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.
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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
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 \(p\in N\),
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 \(\partial M\). 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
which is called the Lorentzian energy of the map (u, v) on M. Critical points (u, v) in \(C^2(M,N\times {\mathbb {R}})\) of the functional (1.2) are called Lorentzian harmonic maps from (M, h) into the Lorentzian manifold \((N\times {\mathbb {R}},g)\). Besides the Lorentzian energy \(E_g(u,v)\), we also consider
and call it the positive energy of u and (u, v) 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),
with the boundary data
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 (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 \(\omega \equiv 0\) 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 (u, v) satisfies
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
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
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
where \(B(u):=(B^1, B^2, \ldots , B^K)\) with
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
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
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 \((u,v)\in \mathcal {V}(M_0^{T_1};N\times \mathbb {R})\) (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
Furthermore, there exist sequences \(\{x_n^i\}_{i=1}^L\subset M\) and \(\{r_n^i\}_{i=1}^L\subset \mathbb {R}_+\) such that
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
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
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
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
Moreover, by the Sobolev embedding \(W^{2,p}\hookrightarrow C^0\), we have
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
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
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
Similarly,
First we assume that \(1<p<2\). By standard elliptic estimates and Poincare’s inequality, we obtain
where we use the Sobolev inequality
Taking \(\epsilon _2>0\) sufficiently small, we have
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
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
By direct computations, noting that
we have
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 \(\rho \) from \(r_0\) to \(2r_0\), we get
Corollary 2.4
For (u, v) 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
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
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
By direct computations, we have
and
Noting that
we have
Similarly, multiplying (1.7) by \(r\widetilde{u}_r\) and integrating by parts, we get
Combining (2.4) with (2.6) and (2.7), we obtain the conclusion of the lemma. \(\square \)
Corollary 2.6
For (u, v) 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
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
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:
then there exists a constant \(\epsilon _0>0\) depending on \((N\times \mathbb {R},g)\), such that if either
then (u, v) 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
The standard elliptic theory tells us that
It is easy to get that, if \(\epsilon _0\) is small enough, (u, v) 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
Sending R to infinity yields that (u, v) 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 (u, v) 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 (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 \(v\in W^{2,p}(D^+_{\rho }(0))\) 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])
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
Definition 3.1
For an approximate Lorentzian harmonic map sequence \(\{(u_n, v_n)\}\), define
and
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 \(x\in M\),
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
By taking a subsequence, we can assume that
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 (u, v) 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 (u, v). 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 (u, v) such that
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
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
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
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
and
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
Since we assume that there is only one bubble, we have that, for any \(\epsilon >0\), there holds that
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
Define
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
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 \(\delta =2^{m_n}r_nR\), 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
Since
we get
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
Furthermore, The image
is a connected set.
Proof
Similar to the proof of Lemma 3.2, to prove (3.18) and (3.19) is equivalent to proving
Since \(\Vert \nabla v_n\Vert _{L^p(D)}\le \Lambda \) for some \(p>2\), we get
Since \(\omega \equiv 0\), (3.17) implies that
Combining (3.16), (3.22) with (3.23), we can get (3.21).
To prove (3.20) is equivalent to prove
To prove (3.24), denote \(Q(s):=D_{2^{s_0+s}r_nR}\setminus D_{2^{s_0-s}r_nR}\) and consider
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
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 \(\epsilon \) and \(\delta \) sufficiently small, we get from (3.25) that
which implies
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
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
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
for some \(0<\alpha <1\) which satisfies that
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 (u, v) such that
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.
-
(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
$$\begin{aligned} \mathbb {R}^2_{a^i}:=\{(x^1,x^2)\in \mathbb {R}^2|x^2\ge a^i\}\ and\ \mathbb {R}^{2+}_{a^i}:=\{(x^1,x^2)\in \mathbb {R}^2|x^2> a^i\}; \end{aligned}$$ -
(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
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
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
We have that as \(n\rightarrow \infty \),
and for any \(x\in \{x^2=-a\}\) on the boundary, \(x_n+r_nx\rightarrow 0\).
Define
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
Lemma 2.2 and (3.31) tell us that for any \(D_R(0)\subset \mathbb {R}^2\),
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 \(R>0\),
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
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
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
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) \(\lim _{n\rightarrow \infty }\frac{d_n}{r_n}=a<\infty \).
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].
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
Moreover, for any \(\rho \in [r_nR,\frac{\delta }{2}]\), there holds
Then we get from (3.32) that
and
By Lemma 2.1, we have
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
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
Define
where \((r,\theta )\) is the polar coordinates at \(x_n'\). By (3.35), we have
Similar to the proof of (3.15), we can obtain
By direct computations, one can get that
and
For the boundary terms of the right hand side of (3.36), by the trace theory and Lemma 2.2, we have
Similarly, we have
Combining these two estimates with (3.36), (3.37) and (3.38), we get
Similarly, we have
By Corollary 2.6, we have
Thus, we arrive at
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
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
Moreover, for any \(\rho \in [d_n, \frac{\delta }{2}]\), there holds
By assumption (3.32), we have
and
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
To estimate the energy concentration in \(\Omega _2\), we can use the same arguments as for case (a) to get that
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
Furthermore, the image
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
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
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
To prove the no neck result (3.47) is equivalent to prove
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
By (3.32), we have
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
and
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
when n is big enough, then (3.32) tells us
Combining this with Lemma 2.2, we get
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
and
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
Combining this with (3.44), we obtain
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
Similar to the derivation of (3.15), we can obtain
By direct computations, we obtain
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
Then (3.57) implies
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
Similarly, we can obtain
Taking \(\epsilon \) and \(\delta \) sufficiently small, we have
which gives us
(3.59) implies that
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 \(\Omega _1\cup \Omega _3\), \(\Omega _4\) 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
with the boundary condition \(\widetilde{v}|_{\partial M}=0\). By Theorem 1 in [21], for any \(1<p<\infty \), we have
Thus we have
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
Moreover, for any \(1<p<\infty \), \(t>0\), there holds
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
and
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
On the one hand, by Lemma 4.1 and Young’s inequality, we have
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
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 \(t \uparrow T_1\),
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
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,
Then, we have
where the first term of the second line is zero by integrating by parts and Eq. (1.14). Noting that
by weak convergence, we have
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Communicated by Luigi Ambrosio.
The research is supported by NSFC Nos. 11471014, 11471299 and the Fundamental Research Funds for the Central Universities. The authors would like to thank the referee for detailed and useful comments.
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Han, X., Jost, J., Liu, L. et al. Bubbling analysis for approximate Lorentzian harmonic maps from Riemann surfaces. Calc. Var. 56, 175 (2017). https://doi.org/10.1007/s00526-017-1271-0
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DOI: https://doi.org/10.1007/s00526-017-1271-0