}=\{p_1\}\). Otherwise, we can find a point \(p_2\) where the energy concentration (3.3) happens. Since the energy of the sequence is bounded, this process must stop after finite steps.

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 (uv) in \(W_{loc}^{1,2}(M\setminus {\mathcal {S}})\), where \({\mathcal {S}}=\mathcal {S}_1\cup \mathcal {S}_2=\{p_1, p_2,\ldots , p_I\}\) is a finite set. \(\square \)

We consider the case that the blow-up points are in the interior first. Since the blow-up set \(\mathcal {S}_1\) is finite, we can find small geodesic disks \(D_{\delta _i}\) (by conformal invariance, we can assume that they are flat disks) for each blow-up point \(p_i\) such that \(D_{\delta _i}\cap D_{\delta _j}=\emptyset \) for \(i\ne j\), \(i,j=1,2,\ldots ,I\), and on \(M\setminus \cup _{i=1}^I D_{\delta _i}\), \((u_n,v_n)\) converges strongly to a limit map (uv). Without loss of generality, we discuss the case that there is only one blow-up point \(0\in D_1(0)\) in \({\mathcal {S}_1}\) and the sequence \(\{(u_n,v_n)\}\) satisfies that there is some (uv) such that
$$\begin{aligned} (u_n,v_n)\rightarrow (u,v)\ \ \text {weakly in}\ \ W^{2,2}_{loc}(D_1(0)\setminus \{0\})\ \ \text {as}\ \ n\rightarrow \infty . \end{aligned}$$
(3.4)

Lemma 3.2

Let \(\{(u_n,v_n)\} \in W^{2,2}(D_1(0),N\times \mathbb {R})\) be a sequence of approximate Lorentzian harmonic maps satisfying (3.1) and (3.4). Up to a subsequence which is still denoted by \(\{(u_n,v_n)\}\), there exist a positive integer L and nontrivial Lorentzian harmonic spheres \((\sigma ^i,\xi ^i):\mathbb {R}^2\cup \{\infty \}\rightarrow N\times \mathbb {R}\), \(i=1,\ldots ,L\) satisfying
$$\begin{aligned} \lim _{n\rightarrow \infty }E_g(u_n,v_n;D_1(0))=E_g(u,v;D_1(0)) +\sum _{i=1}^LE_g(\sigma ^i,\xi ^i). \end{aligned}$$
(3.5)

Proof

According to the standard induction argument in [5], we can assume that there is only one bubble at the singular point \(0\in D_1(0)\). To prove (3.5) is equivalent to prove that there exists a Lorentzian harmonic sphere \((\sigma ,\xi )\) such that
$$\begin{aligned} \lim _{\delta \rightarrow 0}\lim _{n\rightarrow \infty } E_g(u_n,v_n,D_\delta )=E_g(\sigma ,\xi ). \end{aligned}$$
(3.6)
By the standard argument of blow-up analysis, for any n, there exist sequences \(x_n\rightarrow 0\) and \(r_n\rightarrow 0\) such that
$$\begin{aligned} E(u_n,v_n;D_{r_n/2}(x_n))=\sup _{\begin{array}{c} x\in D_\delta ,r\le r_n\\ D_r(x)\subset D_\delta \end{array}}E(u_n,v_n;D_{r/2}(x))=\frac{\epsilon _1}{8}. \end{aligned}$$
(3.7)
Without loss of generality, we may assume that \(x_n=0\) and denote \(\tilde{u}_n=u_n(r_n x),\ \ \tilde{v}_n=v_n(r_n x).\) Then we have
$$\begin{aligned} E(\tilde{u}_n,\tilde{v}_n;D_{1/2})=E(u_n,v_n;D_{r_n/2}) =\frac{\epsilon _1}{8}<\epsilon _1 \end{aligned}$$
(3.8)
and
$$\begin{aligned} E(\tilde{u}_n,\tilde{v}_n;D_{R})=E(u_n,v_n;D_{r_n R})<\Lambda . \end{aligned}$$
By (3.7), we can apply Lemma 2.1 on \(D_{R}\) for \(\{(\tilde{u}_n,\tilde{v}_n)\}\) and get that \(\{(\tilde{u}_n,\tilde{v}_n)\}\) converges strongly to some Lorentzian harmonic map \((\sigma ,\xi )\) in \(W^{1,2}(D_R, N\times {\mathbb {R}})\) for any \(R\ge 1\). By stereographic projection and the removable singularity theorem [9], we get a nonconstant harmonic sphere \((\sigma ,\xi )\). Thus we get the first bubble at the blow-up point and to prove (3.6) is equivalent to prove that
$$\begin{aligned} \lim _{R\rightarrow \infty }\lim _{\delta \rightarrow 0}\lim _{n\rightarrow \infty }E_g(u_n,v_n;D_\delta (0)\setminus D_{r_nR}(0))=0. \end{aligned}$$
(3.9)
Since we assume that there is only one bubble, we have that, for any \(\epsilon >0\), there holds that
$$\begin{aligned} E(u_n,v_n;D_{4\rho }\setminus D_{\frac{\rho }{2}})\le \epsilon ^2\ \ \text {for} \ \rho \in [r_nR,\frac{\delta }{2}] \end{aligned}$$
(3.10)
as \(n\rightarrow \infty \), \(R\rightarrow \infty \) and \(\delta \rightarrow 0\). 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 \(\rho \in [r_nR,\frac{\delta }{2}]\), we have
$$\begin{aligned}&\Vert u_n\Vert _{osc(D_{2\rho }\setminus D_\rho )}+\Vert v_n\Vert _{osc(D_{2\rho }\setminus D_\rho )}\nonumber \\&\quad \le C E^{\frac{1}{2}}(u_n,v_n;D_{4\rho }\setminus D_{\frac{\rho }{2}})+C\rho \Vert (\tau _n,\kappa _n)\Vert _{L^2(D_{4\rho }\setminus D_{\frac{\rho }{2}})}. \end{aligned}$$
(3.11)
Define
$$\begin{aligned} u_n^*(r):=\frac{1}{2\pi }\int _0^{2\pi }u_n(r,\theta )d\theta ,\ v_n^*(r):=\frac{1}{2\pi }\int _0^{2\pi }v_n(r,\theta )d\theta . \end{aligned}$$
By (3.11), we know that
$$\begin{aligned} \Vert u_n-u_n^*\Vert _{L^\infty (D_\delta \setminus D_{r_nR})}&=\sup _{r_nR\le t\le \frac{\delta }{2}} \Vert u_n-u_n^*\Vert _{L^\infty (D_{2t}\setminus D_{t})}\nonumber \\&\le \sup _{r_nR\le t\le \frac{\delta }{2}} \Vert u_n\Vert _{osc(D_{2t}\setminus D_{t})}\le C(\epsilon +\delta ) \end{aligned}$$
(3.12)
and similarly,
$$\begin{aligned} \Vert v_n-v_n^*\Vert _{L^\infty (D_\delta \setminus D_{r_nR})}\le C(\epsilon +\delta ). \end{aligned}$$
Then we get by integrating by parts that
$$\begin{aligned}&\int _{D_\delta \setminus D_{r_nR}}-\Delta u_n(u_n-u_n^*)dx\nonumber \\&=\int _{D_\delta \setminus D_{r_nR}}\nabla u_n\nabla (u_n-u_n^*)dx-\int _{\partial D_\delta }\frac{\partial u_n}{\partial r}(u_n-u_n^*)+\int _{\partial D_{r_nR}}\frac{\partial u_n}{\partial r}(u_n-u_n^*) \nonumber \\&\ge \int _{D_\delta \setminus D_{r_nR}}|\nabla u_n|^2dx-\int _{D_\delta \setminus D_{r_nR}}|\frac{\partial u_n}{\partial r}|^2dx -\int _{\partial D_\delta }\frac{\partial u_n}{\partial r}(u_n-u_n^*)\nonumber \\&\quad +\int _{\partial D_{r_nR}}\frac{\partial u_n}{\partial r}(u_n-u_n^*) \nonumber \\&= \int _{D_\delta \setminus D_{r_nR}}|r^{-1} \frac{\partial u_n}{\partial \theta }|^2dx -\int _{\partial D_\delta }\frac{\partial u_n}{\partial r}(u_n-u_n^*)+\int _{\partial D_{r_nR}}\frac{\partial u_n}{\partial r}(u_n-u_n^*). \end{aligned}$$
(3.13)
Since \((u_n,v_n)\) is an approximate harmonic map, we have
$$\begin{aligned} |\Delta u_n|+|\Delta v_n|\le C(\lambda _1,\lambda _2,N)(|\nabla u_n|^2+|\nabla v_n|^2).\end{aligned}$$
(3.14)
Then we get from (3.12), (3.13) and (3.14) that
$$\begin{aligned}&\int _{D_\delta \setminus D_{r_nR}}|r^{-1} \frac{\partial u_n}{\partial \theta }|^2dx\nonumber \\&\quad \le \int _{D_\delta \setminus D_{r_nR}}-\Delta u_n(u_n-u_n^*)dx+\int _{\partial D_\delta }\frac{\partial u_n}{\partial r}(u_n-u_n^*)-\int _{\partial D_{r_nR}}\frac{\partial u_n}{\partial r}(u_n-u_n^*)\nonumber \\&\quad \le C(\epsilon +\delta )\left( \int _{D_\delta \setminus D_{r_nR}}(|\nabla u_n|^2+|\nabla v_n|^2)dx+\int _{\partial D_{\delta }}|\frac{\partial u_n}{\partial r}|+\int _{\partial D_{r_nR}}|\frac{\partial u_n}{\partial r}|\right) . \end{aligned}$$
(3.15)
By Lemma 2.1 and the trace theory, we obtain for the boundary term in (3.15) that
$$\begin{aligned} \int _{\partial D_{\delta }}|\frac{\partial u_n}{\partial r}|&\le C(\Vert \nabla u_n\Vert _{L^2(D_{\frac{3}{2}\delta }\setminus D_\delta )}+\delta \Vert \nabla ^2 u_n\Vert _{L^2(D_{\frac{3}{2}\delta }\setminus D_\delta )})\\&\le C(E^{\frac{1}{2}}(u_n,v_n;D_{2\delta }\setminus D_{\frac{\delta }{2}})+\delta \Vert (\tau _n,\kappa _n)\Vert _{L^2(D_{2\delta }\setminus D_{\frac{\delta }{2}})})\\&\le C(\epsilon +\delta ). \end{aligned}$$
Similarly,
$$\begin{aligned} \int _{\partial D_{r_nR}}|\frac{\partial u_n}{\partial r}|\le C(\epsilon +\delta ). \end{aligned}$$
Combining these, we have
$$\begin{aligned} \int _{D_\delta \setminus D_{r_nR}}|r^{-1} \frac{\partial u_n}{\partial \theta }|^2dx\le C(\epsilon +\delta ). \end{aligned}$$
(3.16)
Similarly, we can obtain that
$$\begin{aligned} \int _{D_\delta \setminus D_{r_nR}}|r^{-1} \frac{\partial v_n}{\partial \theta }|^2dx\le C(\epsilon +\delta ). \end{aligned}$$
Without loss of generality, we may assume \(\delta =2^{m_n}r_nR\), where \(m_n\) is a positive integer which tends to \(\infty \) as \(n\rightarrow \infty \). By Corollary 2.4, for \(i=0,1,\ldots , m_n-1\), we have
$$\begin{aligned}&\int _{D_{2^{i+1}r_nR}\setminus D_{2^ir_nR}}\left( |\frac{\partial u_n}{\partial r}|^2-\beta (u_n)|\frac{\partial v_n}{\partial r}+\omega _j\frac{\partial u^j_n}{\partial r}|^2\right) dx\\&\quad \le C\left( \int _{D_{2^{i+1}r_nR}\setminus D_{2^ir_nR}}|r^{-1}\frac{\partial u_n}{\partial \theta }|^2dx+\int _{D_{2^{i+1}r_nR}\setminus D_{2^ir_nR}}|r^{-1}\frac{\partial v_n}{\partial \theta }|^2dx+2^ir_nR\right) . \end{aligned}$$
Since
$$\begin{aligned} \sum _{i=0}^{m_n-1}2^ir_nR=2^{m_n}r_n R=\delta , \end{aligned}$$
we get
$$\begin{aligned}&\int _{D_\delta \setminus D_{r_nR}}\left( |\frac{\partial u_n}{\partial r}|^2-\beta (u_n)|\frac{\partial v_n}{\partial r}+\omega _j\frac{\partial u^j_n}{\partial r}|^2\right) dx\nonumber \\&\quad =\sum _{i=0}^{m_n-1}\int _{D_{2^{i+1}r_nR}\setminus D_{2^ir_nR}}\left( |\frac{\partial u_n}{\partial r}|^2-\beta (u_n)|\frac{\partial v_n}{\partial r}+\omega _j\frac{\partial u^j_n}{\partial r}|^2)dx\le C(\epsilon +\delta \right) , \end{aligned}$$
(3.17)
from which (3.9) follows immediately. \(\square \)

When the 1-form \(\omega \equiv 0\), 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 \(\omega \equiv 0\) and \(\Vert \nabla v_n\Vert _{L^{p}}\le \Lambda \) for some \(p>2\), then we have that \(\sigma ^i:\mathbb {R}^2\cup \{\infty \}\rightarrow N\) is a nontrivial harmonic sphere, \(\xi ^i\) is a constant map and (3.5) becomes
$$\begin{aligned} \lim _{n\rightarrow \infty }E(u_n;D_1(0))&=E(u;D_1(0))+\sum _{i=1}^LE(\sigma ^i), \end{aligned}$$
(3.18)
$$\begin{aligned} \lim _{n\rightarrow \infty }E(v_n;D_1(0))&=E(v;D_1(0)). \end{aligned}$$
(3.19)
Furthermore, The image
$$\begin{aligned} u(D_1(0))\cup \bigcup _{i=1}^L\sigma ^i(\mathbb {R}^2) \end{aligned}$$
(3.20)
is a connected set.

Proof

Similar to the proof of Lemma 3.2, to prove (3.18) and (3.19) is equivalent to proving
$$\begin{aligned} \lim _{R\rightarrow \infty }\lim _{\delta \rightarrow 0}\lim _{n\rightarrow \infty }E(u_n,v_n;D_\delta (0)\setminus D_{r_nR}(0))=0. \end{aligned}$$
(3.21)
Since \(\Vert \nabla v_n\Vert _{L^p(D)}\le \Lambda \) for some \(p>2\), we get
$$\begin{aligned} \int _{D_\delta \setminus D_{r_nR}}|\nabla v_n|^2dx\le C\delta ^{1-\frac{2}{p}} \left( \int _{D_\delta \setminus D_{r_nR}}|\nabla v_n|^pdx\right) ^\frac{2}{p}\le C\delta ^{1-\frac{2}{p}}. \end{aligned}$$
(3.22)
Since \(\omega \equiv 0\), (3.17) implies that
$$\begin{aligned}&\int _{D_\delta \setminus D_{r_nR}}|\frac{\partial u_n}{\partial r}|^2dx\le C\int _{D_\delta \setminus D_{r_nR}}|\nabla v_n|^2dx+C(\epsilon +\delta ). \end{aligned}$$
(3.23)
Combining (3.16), (3.22) with (3.23), we can get (3.21).
To prove (3.20) is equivalent to prove
$$\begin{aligned} \lim _{R\rightarrow \infty }\lim _{\delta \rightarrow 0}\lim _{n\rightarrow \infty }\Vert u_n\Vert _{osc(D_\delta (0)\setminus D_{r_nR}(0))}=0. \end{aligned}$$
(3.24)
To prove (3.24), denote \(Q(s):=D_{2^{s_0+s}r_nR}\setminus D_{2^{s_0-s}r_nR}\) and consider
$$\begin{aligned} f(s):=\int _{Q(s)}|\nabla u_n|^2dx, \end{aligned}$$
where \(0\le s_0\le m_n\) and \(0\le s\le \min \{s_0,m_n-s_0\}\). Integrating by parts, we get
$$\begin{aligned}&\int _{Q(s)}-\Delta u_n(u_n-u_n^*)dx\\&\quad =\int _{Q(s)}\nabla u_n\nabla (u_n-u_n^*)dx-\int _{\partial Q(s)}\frac{\partial u_n}{\partial r}(u_n-u_n^*)\\&\quad \ge \frac{1}{2}\int _{Q(s)}|\nabla u_n|^2dx+\int _{Q(s)}\left( \frac{1}{2}|\nabla u_n|^2-|\frac{\partial u_n}{\partial r}|^2\right) dx-\int _{\partial Q(s)}\frac{\partial u_n}{\partial r}(u_n-u_n^*). \end{aligned}$$
By (3.12) and (3.14), we obtain
$$\begin{aligned}&\left( \frac{1}{2}-C(\epsilon +\delta )\right) \int _{Q(s)}|\nabla u_n|^2dx\nonumber \\&\quad \le \int _{Q(s)}\left( |\frac{\partial u_n}{\partial r}|^2-\frac{1}{2}|\nabla u_n|^2\right) dx+C(\epsilon +\delta )\int _{Q(s)}|\nabla v_n|^2dx\nonumber \\&\qquad +\int _{\partial D_{2^{s_0+s}r_nR}}\frac{\partial u_n}{\partial r}(u_n-u_n^*)-\int _{\partial D_{2^{s_0-s}r_nR}}\frac{\partial u_n}{\partial r}(u_n-u_n^*). \end{aligned}$$
(3.25)
We deduce from Corollary 2.4 that
$$\begin{aligned} \int _{Q(s)}\left( |\frac{\partial u_n}{\partial r}|^2-\frac{1}{2}|\nabla u_n|^2\right) dx&\le C\int _{Q(s)}|\nabla v_n|^2dx+C2^{s_0+s}r_nR\\&\le C(2^{s_0+s}r_nR)^{1-\frac{2}{p}}\left( \int _{Q(s)}|\nabla v_n|^pdx\right) ^{\frac{2}{p}}+C2^{s_0+s}r_nR\\&\le C(2^{s_0+s}r_nR)^{1-\frac{2}{p}}. \end{aligned}$$
For the boundary term in (3.25), by Hölder’s inequality and Poincare’s inequality, we have
$$\begin{aligned} |\int _{\partial D_{2^{s_0+s}r_nR}}\frac{\partial u_n}{\partial r}(u_n-u_n^*)|&\le \left( \int _{\partial D_{2^{s_0+s}r_nR}}|\frac{\partial u_n}{\partial r}|^2\right) ^{\frac{1}{2}}\left( \int _{\partial D_{2^{s_0+s}r_nR}}|u_n-u_n^*|^2\right) ^{\frac{1}{2}}\\&\le \left( \int _{\partial D_{2^{s_0+s}r_nR}}|\frac{\partial u_n}{\partial r}|^2\right) ^{\frac{1}{2}}\left( 2^{s_0+s}r_nR\int _{\partial D_{2^{s_0+s}r_nR}}|\frac{\partial u_n}{\partial \theta }|^2\right) ^{\frac{1}{2}}\\&\le C2^{s_0+s}r_nR\int _{\partial D_{2^{s_0+s}r_nR}}|\nabla u_n|^2. \end{aligned}$$
Similarly, we also have
$$\begin{aligned} |\int _{\partial D_{2^{s_0-s}r_nR}}\frac{\partial u_n}{\partial r}(u_n-u_n^*)|\le C2^{s_0-s}r_nR\int _{\partial D_{2^{s_0-s}r_nR}}|\nabla u_n|^2. \end{aligned}$$
Taking \(\epsilon \) and \(\delta \) sufficiently small, we get from (3.25) that
$$\begin{aligned} f(s)\le \frac{C}{\ln 2}f'(s)+C(2^{s_0+s}r_nR)^{1-\frac{2}{p}}, \end{aligned}$$
which implies
$$\begin{aligned} \left( 2^{-\frac{s}{C}}f(s)\right) '\ge -C(2^{s_0}r_nR)^{1-\frac{2}{p}}2^{(1-\frac{2}{p}-\frac{1}{C})s}, \end{aligned}$$
where we can take C sufficiently large such that \(1-\frac{2}{p}-\frac{1}{C}>0\). Integrating from 2 to L, we arrive at
$$\begin{aligned} f(2)\le C2^{-\frac{1}{C}L}f(L) +C(2^{s_0}r_nR)^{1-\frac{2}{p}}2^{\left( 1-\frac{2}{p}-\frac{1}{C}\right) L}. \end{aligned}$$
Let \(s_0=i\) and \(L=L_i:=\min \{i,m_n-i\}\). Noting that \(Q(L_i)\subset D_\delta \setminus D_{r_nR}\), we have
$$\begin{aligned}&\int _{D_{2^{i+2}r_nR}\setminus D_{2^{i-2}r_nR}}|\nabla u_n|^2dx\nonumber \\&\quad \le CE(u_n,D_\delta \setminus D_{r_nR})2^{-\frac{1}{C}L_i}+C(2^{i}r_nR)^{1-\frac{2}{p}}2^{\left( 1-\frac{2}{p} -\frac{1}{C}\right) L_i}\nonumber \\&\quad \le CE(u_n,D_\delta \setminus D_{r_nR})2^{-\frac{1}{C}L_i}+C(2^{i}r_nR)^{1-\frac{2}{p}}2^{\left( 1-\frac{2}{p} -\frac{1}{C}\right) (m_n-i)}\nonumber \\&\quad = CE(u_n,D_\delta \setminus D_{r_nR})2^{-\frac{1}{C}L_i}+C(2^{m_n}r_nR)^{1-\frac{2}{p}}2^{-\frac{1}{C}(m_n-i)}\nonumber \\&\quad = CE(u_n,D_\delta \setminus D_{r_nR})2^{-\frac{1}{C}L_i}+C\delta ^{1-\frac{2}{p}}2^{\left( -\frac{1}{C}\right) (m_n-i)}\nonumber \\&\quad \le C\epsilon 2^{-\frac{1}{C}i}+C\delta ^{1-\frac{2}{p}}2^{\frac{1}{C}(i-m_n)}, \end{aligned}$$
(3.26)
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. \(\square \)
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 \(\{(u_n,v_n)\} \in W^{2,2}(D_1^+(0),N\times \mathbb {R})\) with the Dirichlet boundary condition
$$\begin{aligned} (u_n,v_n)|_{\partial ^0D^+_1(0)}=(\varphi ,\psi )\in C^{2+\alpha }(\partial ^0D^+_1(0)) \end{aligned}$$
(3.27)
for some \(0<\alpha <1\) which satisfies that
$$\begin{aligned} \Vert (\nabla u_n,\nabla v_n)\Vert _{L^{2}(D_1^+(0))} +\Vert (\tau _n,\kappa _n)\Vert _{L^{2}(D_1^+(0))}\le \Lambda . \end{aligned}$$
(3.28)
Without loss of generality, we still suppose that there is only one blow-up point \(0\in D_1^+(0)\) and the sequence \(\{(u_n,v_n)\}\) satisfies that there is some (uv) such that
$$\begin{aligned} (u_n,v_n)\rightarrow (u,v)\ \ \text {weakly in}\ \ W^{2,2}_{loc}(D_1^+(0)\setminus \{0\})\ \ \text {as}\ \ n\rightarrow \infty . \end{aligned}$$
(3.29)
For such a sequence, we have

Lemma 3.4

Let \(\{(u_n,v_n)\} \in W^{2,2}(D_1^+(0),N\times \mathbb {R})\) 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 \(x_n^i\in D_1^+(0)\) and \(r_n^i>0\) satisfying \(\ x^i_n\rightarrow 0\) and \(r^i_n\rightarrow 0\), \(i=1,\ldots ,L\) as \(n\rightarrow \infty \) and both of the following two cases may appear during the blow-up process.
  1. (a)
    If \(\frac{dist(x^i_n,\partial ^0D^+_1(0))}{r^i_n}\rightarrow a^i<\infty \), there is a nonconstant Lorentzian harmonic map \((\sigma ^i,\xi ^i):\mathbb {R}^2_{a^i} \rightarrow N\times \mathbb {R}\) with a constant boundary condition which is the weak limit of \((u_n(x^i_n+r^i_nx),v_n(x^i_n+r^i_nx))\) in \(W^{1,2}_{loc}(\mathbb {R}^{2+}_{a^i})\), where
    $$\begin{aligned} \mathbb {R}^2_{a^i}:=\{(x^1,x^2)\in \mathbb {R}^2|x^2\ge a^i\}\ and\ \mathbb {R}^{2+}_{a^i}:=\{(x^1,x^2)\in \mathbb {R}^2|x^2> a^i\}; \end{aligned}$$
     
  2. (b)

    If \(\frac{dist(x^i_n,\partial ^0D^+_1(0))}{r^i_n}\rightarrow \infty \), there is a nontrivial Lorentzian harmonic sphere \((\sigma ^i,\xi ^i):\mathbb {R}^2\cup \{\infty \} \rightarrow N\times \mathbb {R}\) which is the weak limit of \((u_n(x^i_n+r^i_nx),v_n(x^i_n+r^i_nx))\) in \(W^{1,2}_{loc}(\mathbb {R}^2)\).

     
Furthermore, for both of the two cases, there holds the energy identity
$$\begin{aligned} \lim _{n\rightarrow \infty }E_g(u_n,v_n;D^+_1(0)) =E_g(u,v;D^+_1(0))+\sum _{i=1}^LE_g(\sigma ^i,\xi ^i). \end{aligned}$$
(3.30)
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 \(x_n\rightarrow 0\) and \(r_n\rightarrow 0\) such that
$$\begin{aligned} E(u_n,v_n;D^+_{r_n}(x_n))=\sup _{\begin{array}{c} x\in D_\delta ^+,r\le r_n\\ D^+_r(x)\subset D_\delta ^+ \end{array}}E(u_n,v_n;D^+_r(x))=\frac{1}{8}\min \{\epsilon _1,\epsilon _2\}, \end{aligned}$$
(3.31)
where \(D^+_{r_n}(x_n):=\{(y=(y^1,y^2)\in \mathbb {R}^2|\ |y-x_n|\le r_n,\ y^2\ge 0)\}\). Denote \(d_n=dist(x_n,\partial ^0D^+)\). We have that either \(\limsup _{n\rightarrow \infty }\frac{d_n}{r_n}<\infty \) or \(\limsup _{n\rightarrow \infty }\frac{d_n}{r_n}=\infty \). We discuss these two cases respectively.

Case (a) \(\limsup _{n\rightarrow \infty }\frac{d_n}{r_n}<\infty \).

By taking a subsequence, we may assume that \(\lim _{n\rightarrow \infty }\frac{d_n}{r_n}=a\ge 0\). Denote
$$\begin{aligned} B_n:=\left\{ x\in \mathbb {R}^2|x_n+r_nx\in D^+\right\} . \end{aligned}$$
We have that as \(n\rightarrow \infty \),
$$\begin{aligned} B_n\rightarrow \mathbb {R}^2_a:=\{(x^1,x^2)|x^2\ge -a\} \end{aligned}$$
and for any \(x\in \{x^2=-a\}\) on the boundary, \(x_n+r_nx\rightarrow 0\).
Define
$$\begin{aligned} \widetilde{u}_n(x):=u_n(x_n+r_nx),\ \widetilde{v}_n(x):=v_n(x_n+r_nx). \end{aligned}$$
It is easy to get that \((\widetilde{u}_n,\widetilde{v}_n):B_n\rightarrow N\times \mathbb {R}\) is an approximate Lorentzian harmonic map with \((\widetilde{\tau }_n,\widetilde{\kappa }_n)=r_n^2(\tau _n,\kappa _n)\) and
$$\begin{aligned} (\widetilde{u}_n(x),\widetilde{v}_n(x))=(\varphi (x_n+r_nx),\psi (x_n+r_nx)), \text{ if } x_n+r_nx\in \partial ^0D^+. \end{aligned}$$
Lemma 2.2 and (3.31) tell us that for any \(D_R(0)\subset \mathbb {R}^2\),
$$\begin{aligned} \Vert \widetilde{u}_n\Vert _{W^{2,2}(D_R(0)\cap B_n)}+\Vert \widetilde{v}_n\Vert _{W^{2,2}(D_R(0)\cap B_n)}\le C(\lambda _1,\lambda _2,\Lambda ,R,N). \end{aligned}$$
By a similar argument as in Section 4 of [8], after taking a subsequence of \((\widetilde{u}_n,\widetilde{v}_n)\) if necessary (still denoted by \((\widetilde{u}_n,\widetilde{v}_n)\)), there is a Lorentzian harmonic map \((\widetilde{u},\widetilde{v})\in W^{1,2}(\mathbb {R}_a^2,N\times \mathbb {R})\) with the constant boundary condition \((\widetilde{u},\widetilde{v})|_{\partial \mathbb {R}_a^2}=(\phi (0),\psi (0))\) such that, for any \(R>0\),
$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert \nabla \widetilde{u}_n\Vert _{L^{2}(D_R(0)\cap B_n)}= & {} \Vert \nabla \widetilde{u}\Vert _{L^{2}(D_R(0)\cap \mathbb {R}^2_a)},\\ \lim _{n\rightarrow \infty }\Vert \nabla \widetilde{v}_n\Vert _{L^{2}(D_R(0)\cap B_n)}= & {} \Vert \nabla \widetilde{v}\Vert _{L^{2}(D_R(0)\cap \mathbb {R}^2_a)}. \end{aligned}$$
Moreover, since \(E(\widetilde{u},\widetilde{v};D_1(0)\cap \mathbb {R}_a^2) =\frac{1}{8}\min \{\epsilon _1,\epsilon _2\}\), \((\widetilde{u},\widetilde{v})\) is a nontrivial Lorentzian harmonic map with constant boundary \((\phi (0),\psi (0))\).

Case (b) \(\limsup _{n\rightarrow \infty }\frac{d_n}{r_n}=\infty \).

In this case, \((\widetilde{u}_n,\widetilde{v}_n)\) lives in \(B_n\) which tends to \(\mathbb {R}^2\) as \(n\rightarrow \infty \). Moreover, for any \(x\in \mathbb {R}^2\), when n is sufficiently large, by (3.31), we have
$$\begin{aligned} E(\widetilde{u}_n,\widetilde{v}_n;D_1(x))\le \frac{\epsilon _1}{8}. \end{aligned}$$
According to Lemma 2.1, there exist a subsequence of \((\widetilde{u}_n,\widetilde{v}_n)\) which is still denoted by \((\widetilde{u}_n,\widetilde{v}_n)\)) and a Lorentzian harmonic map \((\widetilde{u}(x),\widetilde{v}(x))\in W^{1,2}(\mathbb {R}^2,N\times \mathbb {R})\) such that
$$\begin{aligned} \lim _{n\rightarrow \infty }(\widetilde{u}_n(x),\widetilde{v}_n(x))=(\widetilde{u}(x),\widetilde{v}(x)) \text{ in } W^{1,2}_{loc}(\mathbb {R}^2). \end{aligned}$$
By Theorem 2.8, \((\widetilde{u},\widetilde{v})\) can be extended to a Lorentzian harmonic sphere and (3.31) tells us that it is nontrivial.
We call the Lorentzian harmonic map \((\widetilde{u},\widetilde{v})\) 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 \(0\in D_1^+(0)\). Under this assumption, similar to (3.10), we have that, for any \(\epsilon >0\), there exist constants \(\delta >0\) and \(R>0\) such that
$$\begin{aligned} E(u_n,v_n;D^+_{4\rho }(x_n)\setminus D^+_{\frac{\rho }{2}}(x_n))\le \epsilon ^2\ \text {for any} \ \rho \in [r_nR,\frac{\delta }{2}] \end{aligned}$$
(3.32)
when n is large enough.
Now to prove the energy identity (3.30) is equivalent to prove
$$\begin{aligned} \lim _{R\rightarrow \infty }\lim _{\delta \rightarrow 0}\lim _{n\rightarrow \infty }E_g(u_n,v_n;D^+_\delta (x_n)\setminus D^+_{r_nR}(x_n))=0. \end{aligned}$$
(3.33)
We shall prove (3.33) for the two cases respectively.

For case (a) \(\lim _{n\rightarrow \infty }\frac{d_n}{r_n}=a<\infty \).

For n and R are sufficiently large, we decompose the neck domain \(D^+_\delta (x_n)\setminus D^+_{r_nR}(x_n)\) into three parts which follows the decomposition in [15, 16].
$$\begin{aligned} D^+_\delta (x_n)\setminus D^+_{r_nR}(x_n)&=D^+_\delta (x_n)\setminus D^+_{\frac{\delta }{2}}(x'_n)\cup D^+_{\frac{\delta }{2}}(x'_n)\setminus D^+_{2r_nR}(x'_n)\cup D^+_{2r_nR}(x'_n)\setminus D^+_{r_nR}(x_n)\\&:=\Omega _1\cup \Omega _2\cup \Omega _3. \end{aligned}$$
Here \(x_n'\in \partial ^0 D^+\) is the projection of \(x_n\), i.e. \(d_n=|x_n-x_n'|\).
Since \(\lim _{n\rightarrow \infty }\frac{d_n}{r_n}=a\), when n and R are large enough, it is easy to get that
$$\begin{aligned} \Omega _1\subset D^+_\delta (x_n)\setminus D^+_{\frac{\delta }{4}}(x_n)\quad \text {and} \quad \Omega _3\subset D^+_{4r_nR}(x_n)\setminus D^+_{r_nR}(x_n). \end{aligned}$$
Moreover, for any \(\rho \in [r_nR,\frac{\delta }{2}]\), there holds
$$\begin{aligned} D^+_{2\rho }(x'_n)\setminus D^+_{\rho }(x'_n)\subset D^+_{4\rho }(x_n)\setminus D^+_{\rho /2}(x_n). \end{aligned}$$
Then we get from (3.32) that
$$\begin{aligned} E(u_n,v_n;\Omega _1)+E(u_n,v_n;\Omega _3)\le \epsilon ^2 \end{aligned}$$
(3.34)
and
$$\begin{aligned} E(u_n,v_n;D^+_{2\rho }(x'_n)\setminus D^+_{\rho }(x'_n))\le \epsilon ^2 \text{ for } \text{ any } \rho \in \left[ r_nR,\frac{\delta }{2}\right] . \end{aligned}$$
By Lemma 2.1, we have
$$\begin{aligned}&\Vert u_n\Vert _{osc(D^+_{2\rho }(x'_n)\setminus D^+_{\rho }(x'_n))}+\Vert v_n\Vert _{osc(D^+_{2\rho }(x'_n)\setminus D^+_{\rho }(x'_n))}\nonumber \\&\quad \le C(\Vert (\nabla u_n,\nabla v_n)\Vert _{L^2(D^+_{4\rho }(x'_n)\setminus D^+_{\rho /2}(x'_n))}+\Vert (\nabla \phi ,\nabla \psi )\Vert _{L^2(D^+_{4\rho }(x'_n)\setminus D^+_{\rho /2}(x'_n))}\nonumber \\&\qquad +\rho \Vert (\nabla ^2 \varphi ,\nabla ^2\psi )\Vert _{L^2(D^+_{4\rho }(x'_n)\setminus D^+_{\rho /2}(x'_n))}+\rho \Vert (\tau _n,\kappa _n)\Vert _{L^2(D^+_{4\rho }(x'_n)\setminus D^+_{\rho /2}(x'_n))}) \end{aligned}$$
(3.35)
for any \(\rho \in [r_nR,\frac{\delta }{2}]\).
To estimate the energy concentration in \(\Omega _2\), we define \(\mu _n(x):=u_n(x)-\varphi (x)\) for \(x\in \Omega _2\) and
$$\begin{aligned} \widehat{\mu }_n(x):= {\left\{ \begin{array}{ll} \mu _n(x)\ &{}x\in \Omega _2,\\ -\mu _n(x')\ &{}x\in \widehat{\Omega }_2\setminus \Omega _2, \end{array}\right. } \end{aligned}$$
where \(\widehat{\Omega }_2:=D_{\frac{\delta }{2}}(x'_n)\setminus D_{2r_nR}(x'_n)\), \(x=(x^1,x^2)\) and \(x'=(x^1,-x^2)\). It is easy to get that \(\widehat{\mu }_n(x)\in W^{2,2}(\widehat{\Omega }_2)\) and satisfies
$$\begin{aligned} |\Delta \widehat{\mu }_n(x)|\le {\left\{ \begin{array}{ll} C(|\nabla u_n(x)|^2+|\nabla v_n(x)|^2)+|\tau _n(x)|+|\Delta \varphi (x)|,\ &{}x\in \Omega _2,\\ C(|\nabla u_n(x')|^2+|\nabla v_n(x')|^2)+|\tau _n(x')|+|\Delta \varphi (x')|,\ &{}x\in \widehat{\Omega }_2\setminus \Omega _2. \end{array}\right. } \end{aligned}$$
Define
$$\begin{aligned} \widehat{\mu }_n^*(r):=\frac{1}{2\pi }\int _{0}^{2\pi }\widehat{\mu }_n(r,\theta )d\theta , \end{aligned}$$
where \((r,\theta )\) is the polar coordinates at \(x_n'\). By (3.35), we have
$$\begin{aligned} \Vert \widehat{\mu }_n(x)-\widehat{\mu }_n^*(x)\Vert _{L^\infty (\widehat{\Omega }_2)}&\le \sup _{r_nR\le \rho \le \frac{\delta }{2}}\Vert \widehat{\mu }_n(x)\Vert _{osc(D_{2\rho }(x'_n)\setminus D_{\rho }(x'_n))}\nonumber \\&\le 2\sup _{r_nR\le \rho \le \frac{\delta }{2}}\Vert \mu _n(x)\Vert _{osc(D^+_{2\rho }(x'_n)\setminus D^+_{\rho }(x'_n))}\nonumber \\&\le C( \epsilon +\delta ). \end{aligned}$$
Similar to the proof of (3.15), we can obtain
$$\begin{aligned}&\int _{\widehat{\Omega }_2}|r^{-1} \frac{\partial \widehat{\mu }_n}{\partial \theta }|^2dx\nonumber \\&\quad \le C(\epsilon +\delta )\left( \int _{\widehat{\Omega }_2}|\Delta \widehat{\mu }_n|dx+\int _{\partial D_{\frac{\delta }{2}}(x_n')}|\frac{\partial \widehat{\mu }_n}{\partial r}|+\int _{\partial D_{2r_nR}(x_n')}|\frac{\partial \widehat{\mu }_n}{\partial r}|\right) . \end{aligned}$$
(3.36)
By direct computations, one can get that
$$\begin{aligned} \int _{\widehat{\Omega }_2}|r^{-1} \frac{\partial \widehat{\mu }_n}{\partial \theta }|^2dx&=2\int _{\Omega _2}|r^{-1} \frac{\partial \mu _n}{\partial \theta }|^2dx\nonumber \\&= 2\int _{\Omega _2}|r^{-1} \frac{\partial u_n}{\partial \theta }|^2dx-4\int _{\Omega _2}r^{-2} \frac{\partial u_n}{\partial \theta }\frac{\partial \phi }{\partial \theta }dx+2\int _{\Omega _2}|r^{-1} \frac{\partial \phi }{\partial \theta }|^2dx\nonumber \\&\ge 2\int _{\Omega _2}|r^{-1} \frac{\partial u_n}{\partial \theta }|^2dx-C\delta \end{aligned}$$
(3.37)
and
$$\begin{aligned} \int _{\widehat{\Omega }_2}|\Delta \widehat{\mu }_n|dx&\le C\int _{\Omega _2}(|\nabla u_n|^2+|\nabla v_n|^2)dx+\int _{\widehat{\Omega }_2}(|\tau _n|+|\Delta \phi |)dx\nonumber \\&\le C\int _{\Omega _2}(|\nabla u_n|^2+|\nabla v_n|^2)dx+C\delta . \end{aligned}$$
(3.38)
For the boundary terms of the right hand side of (3.36), by the trace theory and Lemma 2.2, we have
$$\begin{aligned}&\int _{\partial D_{\delta /2}(x_n')}|\frac{\partial \widehat{\mu }_n}{\partial r}| = 2\int _{\partial ^+ D_{\delta /2}(x_n')}|\frac{\partial \widehat{\mu }_n}{\partial r}| \le C\int _{\partial ^+ D_{\delta /2}(x_n')}(|\nabla u_n|+|\nabla \varphi |)\nonumber \\&\quad \le C\big (\Vert (\nabla u_n,\nabla v_n)\Vert _{L^2(D^+_{\delta }(x_n')\setminus D^+_{\frac{1}{4}\delta }(x_n') )}+\delta \Vert (\nabla ^2 u_n,\nabla ^2 v_n)\Vert _{L^2(D^+_{\delta }(x_n')\setminus D^+_{\frac{1}{4}\delta }(x_n') )}+\delta \big )\nonumber \\&\quad \le C\big (\Vert (\nabla u_n,\nabla v_n)\Vert _{L^2(D^+_{\frac{4}{3}\delta }(x_n)\setminus D^+_{\frac{1}{6}\delta }(x_n) )}+\Vert (\nabla \phi ,\nabla \psi )\Vert _{L^2(D^+_{\frac{4}{3}\delta }(x_n)\setminus D^+_{\frac{1}{6}\delta }(x_n) )}\nonumber \\&\qquad +\delta \Vert (\nabla ^2\phi ,\nabla ^2\psi )\Vert _{L^2(D^+_{\frac{4}{3}\delta }(x_n)\setminus D^+_{\frac{1}{6}\delta }(x_n) )}+\delta \Vert (\tau _n,\kappa _n)\Vert _{L^2(D^+_{\frac{4}{3}\delta }(x_n)\setminus D^+_{\frac{1}{6}\delta }(x_n) )}+\delta \big )\nonumber \\&\le C(\epsilon +\delta ). \end{aligned}$$
Similarly, we have
$$\begin{aligned} \int _{\partial D_{2r_nR}(x_n')}|\frac{\partial \widehat{\mu }_n}{\partial r}|\le C(\epsilon +\delta ). \end{aligned}$$
Combining these two estimates with (3.36), (3.37) and (3.38), we get
$$\begin{aligned} \int _{\Omega _2}|r^{-1} \frac{\partial u_n}{\partial \theta }|^2dx\le C(\epsilon +\delta ). \end{aligned}$$
(3.39)
Similarly, we have
$$\begin{aligned} \int _{\Omega _2}|r^{-1} \frac{\partial v_n}{\partial \theta }|^2dx\le C(\epsilon +\delta ). \end{aligned}$$
(3.40)
By Corollary 2.6, we have
$$\begin{aligned}&\int _{D^+_{2^{i+1}r_nR}(x_n')\setminus D^+_{2^ir_nR}(x_n')}\left( |\frac{\partial u_n}{\partial r}|^2-\beta (u_n)|\frac{\partial v_n}{\partial r}+\omega _j\frac{\partial u^j_n}{\partial r}|^2\right) dx\nonumber \\&\quad \le C\int _{D^+_{2^{i+1}r_nR}(x_n')\setminus D^+_{2^ir_nR}(x_n')}|r^{-1}\frac{\partial u_n}{\partial \theta }|^2dx\nonumber \\&\qquad +C\int _{D^+_{2^{i+1}r_nR}(x_n')\setminus D^+_{2^ir_nR}(x_n')}|r^{-1}\frac{\partial v_n}{\partial \theta }|^2dx+2^ir_nR. \end{aligned}$$
(3.41)
Thus, we arrive at
$$\begin{aligned}&\int _{\Omega _2}\left( |\frac{\partial u_n}{\partial r}|^2-\beta (u_n)|\frac{\partial v_n}{\partial r}+\omega _j\frac{\partial u^j_n}{\partial r}|^2\right) dx\nonumber \\&\quad =\sum _{i=0}^{m_n-1}\int _{D^+_{2^{i+1}r_nR}(x_n')\setminus D^+_{2^ir_nR}(x_n')}\left( |\frac{\partial u_n}{\partial r}|^2-\beta (u_n)|\frac{\partial v_n}{\partial r}+\omega _j\frac{\partial u^j_n}{\partial r}|^2\right) dx\nonumber \\&\quad \le C(\epsilon +\delta ). \end{aligned}$$
(3.42)
Then (3.39), (3.40), (3.41), (3.42) and (3.34) imply (3.33).

For case (b) \(\lim _{n\rightarrow \infty }\frac{d_n}{r_n}=\infty \).

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 \(D^+_\delta (x_n)\setminus D^+_{r_nR}(x_n)\) as in [15, 16] as follows
$$\begin{aligned} D^+_\delta (x_n)\setminus D^+_{r_nR}(x_n)&=D^+_\delta (x_n)\setminus D^+_{\frac{\delta }{2}}(x'_n)\cup D^+_{\frac{\delta }{2}}(x'_n)\setminus D^+_{2d_n}(x'_n)\nonumber \\&\quad \cup D^+_{2d_n}(x'_n)\setminus D^+_{d_n}(x_n)\cup D^+_{d_n}(x_n)\setminus D^+_{r_nR}(x_n)\nonumber \\&:=\Omega _1\cup \Omega _2\cup \Omega _3\cup \Omega _4. \end{aligned}$$
(3.43)
Since \(\lim _{n\rightarrow \infty }d_n=0\) and \(\lim _{n\rightarrow \infty }\frac{d_n}{r_n}=\infty \), when n is large enough, it is easy to get that
$$\begin{aligned} \Omega _1\subset D^+_\delta (x_n)\setminus D^+_{\frac{\delta }{4}}(x_n),\quad \text {and} \quad \Omega _3\subset D^+_{4d_n}(x_n)\setminus D^+_{d_n}(x_n). \end{aligned}$$
Moreover, for any \(\rho \in [d_n, \frac{\delta }{2}]\), there holds
$$\begin{aligned} D^+_{2\rho }(x'_n)\setminus D^+_{\rho }(x'_n)\subset D^+_{4\rho }(x_n)\setminus D^+_{\rho /2}(x_n). \end{aligned}$$
By assumption (3.32), we have
$$\begin{aligned} E(u_n;\Omega _1)+E(u_n;\Omega _3)\le \epsilon ^2 \end{aligned}$$
and
$$\begin{aligned} \int _{D^+_{2\rho }(x'_n)\setminus D^+_{\rho }(x'_n)}|\nabla u_n|^2dx\le \epsilon ^2 \text{ for } \text{ any } \rho \in \left( d_n, \frac{\delta }{2}\right) . \end{aligned}$$
Noting that \(\Omega _4=D^+_{d_n}(x_n)\setminus D^+_{r_nR}(x_n)=D_{d_n}(x_n)\setminus D_{r_nR}(x_n)\), by Lemma 3.2, there holds
$$\begin{aligned} \lim _{R\rightarrow \infty }\lim _{n\rightarrow 0}E_g(u_n,v_n;D_{d_n}(x_n)\setminus D_{r_nR}(x_n))=0. \end{aligned}$$
To estimate the energy concentration in \(\Omega _2\), we can use the same arguments as for case (a) to get that
$$\begin{aligned} E_g(u_n,v_n;\Omega _2)\le C(\epsilon +\delta ). \end{aligned}$$
(3.44)
Thus we finish the proof of the lemma. \(\square \)

Similar to Lemma 3.3, when \(\omega \equiv 0\), we have

Lemma 3.5

Assume that \(\{(u_n,v_n)\}\) is an approximate Lorentzian harmonic map sequence as in Lemma 3.4 and additionally, we assume that \(\omega \equiv 0\) and \(\Vert \nabla v_n\Vert _{L^{p}(D^+)}\le \Lambda \) for some \(p>2\), then case (a) in Lemma 3.4 will not happen and in case (b), we have that \(\sigma ^i:\mathbb {R}^2\cup \{\infty \}\rightarrow N\) is a nontrivial harmonic sphere, \(\xi ^i\) is a constant map and (3.30) becomes
$$\begin{aligned} \lim _{n\rightarrow \infty }E(u_n;D^+_1(0))&=E(u;D^+_1(0))+\sum _{i=1}^LE(\sigma ^i), \end{aligned}$$
(3.45)
$$\begin{aligned} \lim _{n\rightarrow \infty }E(v_n;D^+_1(0))&=E(v;D^+_1(0)). \end{aligned}$$
(3.46)
Furthermore, the image
$$\begin{aligned} u(D^+_1(0))\cup \bigcup _{i=1}^L\sigma ^i(\mathbb {R}^2) \end{aligned}$$
(3.47)
is a connected set.

Proof

We use the same symbols as in Lemma 3.4. First, let us show that if \(\omega \equiv 0\), Case (a) will not happen. In fact, since \(\widetilde{v}\) satisfies
$$\begin{aligned}div(\beta (\widetilde{u})\nabla \widetilde{v})=0\ {\text{ i }n}\ \mathbb {R}^2_a\end{aligned}$$
and \(\widetilde{v}|_{\partial \mathbb {R}^2_a}\equiv \psi (0)\), \(\widetilde{v}\) must be a constant map. Thus, \(\widetilde{u}\) is a harmonic map from \(\mathbb {R}^2_a\) with constant boundary data \(\widetilde{u}|_{\partial \mathbb {R}^2_a}=\phi (0)\) which implies that \(\widetilde{u}\) is a constant map [18]. This is a contradiction with \(E(\widetilde{u},\widetilde{v};\mathbb {R}^2_a) \ge \frac{1}{8}\min \{\epsilon _1,\epsilon _2\}\).
For case (b), when \(\omega \equiv 0\), it is clear that \(\widetilde{v}\) satisfies the equation
$$\begin{aligned} div(\beta (\widetilde{u})\nabla \widetilde{v})=0 \end{aligned}$$
in \({\mathbb {S}}^2\) with finite energy \(\Vert \nabla \widetilde{v}\Vert _{L^2({\mathbb {S}}^2)}\le C\) which implies that \(\widetilde{v}\) must be a constant map. Therefore \(\widetilde{u}:{\mathbb {S}}^2\rightarrow N\) is a nontrivial harmonic sphere.
Now to prove the energy identities (3.45) and (3.46) is equivalent to prove
$$\begin{aligned} \lim _{R\rightarrow \infty }\lim _{\delta \rightarrow 0}\lim _{n\rightarrow \infty }E(u_n,v_n;D^+_\delta (x_n)\setminus D^+_{r_nR}(x_n))=0. \end{aligned}$$
(3.48)
To prove the no neck result (3.47) is equivalent to prove
$$\begin{aligned} \lim _{R\rightarrow \infty }\lim _{\delta \rightarrow 0}\lim _{n\rightarrow \infty }\Vert u_n\Vert _{osc(D^+_\delta (x_n)\setminus D^+_{r_nR}(x_n))}=0. \end{aligned}$$
(3.49)
We decompose the neck domain as (3.43). Since \(\lim _{n\rightarrow \infty }d_n=0\) and \(\lim _{n\rightarrow \infty }\frac{d_n}{r_n}=\infty \), when n is large enough, it is easy to see that
$$\begin{aligned} \Omega _1\subset D^+_\delta (x_n)\setminus D^+_{\frac{\delta }{4}}(x_n)\quad \text {and} \quad \Omega _3\subset D^+_{4d_n}(x_n)\setminus D^+_{d_n}(x_n). \end{aligned}$$
By (3.32), we have
$$\begin{aligned} E(u_n;\Omega _1)+E(u_n;\Omega _3)\le \epsilon ^2, \end{aligned}$$
(3.50)
which implies that there is no energy loss on \(\Omega _1\cup \Omega _3\). By Lemmas 2.1, 2.2 and (3.32), we get
$$\begin{aligned}&\Vert u_n\Vert _{Osc(D^+_{\delta }(x_n)\setminus D^+_{\frac{\delta }{4}}(x'_n))}\nonumber \\&\quad \le \Vert u_n\Vert _{Osc(D^+_{\delta }(x_n)\setminus D^+_{\frac{\delta }{5}}(x_n))}\nonumber \\&\quad \le C(\Vert (\nabla u_n,\nabla v_n)\Vert _{L^2(D^+_{\frac{4\delta }{3}}(x_n)\setminus D^+_{\frac{\delta }{6}}(x_n))}+\Vert (\nabla \phi ,\nabla \psi )\Vert _{L^2(D^+_{\frac{4\delta }{3}}(x_n)\setminus D^+_{\frac{\delta }{6}}(x_n))}\nonumber \\&\qquad +\delta \Vert (\nabla ^2\phi ,\nabla ^2\psi )\Vert _{L^2(D^+_{\frac{4\delta }{3}}(x_n)\setminus D^+_{\frac{\delta }{6}}(x_n))}+\delta \Vert (\tau _n,\kappa _n)\Vert _{L^2(D^+_{\frac{4\delta }{3}}(x_n)\setminus D^+_{\frac{\delta }{6}}(x_n))})\nonumber \\&\quad \le C(\epsilon +\delta ) \end{aligned}$$
(3.51)
and
$$\begin{aligned}&\Vert u_n\Vert _{Osc(D^+_{2d_n}(x'_n)\setminus D^+_{d_n}(x_n))}\nonumber \\&\quad \le \Vert u_n\Vert _{Osc(D^+_{3d_n}(x_n)\setminus D^+_{d_n}(x_n))} \nonumber \\&\quad \le C(\Vert (\nabla u_n,\nabla v_n)\Vert _{L^2(D^+_{4d_n}(x_n)\setminus D^+_{\frac{d_n}{2}}(x_n))}+\Vert (\nabla \phi ,\nabla \psi )\Vert _{L^2(D^+_{4d_n}(x_n)\setminus D^+_{\frac{d_n}{2}}(x_n))}\nonumber \\&\qquad +d_n\Vert (\nabla ^2\phi ,\nabla ^2\psi )\Vert _{L^2(D^+_{4d_n}(x_n)\setminus D^+_{\frac{d_n}{2}}(x_n))}+d_n\Vert (\tau _n,\kappa _n)\Vert _{L^2(D^+_{4d_n}(x_n)\setminus D^+_{\frac{d_n}{2}}(x_n))})\nonumber \\&\quad \le C(\epsilon +\delta ), \end{aligned}$$
(3.52)
when n, R are large and \(\delta \) is small, which implies that there is no neck on \(\Omega _1\cup \Omega _3\).
Moreover, for any \(d_n\le \rho \le \frac{\delta }{2}\), there holds
$$\begin{aligned} D^+_{2\rho }(x'_n)\setminus D^+_{\rho }(x'_n)\subset D^+_{4\rho }(x_n)\setminus D^+_{\rho /2}(x_n). \end{aligned}$$
when n is big enough, then (3.32) tells us
$$\begin{aligned} \int _{D^+_{2\rho }(x'_n)\setminus D^+_{\rho }(x'_n)}|\nabla u_n|^2dx\le \epsilon ^2 \text{ for } \text{ any } \rho \in (d_n, \frac{\delta }{2}). \end{aligned}$$
Combining this with Lemma 2.2, we get
$$\begin{aligned}&\Vert u_n\Vert _{osc(D^+_{2\rho }(x'_n)\setminus D^+_{\rho }(x'_n))}\nonumber \\&\quad \le C(\Vert (\nabla u_n,\nabla v_n)\Vert _{L^2(D^+_{4\rho }(x'_n)\setminus D^+_{\rho /2}(x'_n))}+\Vert (\nabla \phi ,\nabla \psi )\Vert _{L^2(D^+_{4\rho }(x'_n)\setminus D^+_{\rho /2}(x'_n))}\nonumber \\&\qquad +\rho \Vert (\nabla ^2 \varphi ,\nabla ^2\psi )\Vert _{L^2(D^+_{4\rho }(x'_n)\setminus D^+_{\rho /2}(x'_n))}+\rho \Vert (\tau _n,\kappa _n)\Vert _{L^2(D^+_{4\rho }(x'_n)\setminus D^+_{\rho /2}(x'_n))}) \end{aligned}$$
for any \(\rho \in (d_n, \frac{\delta }{2})\).
Noting that \(\Omega _4=D^+_{d_n}(x_n)\setminus D^+_{r_nR}(x_n)=D_{d_n}(x_n)\setminus D_{r_nR}(x_n)\), the proofs of (3.48) and (3.49) are reduced to the case in Lemma 3.3 and we have
$$\begin{aligned} \lim _{R\rightarrow \infty }\lim _{n\rightarrow 0}E(u_n,v_n;D_{d_n}(x_n)\setminus D_{r_nR}(x_n))=0 \end{aligned}$$
(3.53)
and
$$\begin{aligned} \lim _{R\rightarrow \infty }\lim _{n\rightarrow 0}osc(u_n)_{D_{d_n}(x_n)\setminus D_{r_nR}(x_n)}=0. \end{aligned}$$
(3.54)
To prove that there is no energy loss on \(\Omega _2\), noting that \(\Vert \nabla v_n\Vert _{L^p(D^+)}\le \Lambda \) for some \(p>2\), we get
$$\begin{aligned} \int _{D^+_\delta (x_n)\setminus D^+_{r_nR}(x_n)}|\nabla v_n|^2dx\le C\delta ^{1-\frac{2}{p}} (\int _{D^+_\delta (x_n)\setminus D^+_{r_nR}(x_n)}|\nabla v_n|^pdx)^\frac{2}{p}\le C\delta ^{1-\frac{2}{p}}. \end{aligned}$$
(3.55)
Combining this with (3.44), we obtain
$$\begin{aligned} E(u_n,\Omega _2)\le CE(v_n,\Omega )+C(\epsilon +\delta )\le C(\epsilon +\delta ^{1-\frac{2}{p}}) . \end{aligned}$$
(3.56)
Then, (3.48) follows from (3.50), (3.53), (3.55) and (3.56).

Now we only need to analyze the neck on \(\Omega _2\).

We denote \(Q(s):=D^+_{2^{s_0+s}2r_nR}(x_n')\setminus D^+_{2^{s_0-s}2r_nR}(x_n')\) and \(\widehat{Q}(s):=D_{2^{s_0+s}2r_nR}(x_n')\setminus D_{2^{s_0-s}2r_nR}(x_n')\), where \(0\le s_0\le m_n\) and \(0\le s\le \min \{s_0,m_n-s_0\}\). Let
$$\begin{aligned} f(s):=\int _{Q(s)}|\nabla u_n|^2dx. \end{aligned}$$
Similar to the derivation of (3.15), we can obtain
$$\begin{aligned}&\int _{\widehat{Q}(s)}|\nabla \widehat{\mu }_n|^2dx-\int _{\widehat{Q}(s)}| \frac{\partial \widehat{\mu }_n}{\partial r}|^2dx\nonumber \\&\quad \le C(\epsilon +\delta )\int _{\widehat{Q}(s)}|\Delta \widehat{\mu }_n|dx+\int _{\partial \widehat{Q}(s)}\frac{\partial \widehat{\mu }_n}{\partial r}(\widehat{\mu }_n-\widehat{\mu _n}^*). \end{aligned}$$
(3.57)
By direct computations, we obtain
$$\begin{aligned}&\int _{\widehat{Q}(s)}|\nabla \widehat{\mu }_n|^2dx-\int _{\widehat{Q}(s)}| \frac{\partial \widehat{\mu }_n}{\partial r}|^2dx=\int _{Q(s)}|\nabla \mu _n|^2dx-2\int _{Q(s)}\left( |\frac{\partial \mu _n}{\partial r}|^2-\frac{1}{2}|\nabla \mu _n|^2\right) dx\\&\quad =\int _{Q(s)}|\nabla u_n|^2dx-2\int _{Q(s)}\left( |\frac{\partial u_n}{\partial r}|^2-\frac{1}{2}|\nabla u_n|^2\right) dx+4\int _{Q(s)}\left( \frac{\partial u_n}{\partial r}\frac{\partial \phi }{\partial r}-\nabla u_n\nabla \phi \right) dx\\&\qquad +2\int _{Q(s)}\left( |\nabla \phi |^2-|\frac{\partial \phi }{\partial r}|^2\right) dx\\&\quad \ge \int _{Q(s)}|\nabla u_n|^2dx-2\int _{Q(s)}\left( |\frac{\partial u_n}{\partial r}|^2-\frac{1}{2}|\nabla u_n|^2\right) dx-C2^{s_0+s}r_nR. \end{aligned}$$
It is easy to check that (3.38) still holds on \(\Omega _2\). Combining this with \(\Vert \nabla v_n\Vert _{L^p(D^+)}\le C\), we have
$$\begin{aligned} \int _{\widehat{Q}(s)}|\Delta \widehat{\mu }_n|dx&\le C\int _{Q(s)}(|\nabla u_n|^2+|\nabla v_n|^2)dx+C2^{s_0+s}r_nR\\&\le C\int _{Q(s)}|\nabla u_n|^2dx+C(2^{s_0+s}r_nR)^{1-\frac{2}{p}}. \end{aligned}$$
Then (3.57) implies
$$\begin{aligned}&(1-C(\epsilon +\delta ))\int _{Q(t)}|\nabla u_n|^2dx\nonumber \\&\quad \le \int _{\partial \widehat{Q}(s)}\frac{\partial \widehat{\mu }_n}{\partial r}(\widehat{\mu }_n-\widehat{\mu }_n^*)+2\int _{Q(s)}\left( |\frac{\partial u_n}{\partial r}|^2-\frac{1}{2}|\nabla u_n|^2\right) dx+C(2^{s_0+s}r_nR)^{1-\frac{2}{p}}\nonumber \\&\quad \le \int _{\partial \widehat{Q}(s)}\frac{\partial \widehat{\mu }_n}{\partial r}(\widehat{\mu }_n-\widehat{\mu }_n^*)+C(2^{s_0+s}r_nR)^{1-\frac{2}{p}}, \end{aligned}$$
(3.58)
where the last inequality follows from Corollary 2.6 and (3.22).
For the boundary term on the right hand side of (3.58), by Hölder’s inequality and Poincare’s inequality, we have
$$\begin{aligned} \int _{\partial D_{2^{s_0+s}2r_nR}(x_n')}\frac{\partial \widehat{\mu }_n}{\partial r}(\widehat{\mu }_n-\widehat{\mu }_n^*)&\le \left( \int _{\partial D_{2^{s_0+s}2r_nR}(x_n')}|\frac{\partial \widehat{\mu }_n}{\partial r}|^2\int _{\partial ^+ D_{2^{s_0+s}2r_nR}(x_n')} |\widehat{\mu }_n-\widehat{\mu }_n^*|^2\right) ^{\frac{1}{2}}\\&\le C\left( \int _{\partial D_{2^{s_0+s}2r_nR}(x_n')}|\frac{\partial \widehat{\mu }_n}{\partial r}|^2\right) ^{\frac{1}{2}}\left( 2^{s_0+s}r_nR\int _0^{2\pi }|\frac{\partial \widehat{\mu }_n}{\partial \theta }|^2\right) ^{\frac{1}{2}}\\&\le C2^{s_0+s}d_n\int _{\partial D_{2^{s_0+s}2r_nR}(x_n')}|\nabla \widehat{\mu }_n|^2\\&\le C2^{s_0+s}d_n\int _{\partial ^+ D^+_{2^{s_0+s}2r_nR}(x_n')}|\nabla \mu _n|^2\\&\le C2^{s_0+s}d_n\int _{\partial ^+ D^+_{2^{s_0+s}2r_nR}(x_n')}|\nabla u_n|^2+C(2^{s_0+s}r_nR)^2. \end{aligned}$$
Similarly, we can obtain
$$\begin{aligned} \int _{\partial D_{2^{s_0-s}2r_nR}(x_n')}\frac{\partial \widehat{\mu }_n}{\partial r}(\widehat{\mu }_n-\widehat{\mu }_n^*)\le C2^{s_0-s}d_n\int _{\partial ^+ D^+_{2^{s_0-s}2r_nR}(x_n')}|\nabla u_n|^2+C(2^{s_0-s}r_nR)^2. \end{aligned}$$
Taking \(\epsilon \) and \(\delta \) sufficiently small, we have
$$\begin{aligned} \int _{Q(s)}|\nabla u_n|^2dx\le&C2^{s_0+s}2d_n\int _{\partial ^+ (D^+_{2^{s_0+s}2r_nR}(x_n'))}|\nabla u_n|^2\\&+C2^{s_0-s}2d_n\int _{\partial ^+ (D^+_{2^{s_0-s}2r_nR}(x_n'))}|\nabla u_n|^2+C(2^{s_0+s}r_nR)^{1-\frac{2}{p}}, \end{aligned}$$
which gives us
$$\begin{aligned} f(s)\le \frac{C}{\log 2}f'(s)+C(2^{s_0+s}r_nR)^{1-\frac{2}{p}}. \end{aligned}$$
(3.59)
(3.59) implies that
$$\begin{aligned} (2^{-\frac{s}{C}}f(s))'\ge -C(2^{s_0}r_nR)^{1-\frac{2}{p}}2^{(1-\frac{2}{p}-\frac{1}{C})s}. \end{aligned}$$
Integrating from 2 to L, we arrive at
$$\begin{aligned} f(2)\le C2^{-\frac{1}{C}L}f(L)+C(2^{s_0}r_nR)^{1-\frac{2}{p}}2^{(1-\frac{2}{p}-\frac{1}{C})L}. \end{aligned}$$
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 \(\Omega _1\cup \Omega _3\), \(\Omega _4\) and \(\Omega _2\) and get (3.48) and (3.49). \(\square \)

We can now prove Theorems 1.1 and  1.2.

Proof of Theorems 1.1 and 1.2

Theorem 1.1 is a direct conclusion of Lemmas 3.2 and 3.4.

If \(\omega \equiv 0\), \(\widetilde{v}=v-\psi \) satisfies
$$\begin{aligned} div(\beta (u)\nabla \widetilde{v})=-div(\beta (u)\nabla \psi )+\kappa \end{aligned}$$
with the boundary condition \(\widetilde{v}|_{\partial M}=0\). By Theorem 1 in [21], for any \(1<p<\infty \), we have
$$\begin{aligned} \Vert \nabla \widetilde{v}\Vert _{L^p(M)}\le C(\Vert \nabla \psi \Vert _{L^p(M)}+\Vert \kappa \Vert _{L^2(M)}). \end{aligned}$$
Thus we have
$$\begin{aligned} \Vert \nabla v\Vert _{L^p(M)}\le C(\Vert \nabla \psi \Vert _{L^p(M)}+\Vert \kappa \Vert _{L^2(M)}). \end{aligned}$$
Then, Theorem 1.2 is a direct conclusion of Lemmas 3.3 and 3.5. \(\square \)

4 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.

Lemma 4.1

(Lemmas 2.1, 2.4 in [8]) Suppose \((u,v)\in \mathcal {V}(M_0^{T_1};N\times \mathbb {R})\) is a solution of (1.14) and (1.15), then the Lorentzian energy \(E_g(u(t), v(t))\) is non-increasing on \([0,T_1)\) and for any \(0\le s\le t<T_1\), there holds
$$\begin{aligned} E_g(u(t), v(t))+\int _s^t\int _M|\partial _tu|^2dxdt\le E_g(u(s), v(s)). \end{aligned}$$
Moreover, for any \(1<p<\infty \), \(t>0\), there holds
$$\begin{aligned} \int _M|\nabla u(\cdot ,t)|^2dx+\int _M|\nabla v(\cdot ,t)|^pdx+\int _0^t\int _M|\partial _t u|^2dxdt\le C(p,\lambda _1,\lambda _2,\phi ,\psi ). \end{aligned}$$

Lemma 4.2

Let \((u,v)\in \mathcal {V}(M_0^{T_1};N\times \mathbb {R})\) be a solution to (1.14) and (1.15). There exists a positive constant \(R_0<1\) such that, for any \(x_0\in M\), \(0\le R\le R_0\) and \(0<s\le t<T_1\), there hold
$$\begin{aligned} E(u(t);B^M_R(x_0))\le E(u(s);B^M_{2R}(x_0))+C\frac{t-s}{R^2}, \end{aligned}$$
(4.1)
and
$$\begin{aligned} E(u(s);B^M_R(x_0))\le E(u(t);B^M_{2R}(x_0))+C\frac{t-s}{R^2}+C\int _s^t\int _{M}|\partial _tu|^2dxdt, \end{aligned}$$
(4.2)
where \(B^M_R(x_0)\subset M\) is the geodesic ball centered at point \(x_0\) with radius R, C is a positive constant depending on \(\lambda _1,\lambda _2,M,N,E(\phi ),\Vert \psi \Vert _{W^{1,4}(M)}\).

Proof

Let \(\eta \in C^\infty _0(B^M_{2R}(x_0))\) be a cut-off function such that \(\eta (x)=\eta (|x-x_0|)\), \(0\le \eta \le 1\), \(\eta |_{B^M_{R}(x_0)}\equiv 1\) and \(|\nabla \eta |\le \frac{C}{R}\). By direct computations, we get
$$\begin{aligned} \frac{d}{dt}\frac{1}{2}\int _{M}|\nabla u|^2\eta ^2= & {} \int _{M}\langle \nabla u,\nabla u_t\rangle \eta ^2\\= & {} \int _{\partial B^M_{2R}(x_0)}\frac{\partial u}{\partial r}\cdot u_t\eta ^2-\int _{M}\langle \Delta u, u_t\rangle \eta ^2-2\int _{M}\nabla u\cdot \nabla \eta \eta u_t\\= & {} \int _{M}\langle -u_t-B^{\top }(u)|\nabla v|^2, u_t\rangle \eta ^2-2\int _{M}\nabla u\cdot \nabla \eta \eta u_t\\= & {} -\int _{M}|u_t|^2\eta ^2-\int _MB^\top (u)|\nabla v|^2\cdot u_t\eta ^2-2\int _{M}\nabla u\cdot \nabla \eta \eta u_t. \end{aligned}$$
On the one hand, by Lemma 4.1 and Young’s inequality, we have
$$\begin{aligned} \frac{d}{dt}\frac{1}{2}\int _{M}|\nabla u|^2\eta ^2&\le -\frac{1}{2}\int _{M}|u_t|^2\eta ^2+C\int _{M}|\nabla u|^2|\nabla \eta |^2+C\int _{M}|\nabla v|^4\eta ^2\le \frac{C}{R^2}. \end{aligned}$$
By integrating the above inequality from s to t, we can get (4.1).
On the other hand, by Lemma 4.1 and Young’s inequality, we also have
$$\begin{aligned} \frac{d}{dt}\frac{1}{2}\int _{M}|\nabla u|^2\eta ^2&\ge -\frac{3}{2}\int _{M}|u_t|^2\eta ^2-C\int _{M}|\nabla u|^2|\nabla \eta |^2-C\int _{M}|\nabla v|^4\eta ^2\\&\ge -\frac{3}{2}\int _{M}|u_t|^2\eta ^2- \frac{C}{R^2}. \end{aligned}$$
Then (4.2) follows immediately from integrating the above inequality from s to t. \(\square \)

With the help of Lemma 4.2, we can apply the standard argument (see Lemma 6.4.10 in [20]) to obtain

Lemma 4.3

Let \((u,v)\in \mathcal {V}(M_0^{T_1};N\times \mathbb {R})\) be a solution to (1.14) and (1.15). Assume that there is only one singular point \(x_0\in M\) at time \(T_1\). Then there exists a positive number \(m>0\) such that, as \(t \uparrow T_1\),
$$\begin{aligned} |\nabla u|^2(x,t)dx\rightarrow m\delta _{x_0}+|\nabla u|^2(x,T_1)dx \ \ \text {as Radon measures}. \end{aligned}$$
(4.3)
Here \(\delta _{x_0}\) denotes the \(\delta -\)mass at \(x_0\).

Now we shall prove Theorems 1.3 and 1.4.

Proof of Theorem 1.3

In fact, Theorem 1.3 is a consequence of Lemma 4.1, Theorems 1.1 and 1.2.

By Lemma 4.1, we can find a positive sequence \(t_n\rightarrow \infty \), such that
$$\begin{aligned} \lim _{n\rightarrow \infty }\int _M|\partial _t u|^2(\cdot ,t_n)dx=0\quad and \quad E(u(\cdot ,t_n),v(\cdot ,t_n))\le C. \end{aligned}$$
Taking the sequence to be \((u_n,v_n)=(u(\cdot ,t_n),v(\cdot ,t_n))\) with \((\tau _n,h_n)=(\partial _tu(\cdot ,t_n),0)\) in Theorems 1.1 and 1.2, the conclusions of Theorem 1.3 follow immediately. \(\square \)

Proof of Theorem 1.4

With the help of Lemmas 4.2, 4.3, Theorems 1.1 and 1.2, the proof of (1.19) is almost the same as the proof for the harmonic map flow and we omit the details here. One can refer to [20] for the interior case and to [15, 16] for the boundary case.

It is not hard to prove that there is a unique weak limit \((u(T_1),v(T_1))\in W^{1,2}(M, N\times \mathbb {R})\) of (u(t), v(t)) in \(W^{1,2}(M)\) as \(t\rightarrow T_1\) (one can refer to the proof of Theorem 1.2 in [14] for a similar argument). Moreover, by Lemma 4.1,
$$\begin{aligned} v(t)\rightharpoonup v(T_1)\ \text {weakly}\ \text {in}\ W^{1,4}(M). \end{aligned}$$
Then, we have
$$\begin{aligned}&\int _M\beta (u(t))|\nabla v(t)|^2dx-\int _M\beta (u(T_1))|\nabla v(T_1)|^2dx\\&\quad =\int _M\beta (u(t))\nabla v(t)\nabla (v(t)-v(T_1)) +\left( \beta (u(t))\nabla v(t)-\beta (u(T_1))\nabla v(T_1)\right) \nabla v(T_1)dx\\&\quad =\int _M\left( \beta (u(t))-\beta (u(T_1))\right) \nabla v(t)\nabla v(T_1)+\beta (u(T_1))\left( \nabla v(t)-\nabla v(T_1)\right) \nabla v(T_1)dx\\&\quad =\mathbb {I}+{\mathbb {II}}, \end{aligned}$$
where the first term of the second line is zero by integrating by parts and Eq. (1.14). Noting that
$$\begin{aligned} \mathbb {I}\le C\Vert \nabla v(t)\Vert _{L^4(M)}\Vert \nabla v(T_1)\Vert _{L^4(M)}\Vert u(t)-u(T_1)\Vert _{L^2(M)}, \end{aligned}$$
by weak convergence, we have
$$\begin{aligned} \lim _{t\rightarrow T_1}\int _M\beta (u(t))|\nabla v(t)|^2dx=\int _M\beta (u(T_1))|\nabla v(T_1)|^2dx. \end{aligned}$$
Combining this with (1.19), we get (1.20). \(\square \)

Notes

Acknowledgements

Open access funding provided by Max Planck Society.

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Authors and Affiliations

  1. 1.Department of Mathematical SciencesTsinghua UniversityBeijingChina
  2. 2.Max Planck Institute for Mathematics in the SciencesLeipzigGermany
  3. 3.School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems of Ministry of EducationBeijing Normal UniversityBeijingChina

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