Convergence to harmonic maps for the Landau-Lifshitz flows on two dimensional hyperbolic spaces

In this paper, we prove that the solution of the Landau-Lifshitz flow $u(t,x)$ from $\mathbb{H}^2$ to $\mathbb{H}^2$ converges to some harmonic map as $t\to\infty$. The essential observation is that although there exist infinite numbers of harmonic maps from $\Bbb H^2$ to $\Bbb H^2$, the heat flow initiated from $u(t,x)$ for any given $t>0$ converges to the same harmonic map as the heat flow initiated from $u(0,x)$. This observation enables us to construct a variant of Tao's caloric gauge to reduce the convergence to harmonic maps for the Landau-Lifshitz flow to the decay of the corresponding heat tension field. The advantage of the strategy used in this paper is that we can see the limit harmonic map directly by evolving $u(0,x)$ along a heat flow without evolving the Landau-Lifshitz flow to the infinite time.


Introduction
Let (M, h) be a Riemannian manifold and (N, J, g) be a Kähler manifold, the Landau-Lifshitz flow is a map u(x, t) : where α ≥ 0, β ∈ R. In the local coordinates (x 1 , x 2 ) for M and (y 1 , y 2 ) for N , τ (u) is given by where h ij dx i dx j is the metric tension for M , (h ij ) is its inverse, Γ l m,n (u) is the Christoffel symbol at u. Usually, α ≥ 0 is called the Gilbert constant. When α = 0, (1.1) is called the Schrödinger flow. When β = 0, α > 0, it reduces to the heat flows of harmonic maps. In this paper, we consider the case when M = H 2 and N = H 2 .
Besides the physical motivation, such as the continuous isotropic Heisenberg spin model, the gauge theory, the dynamics of the magnetization field inside ferromagnetic material (Landau-Lifshitz [26]), the Landau-Lifshitz flow (LL) is also a typical model in the differential geometry.
We recall the following non-exhaustive list of works. The heat flow (HF) case (α > 0, β = 0) has been intensively studied in the past 60 years, for instance Eells and Sampson [11] for HF from closed manifolds to closed manifolds, Hamilton [16] for HF with Dirichlet boundary condition, Li and Tam [29] for HF from complete manifolds to complete manifolds. When α > 0, β ∈ R, the local well-posedness and partial regularity for weak solutions were considered by many authors for instance [12,35,45]. The local well-posedness of Schrödinger flow was studied by Sulem, Sulem and Bardos [40] for S 2 targets, Ding and Wang [10], McGahagan [34] for general Kähler manifolds. Chang, Shatah and Uhlenbeck [7] obtained the global well-posedness of maps into closed Riemann surfaces for small initial data. The global well-posedness for maps from R d into S 2 with small critical Sobolev norms was proved by Bejenaru, Ionescu, Kenig and Tataru [1,2].
The one dimensional case was studied by Rodnianski, Rubinstein and Staffilani [38].
The dynamic behavior of LL is known in some cases. For the equivariant Schrödinger flows from R 2 into S 2 with energy below the ground state and equivariant flows from R 2 into H 2 with initial data of finite energy, the global well-posedness and scattering in the gauge sense were proved by Bejenaru, Ionescu, Kenig and Tataru [3,4]. For the m-equivariant LL from R 2 into S 2 with initial data near the harmonic map, Gustafson, Kang, Tsai [13,14] proved asymptotic stability for m ≥ 4 and later the case m = 3 was proved by Gustafson, Nakanishi, Tsai [15].
Usually the dynamics for flows defined on the Euclidean space and curved space are typically different and of independent interest. For the heat flow, u : [0, T ] × H n → H n is of particular interest because it is closely related to the Schoen-Li-Wang conjecture, namely any quasiconformal boundary map gives rise to a harmonic map of hyperbolic space. (see for instance Lemm, Markovic [27]) The wave map dynamics on curved space especially hyperbolic space were studied in the sequel works of Lawrie, Oh, Shahshahani [21], [22,23,24].
In this paper, we study the long time behaviors of solutions to (1.1) for α > 0. In our previous paper [32], for LL from R 2 to a compact Riemann surface, we proved that any initial data with energy below the critical energy will evolve to a global solution and converge to a constant map in the energy space. For LL from H 2 to H 2 , we aim to prove that the solution exists globally and will converge to a harmonic map as t → ∞. In order to prove the convergence to harmonic maps, one may study the corresponding linearized equation, which works well for the equivariant case. However, the linearization method is not available in the general case because there are infinite numbers of harmonic maps. In this paper, we apply the caloric gauge technique initially introduced by Tao [41].
The caloric gauge originally used by Tao was applied to solve the global regularity of wave maps from R 2+1 to H n in a sequel of papers [42,43]. We briefly recall the main idea of caloric gauge in the wave map setting. The first step is to evolve the solution of the wave map u(t, x) along a heat flow, i.e., solve the heat flow equation for u(s, t, x) with initial data u(t, x). If there exists no non-trivial harmonic map, one can suppose that the corresponding heat flow converges to a fixed point Q. For any given orthonormal frame at the point Q, we can pullback the orthonormal frame parallel with respect to s along the heat flow to obtain the frame at u(s, t, x), especially u(t, x) when s = 0. Then one has a scalar system for the differential fields and connection coefficients after rewriting (1.1) under the constructed frame. The caloric gauge can be seen as a nonlinear Littlewood-Paley decomposition, and it removes some troublesome frequency interactions. Generally the caloric gauge works well below the critical energy, where no harmonic map occurs. In our case, it makes no sense to study the dynamics below the critical energy because for any given λ ∈ (0, ∞), there exists a harmonic map Q λ whose energy is exactly λ. However, there is an obvious advantage of the caloric gauge in our case, i.e., one can a priori see the limit harmonic map without evolving LL to the infinity time. In fact, denote the solution of the heat flow with initial data u(0, x) by U (t, x), then it is known that U (t, x) converges to some harmonic map Q(x) as t → ∞. The key observation is that one can still expect that the solution u(t, x) of (1.1) also converges to the same harmonic map Q(x) as t → ∞. This informal heuristic idea combined with the caloric gauge reduces the convergence of LL to proving the decay of the heat tension filed. There are two main obstacles while applying the caloric gauge in the appearance of non-trivial harmonic maps. One is to guarantee that all the heat flows initiated from u(t, x) for different t converge to the same harmonic map. The other is the lack of global bounds for the derivatives of the induced connection coefficient. In fact, since formally the induced connection coefficient is solved from the infinity time, an L 1 integrability with respect to time is needed, but harmonic maps prevent the energy from decaying to zero as time goes to infinity, then we have no enough decay to gain the integrability.
The first obstacle is overcome by using the structure of LL. In order to construct the caloric gauge, one needs to prove the heat flow with u(t, x) as the initial data converges to the same harmonic map independent of t. We remark that it is not a trivial fact, because it is false if we only consider t as a smooth parameter, i.e., in the homotopy class. Indeed, it is known that there exist a family of harmonic maps {Q λ } which depend smoothly with respect to λ ∈ (0, 1).
Then perturbing the initial data Q λ of the heat flow to be Q λ ′ yields a different limit harmonic map, since any harmonic map remains time-independent under the heat flow. This inspires us to use the structure of LL. The key point is to study the evolution of ∂ t u with respect to the heat flow. By a monotonous property observed first by Hartman [18] and the decay estimates of the heat semigroup, we can prove the distance between the heat flows initiated from u(t 1 ) and u(t 2 ) goes to zero as s → ∞. Thus the limit harmonic map for the heat flow generated from u(x, t) are all the same for different t.
We remove the second obstacle by the smoothing effect. After constructing the caloric gauge, we obtain the gauged equation for the corresponding differential fields φ i and connection coefficients A i . Then it suffices to prove the decay of the corresponding heat tension field governed by a Ginzburg-Landau type system (see (5.1)). The most difficult term in this system is ∇A i , the derivative of the connection coefficients. Because the energy of the solution to (1.1) is strictly away from zero due to the appearance of the harmonic map, one can not expect any integrability of φ i with respect to s. Thus if one tries to obtain a global bound for ∇A i , one has to put ∇∂ s u in L 1 s . (see (3.32) for the expression for A i ) The term ∇∂ s u might be put in L 1 s by constructing a proper auxiliary function which satisfies a proper semiliear heat equation and controls ∇∂ s u and applying the maximum principle to obtain a pointwise estimate. Instead of constructing such an auxiliary function, we use an indirect but more robust method. By applying the smoothing effect of the heat semigroup, it suffices to bound ∇A i φ s inḢ −1 . Then by duality, one can move the derivative from ∂ i A i to φ s . Since we have a a-prior bound for ∇φ s in terms of ∇∂ t u, which is also a-prior bounded in L 2 t,x through some delicate energy arguments for (1.1), the ∂ i A i φ s term can be tackled without studying ∂ i A i itself.
In order to state our main theorem, we introduce some notions. Any analytic function f : C → C with f (D) ⋐ D where D is the Poincare disk is an admissible harmonic map, see Appendix for the proof. The harmonic map studied in Lawrie, Oh, Shahshahani [25] is in fact f (z) = λz with λ ∈ (0, 1).
The main result of this paper is the following.
For any initial data u 0 ∈ H 3 , there exists a global solution to (1.1) and as t → ∞, u(t, x) converges to some harmonic map Q ∞ : Q . This is related to the uniqueness of harmonic maps with prescribed boundary harmonic map. But we will not go into further details on this problem in this paper, one may see [Lemma 2.4, [31]] for details. This paper is organized as follows. In Section 2, we recall some background materials and prove an equivalence lemma for the intrinsic and extrinsic Sobolev norms in some case. In Section 3, we construct the caloric gauge and obtain the estimates of the connection coefficients.
In Section 4, we prove the local and global well-posedness of (1.1) by energy arguments. In Section 5, we study the gauged system and prove the decay of the heat tension filed which concludes the proof of Theorem 1.1.

Preliminaries
In this section, we recall some standard preliminaries on the geometry notions of the hyperbolic spaces along with some Sobolev embedding inequalities. For the study of the well-posedness theories, we need to use both the intrinsic and extrinsic formulations of the Sobolev spaces, thus we establish an equivalence relationship for the two formulations in some case, namely Lemma 2.4.

The global coordinates and definitions of the function spaces
The covariant derivative in T N is denoted by ∇, the covariant derivative induced by u in u * (T N ) is denoted by ∇. The Riemann curvature tension of N is denoted by R. The components of Riemann metric are denoted by h ij for M and g ij for N . And Γ k lj , Γ k lj denote the Christoffel symbols for M and N respectively.
We recall some facts on hyperbolic spaces. Let R 2+1 be the Minkowski space with the The hyperbolic space H 2 is defined as The Riemann metric equipped on H 2 is the pullback of the Minkowski metric by the inclusion map ι : H 2 → R 2+1 . The Iwasawa decomposition gives a global system of coordinates. Define the diffeomorphism Ψ : R × R → H 2 , Ψ(x 1 , x 2 ) = (coshx 2 + e −x 2 |x 1 | 2 /2, sinhx 2 + e −x 2 |x 1 | 2 /2, e −x 2 x 1 ). (2.1) The Riemann metric with respect to this coordinate system is given by The corresponding Christoffel symbols are For any (t, x) and u : [0, T ] × H 2 → H 2 , we define an orthonormal frame at u(t, x) by It is easily seen Θ 2 = JΘ 1 . Let X, Y, Z ∈ T N , recall the identity for Riemannian curvature on where we use the simplicity notation As a direct consequence of this formula and the comparability, one has for X, Y, Z ∈ u * T N that Therefore we obtain a useful identity Let H k (H 2 ; R) be the usual Sobolev space for scalar functions defined on manifolds, see for instance Hebey [19]. It is known that C ∞ c (H 2 ; R) is dense in H k (H 2 ; R). We also recall the norm of H k : where ∇ l f is the covariant derivative, for instance, ∇f and ∇ 2 f in the local coordinates can be written as The norm |∇ l f | is taken by viewing ∇ l f as a (0, l) type tension field on H 2 . For maps u : H 2 → H 2 , the intrinsic Sobolev semi-norm H k is given by Recall the global coordinates given by (2.1), then map u : H 2 → H 2 can be viewed as a vectorvalued function u : H 2 → R 2 . Indeed, for P ∈ H 2 the vector (u 1 (P ), u 2 (P )) is defined by Ψ(u 1 (P ), u 2 (P )) = u(P ). Let Q : H 2 → H 2 be an admissible harmonic map. Then the extrinsic Sobolev space is defined as  Hebey [19]), H k Q coincides with the completion of D, which denotes the maps from H 2 to H 2 coinciding with Q outside of some compact subset of M = H 2 , under the metric given by (2.6). In the following, without of confusing we write H k Q as H k for simplicity.

The Fourier transform on hyperbolic spaces and Sobolev embedding
The Fourier transform takes proper functions defined on H 2 to functions defined on R × S 1 , see for instance Helgason [20]. For ω ∈ S 1 , and λ ∈ C, let b(ω) = (1, ω) ∈ R 3 , we define The Fourier transform of f ∈ C 0 (H 2 ) is defined by The corresponding Fourier inversion formula is given by where c(λ) is the Harish-Chandra c-function on H 2 , which is defined for some suitable constant C by .
The Plancherel theorem is as follows Thus any bounded multiplier m : R → C defines a bounded operator T m on L 2 (H 2 ) by We define the operator (−∆) s 2 by the Fourier multiplier λ → ( 1 4 + λ 2 ) s 2 . We now recall the Sobolev inequalities of functions in H k .
We also recall the standard diamagnetic inequality which sometimes refers to Kato's inequality as well. (2.12)

Equivalence Lemma
In this subsection, we prove the equivalence of the intrinsic Sobolev space and the extrinsic Sobolev space in some case. Suppose that Q is an admissible harmonic map in Definition 1.1. Although the geodesic distance between (0, 0) and (x 1 , x 2 ) in the coordinate (2.1) is not we use the quantity |x 1 | + |x 2 | in the following three lemmas for simplicity. It suffices to remind ourself that the true distance is bounded by a function of (|x 1 |, |x 2 |).
As a preparation, we give the following lemma which shows the intrinsic formulation is equivalent to the extrinsic one if the image of the map is compact.
Lemma 2.4. Let Q be an admissible harmonic map with Q(H 2 ) contained in a geodesic ball of in the sense that there exist continuous functions Q and P such that Proof. We first prove (2.22). By (2.17)-(2.20), Then Sobolev embedding (2.9) and diamagnetic inequality show Recall the definition in (2.5), then (2.24), (2.12) with Q ∈ H 2 yield Then by Sobolev embedding, Thus Γ j ik and g ik are bounded by the compactness of Q(H 2 ) and (2.2). Applying similar arguments to u 1 with the boundedness of Γ j ik and g ij implies where the implicit constant has an up bound e 8 2 By (2.25) and Sobolev embedding, one has u 1 − Q 1 ∞ < ∞. Then the compactness of Q(H 2 ) indicates u(H 2 ) is covered by a geodesic ball of radius CR 0 + C 2 k=1 ∇dQ k L 2 + ∇du k L 2 , where C is some universal constant. Therefore careful calculations give The third order derivatives can be written as and Then it suffices to bound Thus (2.22) follows from Sobolev embedding. The inverse direction of (2.22), i.e., (2.23) is obtained along the same path. Indeed, Sobolev embedding yields for l = 1, 2 Thus one has for l = 1, 2 Then the k = 2 case of (2.23) follows by Lemma 2.3 with R = R 0 +C u H 2 . Further calculations give the k = 3 case of (2.23).
Remark 2.2. As a byproduct of the proof in Lemma 2.4, we have for some continuous function

Smoothing effects for heat equations on hyperbolic spaces
We also need a smoothing effect lemma for the semigroup e z∆ H 2 in Section 5.
with initial data f 0 is a solution to the linear inhomogeneous equation then we have Proof. Taking inner product with f on both sides of (2.31), the real part yields Then Lemma 2.5 follows by (2.11) and The pointwise estimate of the heat kernel in H 2 is obtained by Davies and Mandouvalos [9].
Lemma 2.6. The heat kernel on H 2 denoted by K 2 (t, ρ) satisfies the pointwise estimate Particularly, we have the decay estimate Proof. (2.34) can be found in [8]. (2.33) follows directly by the upper bound for the heat kernel given above.
The following lemma for the heat semigroup in R 2 was obtained in Lemma 2.5 of Tao [43].
We remark that the same arguments work in the H 2 case, because the proof in [43] only uses the decay estimate (2.34) and the self-ajointness of e t∆ , which are also satisfied by e t∆ H 2 .

The caloric gauge
In order to study the asymptotic behaviors, we need to rewrite (1.1) under a gauge. Let The isomorphism of R 2 to C induces a complex valued function defined by Then u induces a covariant derivative on the trivial complex vector bundle where the induced connection coefficients are defined by 2 1 by A i in the following. It is easy to check the torsion free identity The commutator identity is given by We have a gauge freedom to choose the frame {e 1 , Je 1 }. In fact, given any real valued function Lemma 3.1. With the notions and notations given above, (1.1) can be written as Proof. We first rewrite the tension field τ (u) under the gauge. Recall that By the definition, we have Then the complex valued function φ t satisfies The caloric gauge was first introduced by Tao [42] for the wave maps from R 2+1 to H n . We give the definition of the caloric gauge in our setting.
that the heat flow with u 0 as the initial data converges to a harmonic map Q ∞ from H 2 to H 2 . For a given orthonormal frame where the convergence of frames is defined by The remaining part of this section is devoted to the existence of the caloric gauge. The equation of the heat flow is given by We recall the definition of the energy density e, The following two lemmas are essentially due to Eells, Sampson [11] and Li, Tam [29].
Corollary 3.1. If u is the solution in Lemma 3.2, then for some C > 0 K|du| 2 .
It is known in the heat flow literature that Harnack inequality for the linear heat equation is useful to obtain L ∞ bounds. The Harnack type inequality for complete manifolds was initially proved by Li and Yau [30]. The following form of Harnack inequality which is convenient in our case was obtained by Li and Xu [28].
where dist(x 1 , x 2 ) is the distance between x 1 and x 2 , B 1 (t 1 , t 2 ) = e 2kt 2 −2kt 2 −1 e 2kt 1 −2kt 1 −1 n 4 , and B 2 (t 1 , t 2 ) = Now we consider the heat flow from H 2 to H 2 with a parameter , then there exists a harmonic map Therefore for any 1 < s 1 < s 2 < ∞ we have which implies u(s, t, x) converges uniformly on (t, x) ∈ [0, T ] × H 2 to some map Q ∞ (t, x). By Theorem 5.2 in [29], for any fixed t, Q ∞ (t, x) is a harmonic map form H 2 → H 2 with respect to x. It remains to prove Q ∞ (t, x) is indeed independent of t. We consider the evolution of |∂ t u| 2 with respect to s. In fact, |∂ t u| 2 satisfies Hence the maximum principle and (2.33) imply Consequently, we obtain for 0 ≤ t 1 < t 2 ≤ T , x)) = 0, thus finishing the proof.
In order to obtain a global bound independent of s for the quantities related to the heat flow (3.13), we need to get estimates which only depend on the energy of u(t, x).

20)
where e and e 0 are the energy density of u and u 0 respectively.
Then Lemma 2.7 yields Thus (  On the other hand, we have thus by maximum principle for any s ≥ s 1 − λ/2, e(s) ≤ e sK w(s).
Therefore  Proof. We first show the existence part. Choose an arbitrary orthonormal frame E 0 (t, x) Denote the evolved frame as E s {Ω 1 (s, t, x), JΩ 1 (s, t, x)}. Then Ω 2 JΩ 1 (s, t, x) satisfies ∇ s Ω 2 (s, t, x) = 0 as well. We claim that there exists some orthonormal frame E ∞ Therefore choosing U (t, x) such that U (t, x)E ∞ = Ξ, where E ∞ is the limit frame obtained by (3.27), suffices for our purpose. The uniqueness of the gauge follows from the identity where (Ψ 1 , JΨ 1 ) and (Ψ 2 , JΨ 2 ) are two caloric gauges satisfying (3.5).

Then (3.2) reduces to
The following lemma gives the bounds for the connection coefficients matrix A x,t .
Lemma 3.6. Suppose that Ω(s, t, x) is the caloric gauge constructed in Proposition 3.1, then we have for i = 1, 2 Particularly, we have for i = 1, 2, s > 0, Moreover, let Ξ(x) = Θ(Q ∞ ) in Proposition 3.1, we have the bounds for A x,t : Since u(s, t, x) converges to Q ∞ (x) in C 1 loc and d u(s, t, x) L ∞ du(t, x) L 2 x uniformly for s ≥ 1 shown by (3.20), we obtain Thus (3.36), (3.37) give First, we prove (3.32) under the assumption of (3.30). From the identity (3.29), we obtain Therefore, we infer from (3.43) and (3.40) that for some a 1 (t, x) and a 2 (t, x) Recall the facts that u(s, t, x) converges to Q ∞ (x) in the C 1 loc norm as s → ∞ (Theorem 4.3 of [29]), and the frame Ω converges to Ξ in the sense of (3.6). Then by the definition of A i , in order to prove (3.30) it suffices to verify (3.42) By the identity By integration by parts , u(s, t, x) converges to Q ∞ (x) in the C 0 norm as s → ∞ and (3.6), the first term in the right hand side of (3.43) converges to By the explicit expression of Θ and the fact that u(s, t, x) converges to Q ∞ (x) in the C 1 loc norm, the second term in the right hand side of (3.43) converges to Since (3.17) shows and direct calculation gives we conclude that and further Therefore it suffices to prove (3.44) and (3.45). Since it has been verified in Proposition 3.1 that Then we infer from (3.15), (3.17) that Hence Therefore ∇ t Ω 1 , Θ i converges as s → ∞. Hence the convergence of ∂ t Ω 1 , Θ i as s → ∞ follows from (3.46), |∇ t Θ i | ≤ |∂ t u| e − s 8 . Thus (3.44) is obtained, and similar arguments yield (3.45) which ends the proof of (3.31).

The Local and global well-posedness
If u(t, x) solves (1.1), then it is easy to see (4.1) In order to prove the local well-posedness in H 3 , we apply the approximating scheme introduced by McGahagan [35]. The novel idea in [35] is that one can use the wave map to approximate the Schrödinger map. For any δ > 0, we introduce the wave map model equation: Formally, we can view (4.2) as the approximate scheme for (1.1) because of (4.1). We remark that different choices of g δ 0 will give the same solution to (1.1), thus it is unnecessary to specify a concrete g δ 0 . The local wellposedness of (1.1) is given by the following lemma. Proof. Rigorously, the volume element of H 2 should be written as dvol h , for convenience, we use dx instead in the following proof. And we drop the symbol δ in g δ 0 sometimes. Fix some ε > 0 sufficiently small. In the coordinates given by (2.1), (4.2) can be written as the following semi-linear wave equation Using the identity J ∂ ∂y 1 = e −y 2 ∂ ∂y 2 , the above equation can be further reduced to a semilinear equation.
Step 1. Local solution for approximate equations. Step 2. Uniform a-prior estimates for approximate equations. We claim that there exists a T depending only on u 0 H 3 such that for all δ > 0, t ∈ [0, T ] there exists a a-prior bound u δ (t, x) H 3 < C, (4.5) for some C > 0 independent of δ. Direct calculation gives the following identity which is useful to close the energy estimates where λ = α 2 +β 2 α 3 +β 2 α > 0. And we will frequently use the identity JX, X = 0 and (2.4). Careful calculations with integration by parts imply for any X ∈ u * (T H 2 ), one has (4.7) Define the energy functional by Then by (4.2), we have d dt Thus the energy is decreasing with respect to t and Using (4.6), we have We can gain a key negative dominate term from the first term in the right hand side of (4.9).
In fact, by the non-positiveness of the sectional curvature of the target and expanding τ (u) to tr(∇du), we have Then integration by parts shows (4.10) Similarly, one has (4.11) As a consequence of (4.9), (4.10) and (4.11), we obtain from Hölder inequality that Gagliardo-Nirenberg inequality and diamagnetic inequality imply Therefore, Young's inequality and (4.12) yield By (4.8) and Gronwall inequality, we conclude that where C is independent of δ. Using the non-positiveness of the sectional curvature of the target, we have x . (4.14) Define the second order energy functional by Then by (4.2), we obtain Applying (4.2) again for the second term in (4.15) yields Thus we have from (4.15), (4.16) that Then Young's inequality and Gagliardo-Nirenberg inequality show By (4.14), (4.8), we see that Hence (4.14), (4.17) give where F : R → R + is some C 2 function. Then we have from Gronwall inequality that there which combined with (4.18) yields Define the third order energy functional by Before calculating the differentiation with respect to t of E 3 , we first point out a useful inequality which can be verified by integration by parts Thus (4.20), Gagliardo-Nirenberg inequality and Young inequality further yield where P(x) is some polynomial. Integration by parts and (4.2) give By Gagliardo-Nirenberg inequality and Young inequality, we have Similarly, we have Therefore,we conclude from (4.23) and (4.21) that where G : R → R + is a C 2 function. Thus by Gronwall inequality, we have there exists T 1 > 0 such that for any δ > 0, t ∈ [0, T 1 ] such that (4.2) has a local solution in L ∞ ([0, T 1 ]; H 3 ) and By Lemma 2.4, this implies the uniform a-prior bound for u δ H 3 . Now letting δ → 0, we have a sequence of solutions u δn of (4.2), which converge to u ∈ L ∞ ([0, T ]; H 3 ) in the weak * sense.
The global well-posedness is given by the following proposition. Proof. By the local well-posedness in Lemma 4.1, it suffices to obtain a uniform bound for u H 3 with respect to t ∈ [0, T ]. As before we introduce three energy functionals: By integration by parts and (1.1), we have Thus the energy is decreasing with respect to t and Since u satisfies (1.1), then By (1.1) and integration by parts, one has Hölder and Gagliardo-Nirenberg inequality imply for some fixed sufficiently small ε > 0 Therefore, we deduce from (4.27) that Then applying (4.14), (4.25), we have Gronwall inequality shows Thus by (4.24), we obtain Again using (4.14), (4.25), we conclude Integration by parts and (1.1) yield By Gagliardo-Nirenberg inequality and Young inequality, we see Similarly we have The remaining three terms are easier to bound: Thus we conclude that Then Gronwall inequality, (4.29) and (4.20) imply Therefore we obtain again from (4.29) and (4.20) that By Lemma 2.4, we conclude u(t)  Letũ be the solution to (3.13) with initial data u(t, x), then we have (4.32)

Convergence to a harmonic map
In this section, we always fix the frame Ξ in Proposition 3.1 to be Ξ(x) = Θ(Q ∞ (x)).

Estimates of the heat tension field
Recall that the heat tension filed φ s satisfies And we define the LL tension filed by respectively is given by the following lemma.
Lemma 5.1. The tension fields φ i , i = 1, 2, φ s , w satisfy Proof. By the torsion free identity and the commutator identity, we have Thus (5.3) is verified. The rest is to prove (5.4). By (5.2), the torsion free identity and the commutator identity (3.2), we obtain The torsion free identity and the commutator identity further show Combining the two above equalities together yields (5.4).
Lemma 5.2. The heat tension filed φ s satisfies The LL filed w satisfies Proof. (5.6) follows by (3.24) By the definition ∇ψ j s = ∂ i ∂ sũ , e j dx i and the comparability we obtain Then (3.34) and u(t) H 2 ≤ C(u 0 ) proved in Proposition 4.1 yield which combined with (5.10) gives Thus (5.5) follows from (5.6). By the definition w = φ t − zφ s , in order to verify (5.8), it suffices to prove ∇φ t L 2 e Cs ∂ t u L 2 . (5.12) Similar calculations as (5.11) imply Hence (5.8) follows by (4.32) and (3.10). It remains to prove (5.9). Expanding D i in (5.4) as Then we have from Lemma 2.1 that The later three terms on the right hand side of (5.13) are easy to bound by Hölder and (3.34): Furthermore, by (5.7), (3.20), we have the following acceptable bound for (5.14) Therefore it suffices to bound h ii ∂ i A i w Ḣ −1 x . By duality and integration by parts, we have Then (5.9) follows by (3.34), (5.8), (5.7) and (5.15).

Appendix
In the following lemma we collect some important properties of holomorphic harmonic maps between Poincare disks. Lemma 6.1. Denote D = {z : |z| < 1} with the hyperbolic metric to be the Poincare disk. Then any holomorphic map f : D → D is a harmonic map. If we assume that f (z) can be analytically extended into a larger disk than the unit disk, and f (D) ⊂ {z : |z| ≤ µ}, for some 0 < µ < 1, then the harmonic map f satisfies df L 2 < ∞ (6.1) e r(z) df (z) L ∞ < ∞ (6.2)

3)
where r is the geodesic distance between z and the origin in D. Furthermore, if assume in addition that for all z ∈ D and some 0 < µ 1 ≪ 1, |∂ γ x,y f (z)| ≤ µ 1 , (6.4) then df L 2 µ 1 (6.5) e r(z) df (z) L ∞ µ 1 (6.6) Proof. Since f can be analytically extended to a larger disk, we can assume all the derivatives where C 0 now relies on the three order derivatives of f (z) on D. (6.9) and (6.10) yield (6.8). If we assume in addition that (6.4) holds, then let γ = 0 in (6.4), one has |f (z)| < 1 2 . Then all the Christoffel symbols and metric components on the range f (D) are bounded by C 1 for some universal constant C 1 > 0. Thus the constant C 0 c(µ) in (6.8) to (6.10) can be replaced by Cµ 1 for some universal constant C > 0. Hence (6.5) and (6.6) follow by the same line.
The following lemma proves the free torsion identity and the commutator identity in (3.1), (3.2). The proof is standard to people working in geometric dispersive equations, we write the details down on one side for reader's convenience and on the other side to emphasize the curved background in our case. Lemma 6.2.