Local Well-Posedness of Skew Mean Curvature Flow for Small Data in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\ge 4$$\end{document}d≥4 Dimensions

The skew mean curvature flow is an evolution equation for d dimensional manifolds embedded in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {R}}}^{d+2}$$\end{document}Rd+2 (or more generally, in a Riemannian manifold). It can be viewed as a Schrödinger analogue of the mean curvature flow, or alternatively as a quasilinear version of the Schrödinger Map equation. In this article, we prove small data local well-posedness in low-regularity Sobolev spaces for the skew mean curvature flow in dimension \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\ge 4$$\end{document}d≥4.


The (SMCF) equations.
Let d be a d-dimensional oriented manifold, and (N d+2 , g N ) be a d + 2-dimensional oriented Riemannian manifold. Let I = [0, T ] be an interval and F : I × d → N be a one parameter family of immersions. This induces a time dependent Riemannian structure on d . For each t ∈ I , we denote the submanifold by t = F(t, ), its tangent bundle by T t , and its normal bundle by N t respectively. For an arbitrary vector Z at F we denote by Z ⊥ its orthogonal projection onto N t . The mean curvature H(F) of t can be identified naturally with a section of the normal bundle N t .
The normal bundle N t is a rank two vector bundle with a naturally induced complex structure J (F) which simply rotates a vector in the normal space by π/2 positively. Namely, for any point y = F(t, x) ∈ t and any normal vector ν ∈ N y t , we define J (F)ν ∈ N y t as the unique vector with the same length so that which evolves a codimension two submanifold along its binormal direction with a speed given by its mean curvature. The (SMCF) was derived both in physics and mathematics. The one-dimensional (SMCF) in the Euclidean space R 3 is the well-known vortex filament equation (VFE) where γ is a time-dependent space curve, s is its arc-length parameter and × denotes the cross product in R 3 . The (VFE) was first discovered by Da Rios [6] in 1906 in the study of the free motion of a vortex filament. The (SMCF) also arises in the study of asymptotic dynamics of vortices in the context of superfluidity and superconductivity. For the Gross-Pitaevskii equation, which models the wave function associated with a Bose-Einstein condensate, physics evidence indicates that the vortices would evolve along the (SMCF). An incomplete verification was attempted by Lin [20] for the vortex filaments in three space dimensions. For higher dimensions, Jerrard [14] proved this conjecture when the initial singular set is a codimension two sphere with multiplicity one.
The other motivation is that the (SMCF) naturally arises in the study of the hydrodynamical Euler equation. A singular vortex in a fluid is called a vortex membrane in higher dimensions if it is supported on a codimension two subset. The law of locally induced motion of a vortex membrane can be deduced from the Euler equation by applying the Biot-Savart formula. Shashikanth [24] first investigated the motion of a vortex membrane in R 4 and showed that it is governed by the two dimensional (SMCF), while Khesin [18] then generalized this conclusion to any dimensional vortex membranes in Euclidean spaces.
From a mathematical standpoint, the (SMCF) equation is a canonical geometric flow for codimension two submanifolds which can be viewed as the Schrödinger analogue of the well studied mean curvature flow. In fact, the infinite-dimensional space of codimension two immersions of a Riemannian manifold admits a generalized Marsden-Weinstein sympletic structure, and hence the Hamiltonian flow of the volume functional on this space is verified to be the (SMCF). Haller-Vizman [12] noted this fact where they studied the nonlinear Grassmannians. For a detailed mathematical derivation of these equations we refer the reader to the article [28, Section 2.1].
The study of higher dimensional (SMCF) is still at its infancy compared with its onedimensional case. For the 1-d case, we refer the reader to the survey article of Vega [29]. For the higher dimensional case, Song-Sun [28] proved the local existence of (SMCF) with a smooth, compact oriented surface as the initial data in two dimensions, then Song [27] generalized this result to compact oriented manifolds for all d ≥ 2 and also proved a corresponding uniqueness result. Recently, Li [19] considered the transversal small pertubations of Euclidean planes under the (SMCF) and proved the global regularity for small initial data. In addition, Song [26] also proved that the Gauss map of a d dimensional (SMCF) in R d+2 satisfies a Schrödinger Map type equation but relative to the varying metric. We remark that in one space dimension this is exactly the classical Schrödinger Map type equation, provided that one chooses suitable coordinates, i.e. the arclength parametrization.
As written above, the (SMCF) equations are independent of the choice of coordinates in I × ; here we include the time interval I to emphasize that coordinates may be chosen in a time dependent fashion. The manifold d simply serves to provide a parametrization for the moving manifold t ; it determines the topology of t , but nothing else. Thus, the (SMCF) system written in the form (1.1) should be seen as a geometric evolution, with a large gauge group, namely the group of time dependent changes of coordinates in I × . In particular, interpreting the equations (1.1) as a nonlinear Schrödinger equation will require a good gauge choice. This is further discussed in Sect. 2. In this article we will restrict ourselves to the case when d = R d , i.e. where t has a trivial topology. We will further restrict to the case when N d+2 is the Euclidean space R d+2 . Thus, the reader should visualize t as an asymptotically flat codimension two submanifold of R d+2 .

Scaling and function spaces.
To understand what are the natural thresholds for local well-posedness, it is interesting to consider the scaling properties of the solutions. As one might expect, a clean scaling law is obtained when d = R d and N d+2 = R d+2 . Then we have the following Proposition 1.1 (Scale invariance for (SMCF)). Assume that F is a solution of (1.1) with initial data F(0) = F 0 . If λ > 0 thenF(t, x) := λ −1 F(λ 2 t, λx) is a solution of (1.1) with initial dataF(0) = λ −1 F 0 (λx).
Proof. Since the induced metric and Christoffel symbols of the immersionF arẽ g αβ (t, x) = ∂ αF , ∂ βF = g αβ (λ 2 t, λx), The above scaling would suggest the critical Sobolev space for our moving surfaces t to beḢ d 2 +1 . However, instead of working directly with the surfaces, it is far more convenient to track the regularity at the level of the curvature H( t ), which scales at the level ofḢ d 2 −1 .

The main result.
Our objective in this paper is to establish the local well-posedness of skew mean curvature flow for small data at low regularity. A key observation is that providing a rigorous description of fractional Sobolev spaces for functions (tensors) on a rough manifold is a delicate matter, which a-priori requires both a good choice of coordinates on the manifold and a good frame on the vector bundle (the normal bundle in our case). This is done in the next section, where we fix the gauge and write the equation as a quasilinear Schrödinger evolution in a good gauge. At this point, we content ourselves with a less precise formulation of the main result:  [7], F. J. Almgren, Jr. [1], and J. Simons [25] led to the following theorem (see Theorem 4,2,[3]): "If u : R n−1 → R is an entire solution to the minimal surface equation and n ≤ 8, then u is an affine function." However, in 1969 E. Bombieri, De Giorgi, and E. Giusti [2] constructed entire non-affine solutions to the minimal surface equation in R 9 . Hence the bound H 0 H s ( 0 ) ≤ 0 on the mean curvature does not necessarily imply that the sub-manifold is almost flat.
Here we only prove the small data local well-posedness, which means that the initial submanifold 0 should be a perturbation of Euclidean plane R d . Hence, the bound on metric ∂ x (g 0 − I d ) H s ≤ 0 is also necessary in our main result, at least in very high dimension. This condition on metric will insure the existence of global harmonic coordinates (see Proposition 8.2). Later, the mean curvature bound will also yield an estimate ∂ x (g 0 − I d ) H s+1 0 in harmonic coordinates.
Unlike any of the prior results, which prove only existence and uniqueness for smooth data, here we consider rough data and provide a full, Hadamard style well-posedness result based on a more modern, frequency envelope approach and using a paradifferential form for both the full and the linearized equations. For an overview of these ideas we refer the reader to the expository paper [13]. While, for technical reasons, this result is limited to dimensions d ≥ 4, we expect the same strategy to also work in lower dimension; the lower dimensional case will be considered in forthcoming work.
The favourable gauge mentioned in the theorem, defined in the next section, will have two components: • The harmonic coordinates on the manifolds t .
• The Coulomb gauge for the orthonormal frame on the normal bundle.
In the next section we reformulate the (SMCF) equations as a quasilinear Schrödinger evolution for a good scalar complex variable ψ, which is exactly the mean curvature but represented in the good gauge. There we provide an alternate formulation of the above result, as a well-posedness result for the ψ equation. In the final section of the paper we close the circle and show that one can reconstruct the full (SMCF) flow starting from the good variable ψ.
One may compare our gauge choices with the prior work in [28] and [27]. There the tangential component of ∂ t F in (1.1) is omitted, and the coordinates on the manifold t are simply those transported from the initial time. The difficulty with such a choice is that the regularity of the map F is no longer determined by the regularity of the second fundamental form, and instead there is a loss of derivatives which may only be avoided if the initial data is assumed to have extra regularity. This loss is what prevents a complete low regularity theory in that approach.
Once our problem is rephrased as a nonlinear Schrödinger evolution, one may compare its study with earlier results on general quasilinear Schrödinger evolutions. This story begins with the classical work of Kenig-Ponce-Vega [15][16][17], where local wellposedness is established for more regular and localized data. Lower regularity results in translation invariant Sobolev spaces were later established by Marzuola-Metcalfe-Tataru [21][22][23]. The local energy decay properties of the Schrödinger equation, as developed earlier in [4,5,8,9] play a key role in these results. While here we are using some of the ideas in the above papers, the present problem is both more complex and exhibits additional structure. Because of this, new ideas and more work are required in order to close the estimates required for both the full problem and for its linearization.

An overview of the paper.
Our first objective in this article will be to provide a self-contained formulation of the (SMCF) flow, interpreted as a nonlinear Schrödinger equation for a single independent variable. This independent variable, denoted by ψ, represents the trace of the second fundamental form on t , in complex notation. In addition to the independent variables, we will use several dependent variables, as follows: • The Riemannian metric g on t .
• The (complex) second fundamental form λ for t . • The magnetic potential A, associated to the natural connection on the normal bundle N t , and the corresponding temporal component B.
• The advection vector field V , associated to the time dependence of our choice of coordinates.
These additional variables will be viewed as uniquely determined by our independent variable ψ, provided that a suitable gauge choice was made. The gauge choice involves two steps: (i) The choice of coordinates on t ; here we use harmonic coordinates, with suitable boundary conditions at infinity. (ii) The choice of the orthonormal frame on N t ; here we use the Coulomb gauge, again assuming flatness at infinity.
To begin this analysis, in the next section we describe the gauge choices, so that by the end we obtain Setting the stage to solve these equations, in Sect. 3 we describe the function spaces for both ψ and S. This is done at two levels, first at fixed time, which is useful in solving the elliptic system (2.36), and then using in the space-time setting, which is needed in order to solve the Schrödinger evolution. The fixed time spaces are classical Sobolev spaces, with matched regularities for all the components. The space-time norms are the so called local energy spaces, as developed in [21][22][23].
Using these spaces, in Sect. 4 we consider the solvability of the elliptic system (2.36). This is first considered and solved without reference to the constraint equations, but then we prove that the constraints are indeed satisfied.
Finally, we turn our attention to the Schrödinger system (2.35), in several stages. In Sect. 5 we establish several multilinear and nonlinear estimates in our space-time function spaces. These are then used in Sect. 6 in order to prove local energy decay bounds first for the linear paradifferential Schrödinger flow, and then for a full linear Schrödinger flow associated to the linearization of our main evolution. The analysis is completed in Sect. 7, where we use the linear Schrödinger bounds in order to (i) construct solutions for the full nonlinear Schrödinger flow, and (ii) to prove the uniqueness and continuous dependence of the solutions. The analysis here broadly follows the ideas introduced in [21][22][23], but a number of improvements are needed which allow us to take better advantage of the structure of the (SMCF) equations.
Last but not least, in the last section we prove that the full set of variables (g, λ, V, A, B) suffice in order to uniquely reconstruct the defining function F for the evolving surfaces t , as H s+2 loc manifolds. More precisely, with respect to the parametrization provided by our chosen gauge, F has regularity

The Differentiated Equations and the Gauge Choice
The goal of this section is to introduce our main independent variable ψ, which represents the trace of the second fundamental form in complex notation, as well as the following auxiliary variables: the metric g, the second fundamental form λ, the connection coefficients A, B for the normal bundle as well as the advection vector field V . For ψ we start with (1.1) and derive a nonlinear Schödinger type system (2.35), with coefficients depending on S = (λ, h, V, A, B), where h = g − I d . Under suitable gauge conditions, the auxiliary variables S are shown to satisfy an elliptic system (2.36), as well as a natural set of constraints. We conclude the section with a gauge formulation of our main result, see Theorem 2.7. We remark that H. Gomez ([11,Chapter 4]) introduced the language of gauge fields as an appropriate framework for presenting the structural properties of the surface and the evolution equations of its geometric quantities, and showed that the complex mean curvature of the evolving surface satisfies a nonlinear Schrödinger-type equation. Here we will further derive the self-contained modified Schrödinger system under harmonic coordinate conditions and Coulomb gauge.

The second fundamental form.
Let∇ be the Levi-Civita connection in (R d+2 , g R d+2 ) and let h be the second fundamental form for as an embedded manifold. For any vector fields u, v ∈ T * , the Gauss relation is Then we have By¯ j kl = 0, this gives the mean curvature H at F(x), Hence, the F-equation in (1.1) is rewritten as This equation is still independent of the choice of coordinates in d , which at this point are allowed to fully depend on t.

The complex structure equations.
Here we introduce a complex structure on the normal bundle N t . This is achieved by choosing {ν 1 , ν 2 } to be an orthonormal basis of N t such that If we differentiate the frame, we obtain a set of structure equations of the following type where the tensors κ αβ , τ αβ and the connection coefficients A α are defined by The mean curvature H can be expressed in term of κ αβ and τ αβ , i.e.
Then we define the complex scalar mean curvature ψ as the trace of the second fundamental form, ψ := tr λ = g αβ λ αβ . (2.4) Our objective for the rest of this section will be to interpret the (SMCF) equation as a nonlinear Schrödinger evolution for ψ, by making suitable gauge choices. We remark that the action of sections of the SU (1) bundle is given by for a real valued function θ . We use the convention for the inner product of two complex vectors, say a and b, as where a j and b j are the complex components of a and b respectively. Then we get the following relations for the complex vector m, From these relations we obtain Then the structure equations (2.3) are rewritten as where

The Gauss and Codazzi relations.
The Gauss and Codazzi equations are derived from the equality of second derivatives ∂ α ∂ β F γ = ∂ β ∂ α F γ for the tangent vectors on the submanifold and for the normal vectors respectively. Here we use the Gauss and Codazzi relations to derive the Riemannian curvature, the first compatibility condition and a symmetry. By the structure equations (2.6), we get Then in view of ∂ α ∂ β F γ = ∂ β ∂ α F γ and equating the coefficients of the tangent vectors, we obtain This gives the Riemannian curvature which is a complex formulation of the Gauss equation. Correspondingly we obtain the the Ricci curvature After equating the coefficients of the vector m in (2.7), we obtain By the definition of covariant derivatives, i.e.
This implies the complex formulation of the Codazzi equation, namely As a consequence of this equality, we obtain Lemma 2.1. The second fundamental form λ satisfies the Codazzi relations Proof. Here we prove the last equality. By ∇ β g γ σ = 0 and (2.10) we have The first equality can be proved similarly.
Next, we use the relation ∂ α ∂ β m = ∂ β ∂ α m in order to derive a compatibility condition between the connection A in the normal bundle and the second fundamental form. Indeed, from ∂ α ∂ β m = ∂ β ∂ α m we obtain the commutation relation Then multiplying (2.12) by m yields This gives the compatibility condition for the curvature of A, Using covariant derivative, this can be written as which can be seen as the complex form of the Ricci equations. We remark that, by equating the coefficients of the tangent vectors in (2.12), we also obtain and hence which is the same as (2.11). Next, we state an elliptic system for the second fundamental form λ αβ in terms of ψ, using the Codazzi relations (2.11). Lemma 2.2 (Div-curl system for λ). The second fundamental form λ satisfies (2.14) We remark that a-priori solutions λ to the above system are not guaranteed to be symmetric, so we record this as a separate property: (2.15) Finally, we turn our attention to the connection A, for which we have the curvature relations (2.13) together with the gauge group (2.5). In order to both fix the gauge and obtain an elliptic system for A, we impose the Coulomb gauge condition (2.16) Next, we derive the elliptic A-equations from the Ricci equations (2.13).

Lemma 2.3 (Elliptic equations for A). Under the Coulomb gauge condition, the connection A solves
Proof. Applying ∇ β to (2.13), by curvature and (2.16) we obtain Then the equation (2.17) for A is obtained from (2.9).

The elliptic equation for the metric g in harmonic coordinates.
Here we take the next step towards fixing the gauge, by choosing to work in harmonic coordinates. Precisely, we will require the coordinate functions {x α , α = 1, . . . , d} to be globally Lipschitz solutions of the elliptic equations g x α = 0. (2.18) This determines the coordinates uniquely modulo time dependent affine transformations. This remaining ambiguity will be removed later on by imposing suitable boundary conditions at infinity. After this, the only remaining degrees of freedom in the choice of coordinates will be given by time independent translations and rigid rotations. Thus, once a choice is made at the initial time, the coordinates will be uniquely determined later on (see also Remark 2.5.1). Here we will interpret the above harmonic coordinate condition at fixed time as an elliptic equation for the metric g (see e.g. [10], [30, P161]). The equations (2.18) may be expressed in terms of the Christoffel symbols , which must satisfy the condition g αβ γ αβ = 0, for γ = 1, . . . , d. (2.19) This implies Then we also have This leads to an equation for the metric g: Lemma 2.4 (Elliptic equations of g). In harmonic coordinates, the metric g satisfies We compute the first term I . By the definition of αβ,γ in (2.21), we have Since, by (2.20) we have Then Hence, By (2.9) this concludes the proof of the Lemma.
where V γ is a vector field on the manifold , which in general depends on the choice of coordinates. By the definition of m and λ αβ , we get Hence, the above F-equation (2.23) is rewritten as Then we use this to derive the equations of motion for the frame. Applying ∂ α to (2.24), by the structure equations (2.6) we obtain By the orthogonality relation m⊥F α = 0, this implies In order to describe the normal component of the time derivative of m, we also need the temporal component of the connection in the normal bundle. This is defined by We have Then we get which can be further rewritten as Therefore, we obtain the following equations of motion for the frame (2.25) From this we obtain the evolution equation for the metric g. By the definition of the induced metric g (2.1) and (2.25), we have which we record for later reference: (2.26) Then we also obtain where G αβ are defined by So far, the choice of V has been unspecified; it depends on the choice of coordinates on our manifold as the time varies. However, once the latter is fixed via the harmonic coordinate condition (2.19), we can also derive an elliptic equation for the advection field V : Lemma 2.5 (Elliptic equation for the vector field V ). Under the harmonic coordinate condition (2.19), the advection field V solves (2.30) Proof. Applying ∂ t to g αβ γ γβ , by (2.27) and (2.28) we have Since Here we uniquely determine the evolution of the coordinates as the time varies by choosing the advection vector field V , precisely so that it satisfies the V -equation (2.30). For this choice we obtain ∂ t (g αβ γ αβ ) = 0. This implies that g αβ γ αβ is conserved for any x ∈ R d , and thus the harmonic gauge condition is propagated in time.

Derivation of the modified Schrödinger system from SMCF.
Here we derive the main Schrödinger equation and the second compatibility condition. We consider the commutation relation In order, for the left-hand side, by (2.6) and (2.25) we have . Then by the above three equalities, equating the coefficients of the tangent vectors and the normal vector m, we obtain the evolution equation for λ as well as the compatibility condition (curvature relation) = Re(λ γ α∂ A γψ ) − Im(λ γ αλ γ σ )V σ , which we record for later reference: This in turn allows us to use the Coulomb gauge condition (2.16) in order to obtain an elliptic equation for B:

Lemma 2.6 (Elliptic equation of B). The temporal connection coefficient B solves
By the harmonic coordinates condition (2.19), (2.27) and the Coulomb gauge condition (2.16) the first term in the right hand side is written as We then obtain the B-equation.
Next, we use (2.31) to derive the main equation, i.e. the Schrödinger equation for ψ. By (2.10), the right-hand side of (2.31) is rewritten as Hence, we have and then contracting this yields This can be further written as Hence, under the harmonic coordinates condition (2.19) and the Coulomb gauge condition (2.16) we obtain the main Schrödinger equation In conclusion, under the Coulomb gauge condition ∇ α A α = 0 and the harmonic coordinate condition g αβ γ αβ = 0, by (2.34), (2.14), (2.22), (2.30), (2.17) and (2.33), we obtain the Schrödinger equation for the complex mean curvature ψ where the metric g, curvature tensor λ, the advection field V , connection coefficients A and B are determined at fixed time in an elliptic fashion via the following equations Fixing the remaining degrees of freedom (i.e. the affine group for the choice of the coordinates as well as the time dependence of the SU (1) connection) we can assume that the following conditions hold at infinity in an averaged sense: These are needed to insure the unique solvability of the above elliptic equations in a suitable class of functions. For the metric g it will be useful to use the representation Finally, we note that the above system ( (ix) The time evolution (2.32) for A . These conditions will all be shown to be satisfied for small solutions to the nonlinear elliptic system (2.35). Now we can restate here the small data local well-posedness result for the (SMCF) system in Theorem 1.2 in terms of the above system:  Here the solution ψ satisfies in particular the expected bounds The spaces l 2 X s and E s , defined in the next section, contain a more complete description of the full set of variables ψ, λ, h, V, A, B, which includes both Sobolev regularity and local energy bounds.
In the above theorem, by well-posedness we mean a full Hadamard-type wellposedness, including the following properties:

Function Spaces and Notations
The goal of this section is to define the function spaces where we aim to solve the (SMCF) system in the good gauge, given by (2.35). Both the spaces and the notation presented in this section are similar to those introduced in [21 -23]. All the function spaces described below will be used with respect to harmonic coordinates determined by our gauge choices described in the previous section. We neither attempt nor need to transfer these spaces to other coordinate frames.
For a function u(t, x) or u(x), letû = Fu denote the Fourier transform in the spatial variable x. Fix a smooth radial function ϕ : R d → [0, 1] supported in [−2, 2] and equal to 1 in [−1, 1], and for any i ∈ Z, let We then have the spatial Littlewood-Paley decomposition, where P i localizes to frequency 2 i for i ∈ Z, i.e, For simplicity of notation, we set For each j ∈ N, let Q j denote a partition of R d into cubes of side length 2 j , and let {χ Q } denote an associated partition of unity. For a translation-invariant Sobolev-type space U , set l p j U to be the Banach space with associated norm with the obvious modification for p = ∞.
Next we define the l 2 X s and l 2 N s spaces, which will be used for the primary variable ψ, respectively for the source term in the Schrödinger equation for ψ. Following [21][22][23], we first define the X -norm as Here and throughout, L p L q represents L p t L q x . To measure the source term, we use an Then we define N as linear combinations of the form For solutions which are localized to frequency 2 j with j ≥ 0, we will work in the space One way to assemble the X j norms is via the X s space But we will also add the l p spatial summation on the 2 j scale to X j , in order to obtain the space l p j X j with norm We then define the space l p X s by For the solutions of Schrödinger equation in (2.35), we will be working primarily in l 2 X s , which is defined by We note that the second component, introduced here for the first time, serves the purpose of providing better bounds at low frequencies j ≤ 0. We analogously define which has norm Here we shall be working primarily with l 2 N s . We also note that for any j ∈ N, we have This bound will come in handy at several places later on. For the elliptic system (2.36), at a fixed time we define the H s norm, In addition to the fixed time norms, for the study of the Schrödinger equation for ψ we will also need to bound time dependent norms E s and E s for the elliptic system (2.36), in terms of similar norms for ψ. For simplicity of notation, we define Then the Z σ,s spaces are defined by For the λ, V , A and B-equations in (2.36), we will be working primarily in Z 0,s , Z 1,s+1 , Z 1,s+1 and Z 1,s , respectively.
On the other hand, for the metric component h = g − I d we need to introduce some additional structure which is associated to spatial scales larger than the frequency. Precisely, to measure the portion of h which is localized to frequency 2 j , j ∈ Z, we decompose P j h as an atomic summation of components h j,l associated to spatial scales 2 l with l ≥ | j|, where h j,l still localizes to frequency 2 j , i.e., Then we define the Y j -norm by Assembling together the dyadic pieces in an l 2 Besov fashion, we obtain the Y σ,s space with norm given by Then for h-equation in (2.35), we will be working primarily in Y s+2 , whose norm is defined by Collecting all the components defined above, for the elliptic system (2.36), we define the E s norm as and the E s norm as Since we often use Littlewood-Paley decompositions, the next lemma is a convenient tool to see that our function spaces are invariant under the action of some standard classes of multipliers: We will also need the following Bernstein-type inequality: Proof. We begin with the Bernstein-type inequality (3.1). Using the properies of the Fourier transform, P k f is rewritten as Then Then from Young's inequality and 1 + 1/ p = 1/q + 1/q we can bound I by On the other hand, since |K (x)| x −N for any large N , for I I we have which can be absorbed by the term on the left. These imply the bound (3.1).
Next, we prove the estimate (3.2). The left hand side of (3.2) is decomposed as Then by (3.1) we bound I 1 by On the other hand, by Hölder's inequality and (3.1), we bound I 2 by Finally, we define the frequency envelopes as in [21][22][23] which will be used in multilinear estimates. Consider a Sobolev-type space U for which we have We shall only permit slowly varying frequency envelopes. Thus, we require a 0 ≈ u U and The constant δ only depends on s and the dimension d. Such frequency envelopes always exist. For example, one may choose

Elliptic Estimates
Here we consider the solvability of the elliptic system (2.36), together with the constraints (2.4), (2.8), (2.15), (2.13), (2.19) and (2.16). We will do this in two steps. First we prove that this system is solvable in Sobolev spaces at fixed time. Then we prove space-time bounds in local energy spaces; the latter will be needed in the study of the Schrödinger evolution (2.35).
For simplicity of notations, we define the set of elliptic variables by Later when we compare two solutions for (2.36), we will denote the differences of two solutions or the linearized variable by δS = (δλ, δh, δV, δ A, δB).
Our fixed time result is as follows: Moreover, assume thatp k and s k are admissible frequency envelopes for ψ ∈ H σ , S ∈ H s respectively. Then we have c) We also have a similar bound for the Hessian of the solution map, where = d α=1 ∂ 2 α and the nonlinear source terms are given by In order to prove the existence of solutions to (4.5) at a fixed time for small ψ ∈ H s , we construct solutions to (4.5) iteratively. We define the sets of elliptic variables with the trivial initialization We will inductively show that with a large universal constant C. This trivially holds for our initialization. Then using a standard Littlewood-Paley decomposition, Bernstein's inequality and the smallness of our data ψ ∈ H s in order to estimate the source terms H From the iterative scheme (4.6) and ψ ∈ H s small, we can repeat the same analysis for successive differences in order to obtain a small Lipschitz constant, We need to show that these constraints are satisfied for solutions to the elliptic system (2.36). We can disregard the B and V equations, which are unneeded here.
To shorten the notations, we define Here C 2 and C 3 are antisymmetric, C 6 is symmetric and C 7 inherits all the linear symmetries of the curvature tensor.
Our goal is to show that all these functions vanish. We will prove this by showing that they solve a coupled linear homogeneous elliptic system of the form Here the covariant Laplace operators g , respectively A g are symmetric and coercive inḢ 1 . We consider these equations as a system in the space usingḢ 1 bounds for the Laplace operator in the second to fifth equations, and interpreting the last two equations as an elliptic div-curl system in L 2 , with anḢ −1 source term.
Since the coefficients are all small, the right hand side terms are perturbative and 0 is the unique solution for this system. The details are left for the reader, as they only involve Sobolev embeddings and Hölder's inequality.
To complete the argument, we now successively derive the equations in the above system. In the computations below, it is convenient to introduce several auxiliary notations. The curvature of the connection A acting on complex valued functions is denoted by We also set The equation for C 1 This equation has the exact form This is obtained by (2.14) directly.
The equation for C 2 The full system for C 2 has the form By λ-equation (2.14) we have Then we use C 6 , C 7 and C 3 to give Hence, the C 2 -equation (4.7) follows.
The equation for C 3 This has the form To prove this, it is convenient to separate the left hand side into two terms, For the commutator we use the Bianchi identities to compute On the other hand for the second term we use the A equation in (2.36) to write The first term I I 1 combines directly with the first two terms in I . For the second we commute Summing up the expressions for I and I I we obtain (4.8).
The equation for C 4 This has the form To prove it we commute g with ∇ α In the last term we can symmetrize in α and γ , and the desired equation (4.9) follows.
The equation for C 5 Here we compute We can rewrite the g equation (2.22) as which by contraction yields R =R + ∇ α C 5 α . To get to g C 5 , by the above two equalities we write The first term drops by twice contracted Bianchi, and the last one is quadratic in λ and yields C 1 and C 2 terms, This completes the derivation of (4.10).
The equation for C 6 This has the form Indeed, by the g-equation in (2.36) and its proof, we recover the Ricci curvature This implies the relation (4.11) immediately.
The equation for C 7 By the second Bianchi identities of Riemannian curvature and the following equality we have the counterpart of the second Bianchi identities ∇ δ C 7 σ γ αβ + ∇ σ C 7 γ δαβ + ∇ γ C 7 δσ αβ = 0, which combine with the algebraic symmetries of the same tensor to yield an elliptic system for C 7 . Precisely, using the above relation we have which combined with the previous one yields the desired elliptic system, with C 6 viewed as a source term. b) Assume thats k and s k are admissible frequency envelopes for δS ∈ H σ and S ∈ H s , respectively. In view of the bound (4.1) and of the smallness of ψ H s , it suffices to prove the difference or linearized estimate Now we turn our attention to the proof of (4.12). Here we first prove the estimates for δλ. By λ-equations in (4.5) it suffices to consider the following form ∂ α δλ αβ = ∂ β δψ + δ Aψ + Aδψ + δh∇λ + h∇δλ + ∇δhλ + ∇hδλ, ∂ α δλ βγ − ∂ β δλ αγ = δ Aλ + Aδλ + ∇δhλ + ∇hδλ.

By the relation
we obtain Next we provide the estimate for δ A; the other estimates can be proved similarly. By A-equation in (4.5) and Lemma 4.2, it suffices to consider the following form Using Littlewood-Paley trichotomy and Bernstein inequality, we bound all the nonlinearities except ∇λδλ and λ∇δλ by For the remainder terms, we can also bound their low-frequency part by and bound their high-frequency part S k for k > 0 by This completes the proof of (4.2). c) Using the similar argument to b), we have Then by the smallness of ψ ∈ H s , (4.2) and interpolation, the above two bounds imply This completes the proof of (4.4).
Next we establish bounds for the above solutions in space-time local energy spaces: with Lipschitz dependence on the initial data in these topologies. Moreover, assume that p k is an admissible frequency envelope for ψ ∈ l 2 X s , we have the frequency envelope version b) In addition, for the linearization of the elliptic system (2.36) we have the bounds Proof of Theorem 4.3. For the elliptic system (4.5), we will prove the bound for differences δS If this is true, by a continuity argument the bounds (4.14) and (4.16) follow. Assume thats k and s k are admissible frequency envelopes for δS ∈ E σ and S ∈ E s , respectively. We can separate the bound (4.17) into two parts, namely As an intermediate step in the proof of (4.18), we collect in the next Lemma several bilinear estimates and equivalent relations.
As consequences of these bounds, for Proof of Lemma 4.4. We do this in several steps: Proof of the bound (4.19). First, we consider the Y -norm estimates. For the high-low interaction, for any decomposition P jh = l≥| j|h j,l , we have l≥| j| Taking the infimum over the decomposition ofh j yields l≥| j| which is acceptable. Similarly, for the low-high interaction, we have l≥| j| which is acceptable. Next, for the high-high interaction, when j < 0 we rewrite it as Then we bound the first term by We bound the second term by When j ≥ 0, we have which is acceptable. Secondly, we consider the Z 1,σ +1 -norm estimates. For the low-frequency part, we have For the high frequency part, by Littlewood-Paley dichotomy, we have which is acceptable. This completes the proof of (4.19).
Proof of the bound (4.20). First we consider the Z δ,σ -norm estimates. For the lowfrequency part we have For the high-frequency part, by the Littlewood-Paley dichotomy, we have

Proof of the bound (4.21). For the low-frequency part, by Bernstein's inequality we have
For the high-frequency part, by Littlewood-Paley dichotomy we bound the high-low and low-high interactions by which is acceptable. We bound the high-high interaction by which is also acceptable. Hence, we conclude the proof of the bound (4.21).
We now turn our attention to the proof of (4.18).
By the relation (4.13) we have for any k > 0 In order to bound the low frequency part k = 0, we use the relation Then we have Using this idea, by Sobolev embeddings we have The high frequency part is obtained by a standard Littlewood-Paley decomposition and Bernstein inequality. This gives the elliptic estimate for the δλ-equation.
Step 2. Proof of the elliptic estimates for V , A and B equations. By the V, A, Bequations and Proposition 4.4, it suffices to consider the following form The proofs of the three elliptic estimates for the above equations are similar, so we only prove the elliptic estimate for the linearization of A-equation in detail, i.e.
We bound all the nonlinearities except ∇λδλ and λ∇δλ by for σ ∈ (d/2 − 1, s]. All terms are estimated in a similar fashion, so we only bound the first term δh∇ 2 A. For the low-frequency part we use the relation (4.22) to bound the second term δh∇ 2 A by A minor modification of this argument also yields For the high-frequency part, by Littlewood-Paley dichotomy and Bernstein's inequality (3.1), we have Finally, we bound the last two terms ∇λδλ and λ∇δλ. For low-frequency part, using d ≥ 4 we have We also obtain For the high-frequency part, we have We can also bound the term λ∇δλ similarly. This gives the elliptic estimate for δ Aequation.
The proof of the Z 1,σ +2 bound is similar to the estimates for V, A, B equations in Step 2, hence we only bound of the Y d/2−1−δ,σ +2 -norm. We prove that the following frequency envelope version holds: Case 1. The contribution of δλλ. By the Littlewood-Paley dichotomy, it suffices to consider the high-low, low-high and high-high cases for any j ∈ Z l< j+O (1) P j (P j δλP l λ), l< j+O (1) P j (P l δλP j λ), l> j+O (1) P j (P l δλP l λ).

Case 1(a). The contribution of high-low and low-high interaction.
The two cases are proved similarly, so we only consider the worst case, namely the low-high interaction. When j ≤ 0, by the definition of the Y j -norm we have When j > 0, we further divide the low-high interaction into l< j P j (P l δλ · P j λ) = − j≤l< j P j (P l δλ · P j λ) + l<− j P j (P l δλ · P j λ).
For the first term, by Bernstein's inequality we have For the second term we have

Case 1(b). The contribution of high-high interactions.
When j < 0, we divide this into l> j P j (P l δλ · P l λ) = − j≥l> j P j (P l δλ · P l λ) + l>− j P j (P l δλ · P l λ).
Then we bound the first term by Using the Y j norm we can also bound the second term by Finally, when j > 0, using again the Y j norm we have Case 2. The contribution of δh∇ 2 h, h∇ 2 δh and ∇δh∇h. It suffices to prove that For the high-low interactions, it suffices to consider the worst case ∇ 2 P j δh · P ≤ j h. For any decomposition P j δh = l≥| j| δh j,l , we have Taking the infimum over the decomposition of P j h yields which is acceptable. The low-high interactions is similar and omitted.
For the high-high interaction, it suffices to estimate l> j P j (P l ∇δh P l ∇h). By Bernstein's inequality we have

For the low-frequency part, By Bernstein's inequality and d ≥ 4 we have
For the high-frequency part, by Bernstein's inequality we also have Thus this completes the proof of Y d/2−1−δ,σ +2 bound.

Multilinear and Nonlinear Estimates
This section contains our main multilinear estimates which are needed for the analysis of the Schrödinger equation in (2.35). We begin with the following low-high bilinear estimates of ∇h∇ψ.
In addition, if −s ≤ σ ≤ s − 1 then we have

Proof. a) The estimates (5.1) and (5.3). The proof of second bound (5.3) is similar to
the first, so we only prove the first bound in detail. By duality, it suffices to estimate for any z k := S k z ∈ l 2 k X k with z k l 2 k X k ≤ 1. For I j and any decomposition P j h = l≥| j| h j,l , by duality and Bernstein inequality, we have Then taking the infimum over the decomposition of P j h and incorporating the summation over j yield for any > 0. If −s ≤ σ ≤ d/2, we also have Thus the bound (5.1) follows. Estimate (5.2). By duality, it suffices to bound for any z k ∈ l 2 k X k with z k l 2 k X k ≤ 1. For any decomposition P j h = l≥| j| h j,l , by |∇a|(x) x −1 , we consider the two cases |x| ≥ 2 j/2 and |x| < 2 j/2 respectively and then obtain The first term is bounded by The second term is bounded by Then we obtain Thus the bound (5.2) follows.
We next prove the remaining bilinear estimates and trilinear estimates.

Proposition 5.2 (Nonlinear estimates)
. a) Let s > d 2 and d ≥ 3, assume that p k and s k are admissible frequency envelopes for ψ ∈ l 2 X s , S ∈ E s respectively. Then we have b) Assume thatp k ands k are admissible frequency envelopes for ψ ∈ l 2 X σ , S ∈ E σ respectively. Then for −s ≤ σ ≤ s we have Proof. We first prove (5.7) and (5.8). These two bounds are proved similarly, here we only prove the first bound in detail. For the high-low case, by (3.1) we have For the high-high case, when σ + d/2 + 1 > δ we have and when σ + d/2 + 1 ≤ δ we have Next, we prove the bounds (5.4)-(5.6) and (5.9)-(5.11). These bounds can be estimated similarly, we only prove (5.4) and (5.9) in detail. Indeed, by duality we have Then using Littlewood-Paley dichotomy to divide this into low-high, high-low and highhigh cases. For the low-high case, by Sobolev embedding we have The high-low case can be estimated similarly. For the high-high case, by Sobolev embedding when σ + d/2 ≥ 0 we have These imply the bound (5.4) and (5.9). Finally, we prove the bound (5.12). If σ > d/2 − 1 + δ, by duality and Sobolev embedding, we have Then the bound (5.12) follows. Hence this completes the proof of the lemma. We shall also require the following bounds on commutators.

Proposition 5.3 (Commutator bounds). Let s >
Let m(D) be a multiplier with symbol m ∈ S 0 . Assume h ∈ Y s+2 , A ∈ Z 1,s+1 and ψ k ∈ l 2 X s , frequency localized at frequency 2 k . If −s ≤ σ ≤ s we have Proof. First we estimate (5.13). In [21,Proposition 3.2], it was shown that where L is a translation invariant operator satisfying Given this representation, as we are working in translation-invariant spaces, by (5.1) the bound (5.13) follows. Next, for the bound (5.14). Since By translation-invariance and the similar argument to (5.9), the bound (5.14) follows. This completes the proof of the lemma.

Local Energy Decay and the Linearized Problem
In this section, we consider a linear Schrödinger equation and, under suitable assumptions on the coefficients, we prove that the solution satisfies suitable energy and local energy bounds.
6.1. The linear paradifferential Schrödinger flow. As an intermediate step, here we prove energy and local energy bounds for a frequency localized linear paradifferential Schrödinger equation We begin with the energy estimates, which are fairly standard: Proof. By (6.2), we have and notice that for each t ∈ [0, 1] we have by duality and Sobolev embedding We take the supremum over t on the left hand side and the conclusion follows.
Next, we prove the main result of this section, namely the local energy estimates for solutions to (6.2): Proposition 6.2 (Local energy decay). Let d ≥ 3, assume that the coefficients g αβ = δ αβ + h αβ and A α in (6.2) satisfy for some s > d 2 . Let ψ k be a solution to (6.2) which is localized at frequency 2 k . Then the following estimate holds: Proof. The proof is closely related to that given in [21,22]. However, here we are able to relax the assumptions both on the metric g and on the magnetic potential A. In the latter case, unlike in [21,22], we treat the magnetic term 2i A α <k−4 ∂ α ψ k as a part of the linear equation, which allows us to avoid bilinear estimates for this term and use only the bound for A in Z 1,s+1 .
As an intermediate step in the proof, we will establish a local energy decay bound in a cube Q ∈ Q l with 0 ≤ l ≤ k: (6.6) The proof of this bound is based on a positive commutator argument using a well chosen multiplier M. This will be first-order differential operator with smooth coefficients which are localized at frequency 1. Precisely, we will use a multiplier M which is a sef-adjoint differential operator having the form with uniform bounds on a and its derivatives. Before proving (6.5), we need the following lemma which is used to dismiss the (g − I d ) contribution to the commutator [∂ α g αβ ∂ β , M]. Lemma 6.3. Let s > d 2 and d ≥ 3, assume that h ∈ Y s+2 , A ∈ Z 1,s+1 and ψ ∈ l 2 k X k , let M be as (6.7). Then we have Proof of Lemma 6.3. By (6.7) and directly computations, we get Then it suffices to estimate The first integral is estimated by (5.1) and (5.2). Using Sobolev embedding, the second integral is bounded by Hence, the bound (6.8) follows. For the second bound (6.9), by (6.7) and integration by parts we rewrite the following term as Then we bound the left-hand side of (6.9) by This implies the bound (6.9), and hence completes the proof of the lemma.
Returning to the proof of (6.6), for the self-adjoint multiplier M we compute We then use the multiplier M as in [21,22] so that the following three properties hold: (1) Boundedness on frequency 2 k localized functions, (2) Boundedness in X , (3) Positive commutator, If these three properties hold for u = ψ k , then by (6.9) and (6.4) the bound (6.6) follows. We first do this when the Fourier transform of the solution ψ k is restricted to a small angle supp ψ k ⊂ {|ξ | ξ 1 }. (6.10) Without loss of generality due to translation invariance, Q = {|x j | ≤ 2 l : j = 1, . . . , d}, and we set m to be a smooth, bounded, increasing function such that m (s) = ϕ 2 (s) where ϕ is a Schwartz function localized at frequencies 1, and ϕ ≈ 1 for |s| ≤ 1. We rescale m and set m l (s) = m(2 −l s). Then, we fix The properties (1) and (2) are immediate due to the frequency localization of u = ψ k and m l as well as the boundedness of m l . By (6.8) it suffices to consider the property (3) for the operator This yields Utilizing our assumption (6.10), it follows that which yields (3) when combined with (6.8).
We proceed to reduce the problem to the case when (6.10) holds. We let {θ j (ω)} d j=1 be a partition of unity, where θ j (ω) is supported in a small angle about the j-th coordinate axis. Then, we can set ψ k, j = k, j ψ k where We see that By applying M, suitably adapted to the correct coordinate axis, to ψ k, j and summing over j, we obtain The commutator is done via (5.13) and (5.14). Then (6.6) follows. Next we use the bound (6.6) to complete the proof of Proposition 6.2. Taking the supremum in (6.6) over Q ∈ Q l and over l, we obtain Combined with (6.3), we get We now finish the proof by incorporating the summation over cubes. We let {χ Q } denote a partition via functions which are localized to frequencies 1 which are associated to cubes Q of scale M2 k . We also assume that |∇ l χ Q | (2 k M) −l , l = 1, 2. Thus, But by (6.4) we have and also For M sufficiently large, we can bootstrap the commutator terms, and, after a straightforward transition to cubes of scale 2 k rather than M2 k , we observe that We now apply (6.11) to χ Q ψ k , and then by (6.12) and (6.13) we see that For M 1, we have By (6.4), for k sufficiently large (depending on M), we may absorb the last terms in the right-hand side into the left, i.e On the other hand, for the remaining bounded range of k, we have and then (6.14) and (6.4) gives which finishes the proof of (6.5).

The full linear problem.
Here we use the bounds for the paradifferential equation in the previous subsection in order to prove similar bounds for the full equation (6.1): Proposition 6.4 (Well-posedness). Let s > d 2 , d ≥ 3 and h = g − I d ∈ Y s+2 , assume that the metric g, and the magnetic potential A satisfy Then the equation (6.1) is well-posed for initial data ψ 0 ∈ H σ with −s ≤ σ ≤ s, and we have the estimate Moreover, for 0 ≤ σ ≤ s we have the estimate Proof. The well-posedness follows in a standard fashion from a similar energy estimate for the adjoint equation. Since the adjoint equation has a similar form, with similar bounds on the coefficients, such an estimate follows directly from (6.15). Thus, we now focus on the proof of the bound (6.15). For ψ solving (6.1), we see that ψ k solves If we apply Proposition 6.2 to each of these equations, we see that Indeed, the bound for the terms in H k follows from (5.7), (5.13), (5.14), (5.8), respectively. Then by the above two bounds, we obtain the estimate (6.15). Finally, by the ψ-equation (6.1), for time derivative bound it suffices to consider the form Then by the standard Littlewood-Paley dichotomy and Bernstein's inequality, for 0 ≤ σ ≤ s we have the following estimates This, combined with (6.15), yields the bound (6.16), and then completes the proof of the Lemma.

The linearized problem.
Here we consider the linearized equation: and we prove the following.
, assume that is a solution of (6.17), the metric g and A satisfy 1.
Then we have the estimate Proof. For solving (6.17), we see that k solves The proof of (6.18) is similar to that of (6.16). Here it suffices to prove Indeed, the bound for the terms in G k follows from (5.7), (5.3), (5.8) and (5.12). The second bound follows from a standard Littlewood-Paley decomposition and Bernstein's inequality. This completes the proof of the Lemma.

Well-Posedness in the Good Gauge
In this section we use the elliptic results in Sect. 4, the multilinear estimates in Sect. 5 and the linear local energy decay bounds in Sect. 6 in order to prove the good gauge formulation of our main result, namely Theorem 2.7.

The iteration scheme: uniform bounds.
Here we seek to construct solutions to (2.35) iteratively, based on the scheme with the trivial initialization where the nonlinearities are and ) are the solutions of elliptic equations (2.36) with ψ = ψ (n) . We assume that ψ 0 is small in H s . Due to the above trivial initialization, we also inductively assume that where C is a big constant. Applying the elliptic estimate (4.14) to (2.36) with ψ = ψ (n) at each step, we obtain Applying at each step the local energy bound (6.16) with σ = s we obtain the estimate Here the nonlinear terms in F (n) are estimated using (5.1), (5.7), (5.4), (5.5) and (5.6) with σ = s. Since ψ 0 is small in H s , the above bound gives which closes our induction.

The iteration scheme: weak convergence.
Here we prove that our iteration scheme converges in the weaker H s−1 topology. We denote the differences by Then from (7.1) we obtain the system where the nonlinearities G (n) have the form By (4.16) we obtain Applying (6.18) with σ = s − 1 for the (n+1) equation we have Then by (5.1), (5.7), (5.9), (5.10) and (5.11) with σ = s − 1 we bound the right hand side above by This implies that our iterations ψ (n) converge in l 2 X s−1 to some function ψ. Furthermore, by the uniform bound (7.3) it follows that Interpolating, it follows that ψ (n) converges to ψ in l 2 X s− for all > 0. This allows us to conclude that the auxiliary functions S (n) associated to ψ (n) converge to the functions S associated to ψ, and also to pass to the limit and conclude that ψ solves the (SMCF) equation (2.35). Thus we have established the existence part of our main theorem.

The solves an equation of this form
where the nonlinearity G is (2) .
Applying (6.18) with σ = s − 1 to the equation, we obtain the estimate Then, by the above bound (7.5), we further have Since the initial data ψ (1) 0 and ψ (2) 0 are sufficiently small, we obtain This gives the weak Lipschitz dependence, as well as the uniqueness of solutions for (2.35).

Frequency envelope bounds.
Here we prove a stronger frequency envelope version of estimate (7.4).
Proposition 7.1. Let ψ ∈ l 2 X s be a small data solution to (2.35), which satisfies (7.4). Let { p 0k } be an admissible frequency envelope for the initial data ψ 0 ∈ H s . Then { p 0k } is also frequency envelope for ψ in l 2 X s .
Proof. Let p k and s k be the admissible frequency envelopes for solution (ψ, S) ∈ l 2 X s ×E s . Applying S k to the Schrödinger equation (2.35), we obtain the paradifferential equation and S = (λ, h, V, A, B) is the solution to the elliptic system (2.36). We estimate ψ k = S k ψ using Proposition 6.4. By Proposition 5.2, Lemmas 5.1 and 5.3 we obtain Then by (4.15), the definition of frequency envelope (3.3) and (7.4), this implies By the smallness of ψ ∈ l 2 X s , this further gives p k p 0k , and concludes the proof.

Continuous dependence on the initial data.
Here we show that the map ψ 0 → (ψ, S) is continuous from H s into l 2 X s × E s . By (4.16), it suffices to prove ψ 0 → ψ is continuous from H s to l 2 X s . Suppose that ψ 0k , respectively p 0k the frequency envelopes associated to ψ To compare ψ (n) with ψ we use (7.6) for low frequencies and (7.7) for the high frequencies, Letting n → ∞ we obtain lim sup n→∞ ψ (n) − ψ l 2 X s .

Higher regularity.
Here we prove that the solution (ψ, S) satisfies the bound (ψ, S) l 2 X σ ×E σ ψ 0 H σ , σ ≥ s, (7.8) whenever the right hand side is finite. Differentiating the original Schrödinger equation (2.35) yields where F is defined as in (7.2) without superscript (n). Using Proposition 6.5 we obtain For elliptic equations, by (4.16) we obtain Hence, by (7.4) and the smallness of ψ 0 in H s , these imply Inductively, we can obtain the system for (∇ n ψ, ∇ n S). This leads to which shows that and hence gives the bound (7.8) by the smallness of ψ in l 2 X s . 7.7. The time evolution of (λ, g, A). As part of our derivation of the (SMCF) equations (2.35) for the mean curvature ψ in the good gauge, coupled with the elliptic system (2.36), we have seen that the time evolution of (λ, g, A) is described by the equations ( To shorten the notations, we define the tensors We need to show that T 1 = 0, T 2 = 0, T 3 = 0. To do this, we will show that (T 1 , T 2 , T 3 ) solve a linear homogeneous coupled elliptic system of the form Considering this system for (T 1 , T 2 , T 3 ) ∈Ḣ 1 × L 2 × L 2 , the smallness condition on the coefficients (λ, h, V, A, B) ∈ S insures that this system has the unique solution (T 1 , T 2 , T 3 ) = 0. It remains to derive the system for (T 1 , T 2 , T 3 ).

The equation for T 1 This has the form
We start with the first term in T 1 , and compute the expression g ∂ t g αβ . We have We then use covariant derivatives to write I I as For I , by the g equation (2.22) we have The expression I 1 is written as For I 2 , we first compute By the above computations, we collect the ∇∂ t g terms from I 1 , I 2 and I I ∇ α ∂ t g μν (− μν,β + μβ,ν ) + ∇ β ∂ t g μν (− μν,α + μα,ν ) − ∇ μ ∂ t g νβ ∂ α g μν − ∇ μ ∂ t g να ∂ β g μν where the terms containing ∇∂ t g να and ∇∂ t g νβ vanish, i.e.
Hence, given the expressions of I 3 and I I I , we obtain which combined with (7.11) yields the T 1 -equation (7.9).

The equation for T 2 This has the form
(7.12) We compute the divergence of T 2 in (7.12) first. Applying ∇ A,α to T 2,σ α , we have Three of the terms on the right-hand side are written as We can further use T 1 to rewrite the last two terms on the first line above as V β ] and the following term as . Similarly, we compute the second commutator by Hence, using T 2,α α and the V equation (2.30) we reorganize the expression of ∇ A,α T 2,σ α and obtain Using T 2,α α and the V -equation (2.30), we have γ ψ Im(ψλ σ γ ) − 2∇ A,σ λ αβ Im(ψλ αβ ) Combining these two expressions, we obtain Applying ∇ α to T 3 α , we then use the Coulomb condition ∇ α A α = 0 and the B-equation (2.33) to get The curl of T 3 is obtained by (2.13) directly.

The Reconstruction of the Flow
In this last section we close the circle of ideas in this paper, and prove that one can start from the good gauge solution given by Theorem 2.7, and reconstruct the flow at the level of d-dimensional embedded submanifolds. For completeness, we provide here another, more complete statement of our main theorem: which, when represented in harmonic coordinates, has regularity . and induced metric and mean curvature In addition the mean curvature satisfies the bounds where ψ and λ are expressed using the Coulomb gauge in the normal bundle N t .
We complement the theorem with the following remarks: Once this is done, we have the frame F α in the tangent space and the frame m in the normal bundle. In turn, as described in Sect. 2, these generate the metric g, the second fundamental form λ with trace ψ and the connection A, all at the initial time t = 0.
Moving forward in time, Theorem 2.7 provides us with the time evolution of ψ via the Schödinger flow (2.35), as well as the functions (λ, g, V, A, B)  We now return to the question of constructing the harmonic coordinates at the initial time. In order to state the following proposition, we define some notations. Let F : be an immersion with induced metric g(x). For any change of coordinate y = x + φ(x), we denotẽ F(y) = F(x(y)), and its induced metricg αβ (y) = ∂ y αF , ∂ y βF . We also denote its Christoffel symbol as˜ andh(y) =g(y) − I d .
be an immersion with induced metric g = I d + h. Assume that ∇h(x) and mean curvature H are small in H s (dx), namely Then there exists a unique change of coordinates y = x + φ(x) with lim x→∞ φ(x) = 0 and ∇φ uniformly small, such that the new coordinates {y 1 , . . . , y d } are global harmonic coordinates, namely,g αβ (y)˜ γ αβ (y) = 0, for any y ∈ R d .
Then for the equation (8.9) we have estimates as follows: Next, we turn to prove the bound (8.17). By the change of coordinates, we have the representation ∂ 2 y α y βF as Since ∂ x γ ∂ y β is a function depending on x and has the form ∂ x ∂ y = I d + P(x), we write this as ∂ 2 y α y βF = ∂ 2 σ γ F(I d + P) 2 + ∂ γ F∂ x (I d + P) · (I d + P) = ∂ 2 σ γ F(I d + P) 2 + ∂ γ F∂ x P · (I d + P).
As a vector depends on x, by Sobolev embedding, (8.18)  Then by Lemma 8.5, the bound (8.17) follows.
Step 5: Prove the bound (8.4). Finally, we construct the initial data ψ 0 in the harmonic coordinates and Coulomb gauge. To obtain the Coulomb gauge, we chooseν constant uniformly transversal to T 0 ; such a ν exists because, by Sobolev embeddings, ∂ x F has a small variation in L ∞ . Projectingν on the normal bundle N 0 and normalizing we obtain someν 1 with the same regularity as ∂ F. Then we chooseν 2 in N 0 perpendicular toν 1 . We obtain the orthonormal frame (ν 1 ,ν 2 ) in N 0 , which again has the same regularity and bounds as ∂ x F. Then we rotate the frame to get a Coulomb frame (ν 1 , ν 2 ), i.e. where the Coulomb gauge condition (2.16) is satisfied. Projecting the mean curvature H on the Coulomb frame as in Sect. 2.3 we obtain the complex mean curvature ψ ∈ H s .
We easily have Then it suffices to bound theḢ s norm of λ. If s ∈ N, we have λ Ḣ s ν∈{ν 1 ,ν 2 };n 1 +n 2 =s If s / ∈ N, let 1 p + 1 q = 1 2 we also have We bound the first term by which combined with the smallness of ∂ x g ∈ H s also gives the bound (8.4) for ψ.

The moving frame.
Once we have the initial data ψ 0 which is small in H s , Theorem 2.7 yields the good gauge local solution ψ, along with the associated derived variables (λ, h, V, A, B). But this does not yet give us the actual maps F. Here we undertake the task of reconstructing the frame (F α , m). For this we use the system consisting of (2.6) and (2.25), viewed as a linear ode. We recall these equations here: respectively where (ψ, λ, g, V, A, B) is the unique solution of (2.35)-(2.36) with initial data ψ 0 small. We start with the frame at time t = 0, which already is known to solve (8.19), and has the following properties: (i) Orthogonality, F α ⊥ m, m, m = 2, m,m = 0 and consistency with the metric g αβ = F α , F β . (ii) Integrability, ∂ β F α = ∂ α F β . (iii) Consistency with the second fundamental form and the connection A: Next we extend this frame to times t > 0 by simultaneously solving the pair of equations (8.19) and (8.20). To avoid some technical difficulties, we first do this for regular solutions, i.e. s > d/2 + 2, and then pass to the limit to obtain the frame for rough solutions.
8.2.1. The frame associated to smooth solutions The system consisting of (8.19) and (8.20) is overdetermined, and the necessary and sufficient condition for existence of solutions is provided by Frobenius' theorem. We now verify these compatibility conditions in two steps: a) Compatibility conditions for the system (8.19) at fixed time. Here, by C 2 αβ = 0, C 3 αβ = 0 and C 7 αβμν = 0 we have ∂ α ( σ βγ F σ + Re(λ βγm )) − ∂ β ( σ αγ F σ + Re(λ αγm )) = C 7 σ γ αβ F σ = 0, and ∂ α (i A β m + λ σ β F σ ) − ∂ β (i A α m + λ σ α F σ ) = iC 3 αβ m = 0, as needed. b) Between the system (8.19) and (8.20). By (8.19) and (8.20) we have The first equality is obtained directly. For the second equality (8.21), by (8.19) and (8.20) we compute this by By T 1 and the notation G αβ (2.29) we compute the last term by Then by Bianchi identities and (2.8), we collect the terms above containing V and have From the above expressions the equality (8.21) follows. Once the compatibility conditions in Frobenius' theorem are verified, we obtain the frame (F α , m) for t ∈ [0, 1]. For this we can easily obtain the regularity Finally, we show that the properties (i)-(iii) above also extend to all t ∈ [0, 1]. The properties (ii) and (iii) follow directly from the equations (8.19) and (8.20) once the orthogonality conditions in (i) are verified. For (i) we denotẽ g 00 = m, m ,g α0 = F α , m ,g αβ = F α , F β .
This implies that the F solves (1.1).
Acknowledgements. J. Huang would like to thank Prof. Lifeng Zhao for many inspirations and discussions, and Dr. Ze Li for carefully reading the manuscript, helpful discussions and comments.
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