Global existence of Dirac-wave maps with curvature term on expanding spacetimes

We prove the global existence of Dirac-wave maps with curvature term with small initial data on globally hyperbolic manifolds of arbitrary dimension which satisfy a suitable growth condition. In addition, we also prove a global existence result for wave maps under similar assumptions.


Introduction and results
Wave maps are among the fundamental variational problems in differential geometry. They are defined as critical points of the Dirichlet energy for a map between two manifolds, where one assumes that the domain manifold is Lorentzian and the target manifold is Riemannian. More precisely, let (M, h) be a globally hyperbolic Lorentzian manifold, (P, G) a Riemannian manifold and φ : M → P a map. Squaring the norm of its differential gives rise to the Dirichlet energy, whose critical points are given by the wave map equation, which is τ (φ) = 0, where τ (φ) := −h αβ ∇ ∂ α dφ(∂ β ).
The wave map equation is a second-order semilinear hyperbolic system, that is the natural analogue of the harmonic map equation for maps between Riemannian manifolds. Wave maps are also well-studied in the physics literature, they appear as critical points of the Polyakov action in bosonic string theory for a string with Lorentzian worldsheet.
There are many articles that study wave maps in the case that the domain is Minkowski space. We cannot give an exhaustive list of the results here, but want to mention the influ-Communicated by J. Jost.
The square of the Dirac operator satisfies the Schrödinger-Lichnerowicz formula where scal M denotes the scalar curvature of the manifold M. Note that / ∂ 2 is a wave-type operator.
In quantum field theory one usually considers spinors that are twisted by some additional vector bundle. Here, we consider the case that the spinor bundle is twisted with the pullback of the tangent bundle from the target manifold. More precisely, we are considering the bundle SM ⊗ φ * T P and sections in this bundle are called vector spinors. We obtain a connection on SM ⊗ φ * T P by setting Note that Clifford multiplication on the twisted bundle is defined by acting only on the first factor. This allows us to define the Dirac operator acting on vector spinors as follows We assume that the connection on φ * T P is metric and thus the operator i / D is also self-adjoint with respect to the L 2 -norm. After these considerations we are ready to present the energy functional for Dirac-wave maps with curvature term In the last term the indices are contracted as follows ψ , R P ψ ,ψ ψ which ensures that the energy functional is real-valued. We would like to point out that in the physics literature (see e.g. [1]) one usually considers Grassmann-valued spinors in the analysis of (1.1). However, we want to use methods from the geometric calculus of variation and due to this reason we are employing standard spinors. The critical points of (1.1) can easily be calculated as where τ (φ) := −h αβ ∇ ∂ α dφ(∂ β ) represents the wave map operator. The solutions of (1.2), (1.3) are called Dirac-wave maps with curvature term.
We are able to provide the first existence result for the Dirac-wave map with curvature term system which is as follows: Theorem 1.1 Letg t be a smooth family of complete Riemannian metrics on n−1 , N ∈ C ∞ (R × ) with 0 < A ≤ N ≤ B < ∞ and (M n ,h) = (R × , −N 2 dt 2 +g t ) be a globally hyperbolic Lorentzian spin manifold that satisfies the following condition: There exists a monotonically increasing smooth function s : R → R + with ∞ 0 s −1 dt < ∞, such that the conformal metric h = (N s) −2h = −s −2 dt 2 + g t has bounded geometry, the metrics g t have a uniform Sobolev constant and N C k (g t ) + ∇ ν N C k (g t ) + II C k (g t ) + s II L ∞ ≤ C < ∞ (1.4) for all k ∈ N. Here, ν is the future-directed unit normal of the hypersurfaces {t} × and II is their second fundamental form.
Then if in addition, the Riemannian manifold (P, G) has bounded geometry, there exists for each r ∈ N with r > n−1

Remark 1.2
Under the same assumptions, the proof of Theorem 1.1 also implies global existence of Dirac-wave maps, which satisfy an equation slightly simpler than (1.2), (1.3). In this case, the second term on the right hand side of (1.2) and the right-hand side of (1.3) both vanish.
Along the line of Theorem 1.1 we obtain the following result for the wave map equation generalizing the results from [9,10]:

Theorem 1.3 Letg t be a smooth family of complete Riemannian metrics on
for all k ∈ N. Here, ν is the future-directed unit normal of the hypersurfaces {t} × and II is their second fundamental form.
Then if in addition, the Riemannian manifold (P, G) has bounded geometry, there exists for each r ∈ N with r > n−1 2 an ε > 0 such that if the initial data (φ 0 , φ 1 ) for the wave map equation satisfies the unique wave map with initial data φ| t=0 = φ 0 , ∂ t φ| t=0 = φ 1 exists for all times t ∈ [0, ∞) and satisfies

Remark 1.4
If we compare the assumptions of Theorems 1.1 and 1.3 we observe that one also has to control the R 0i j0 -components of the curvature tensor in the result for Diracwave maps with curvature term. This curvature contribution appears when computing the curvature of the spinor bundle. This explains why we have to assume bounded geometry of the Lorentzian manifold in Theorem 1.1 but only bounded geometry of the Riemannian slices in Theorem 1.3.
In addition, we also have to demand a decay of the second fundamental form (1.4) in Theorem 1.1 which originates from the choice of a positive definite scalar product on the spinor bundle that is no longer metric.

Remark 1.5
It seems that the class of Lorentzian manifolds that we are considering is the appropriate setting that guarantees a nice long-time behavior for solutions of various nonlinear wave equations with small initial data. We therefore think that our approach can be also used to establish existence results for a large class of other second order hyperbolic systems arising in mathematical physics. Example 1. 6 There are many spacetimes that satisfy the assumptions in Theorems 1.1 and 1.3. The simplest class is the class of Robertson-Walker spacetimes −dt 2 + s 2 (t)g with s −1 ∈ L 1 ([0, ∞)) which already contains the de-Sitter space and the power-law inflation metric. More generally, it is believed that generic future geodesically complete solutions of the Einstein equation with positive cosmological constant > 0 satisfy the assumptions of our theorems.
Throughout this article we will employ the following notation: We will use small Greek letters α β γ for space-time indices, small Latin indices i j k for spatial derivatives and capital Latin indices I J K for indices on the Riemannian target. We will denote spatial derivative by D. Moreover, we will make use of the usual summation convention, that is we will always sum over repeated indices.
This article is organized as follows: In the second section we introduce a suitable conformal transformation that we will be using to prove our main results. In the third section we establish the necessary energy estimate which is the key tool to prove the main result.

Conformal Euler-Lagrange equations
In the following we calculate how the energy functional (1.1) transforms under a certain conformal transformation of the metric on M.
Proof Under a conformal change of the metrich = (N s) 2 h the volume elements transform as dVh = (N s) n dV h . The Dirac operators transform as Note that the twisted Dirac operator / D transforms in the same way as the standard Dirac operator / ∂ since the twist bundle φ * T P does not depend on the metric on M. Inserting our choice forψ we get which completes the proof.
For the sake of completeness we calculate the critical points of (2.1).
We want to turn the system (2.2) into a system of two wave-type equations. To this end we recall the following Weitzenböck formula for the twisted Dirac operator / D.

Lemma 2.3
The square of the twisted Dirac operator / D satisfies Proof For a proof the we refer to [19,Theorem 8.17].
Making use of (2.3) we obtain the following rescaled Euler-Lagrange equations where h denotes the corresponding wave-operator. In terms of local coordinates this system acquires the form (I = 1, . . . , dim P)

Energies on general vector bundles
Let n be a manifold, s : R → R + smooth and g t , t ∈ R be a smooth family of Riemannian metrics on . We consider the (globally hyperbolic) Lorentzian manifold The non-vanishing Christoffel symbols of this metric are given by Let V be a Riemannian vector bundle over M which is equipped with a metric connection ∇. As usual, iterating yields a map We define the associated wave operator : (V ) → (V ) by the sign convention such that where π : M → is the canonical projection. In the following, we will write T * instead of π * (T * ) for notational convenience. The covariant derivative naturally extends as a map D : where for each t ∈ R, D X X i denotes the covariant derivative of g t . Furthermore, we define its formal adjoint D * : Observe that this definition makes sense as ∇ ∂ t X i ∈ (T M) is always tangential to due to the structure of the metric h. A quick computation shows that the wave operator can be written as Lemma 3. 1 We use the notations from above. Let ., . be the natural t-dependent scalar product induced on (T * ⊗k ⊗ V ). Let ξ, η ∈ (V ). Then we have the product rule Proof Let t 0 be a fixed time and let {e 1 , . . . , e n } be an orthonormal basis of ( , g t 0 ) at x ∈ . Then the scalar product can also be written as where the scalar products on the right hand side are the ones on V . Now think of {e 1 , . . . , e n } as an orthonormal system in . . , n} so that we in fact obtain orthonormal bases of T x with respect to the metric g t for each t ∈ R. Therefore we get

Remark 3.2
In the above lemma, the special structure of the metric h is essential.
For a section ξ ∈ (V ), we now define the kth energy density which is a nonnegative function on M and the kth energy as where dV g t is the volume element of the metric g t and the integral has to be understood as an integral over {t} × .

Proposition 3.3
Let ξ ∈ (V ) be spacelike compactly supported. Then its kth energy satisfies the evolution equation

Proof
We compute The statement follows by combining both equalities and using (3.1).
In order to derive energy estimates for the vector spinors we have to take into account the special structure of the scalar product of the spinor bundle SM. The natural geometric scalar product which is invariant under the spin group is not positive definite and thus not a good candidate for analytic purposes. To obtain a positive definite scalar product one has to Clifford multiply the second factor with the timelike unit vector field e 0 , see [3], which in our setup is given by e 0 = s∂ t . Using our geometric setup we find Hence, we define the following kth energy density for the vector spinors ψ which is a nonnegative function. In the following we will always employ the positive definite scalar product without mentioning it explicitly. However, since e 0 is not parallel with respect to the spatial coordinates the kth energy of the spinor satisfies Then we have the identity Proof At first, we compute Similarly, we find Summing up, we obtain As this expression is tensorial in X and the X i , we may assume that the components of these vector fields with respect to a chart of are independent of time. By raising an index with respect to g t we can think ofġ t as an endomorphism on T * . Then the formulas for the Christoffel symbols imply ∇ ∂ t X = ∇ X ∂ t = 1 2ġ (X ) and the same holds for X i . Moreover, By putting these facts together, we immediately get the statement of the Lemma.
In the following we will often make use of the so-called notation. More precisely, we will use a to denote various contractions between the objects involved.
We apply the general formula from above in the case where we have a map φ ∈ C ∞ (M, P), More generally, if E = φ * T P in the Lemma above, we get By an iteration argument, we find In the case of a vector spinor ψ ∈ (SM ⊗ φ * T P), the formulas are and By iterating the formula from above, we get where {e i } 1≤i≤n−1 is a local orthonormal frame. Here and throughout the proof, we sum over i.
and D R V is the covariant derivative of the curvature endomorphism on V restricted to vectors tangential to . In other words, In the case φ ∈ C ∞ (M, P), we obtain and for ξ ∈ (T * ⊗k ⊗ φ * T P), we have Thus by iteration we get In the case of vector spinors, that is ψ ∈ (SM ⊗ φ * T P), we obtain D * D, D ψ (X ) = −D Ric(X ) ψ + 2R P (Dφ(X ), Dφ(e i ))(Dψ(e i )) and for ξ ∈ (T * ⊗k ⊗ SM ⊗ φ * T P), we have By iteration, we then obtain We conclude this section with a very important lemma, which we will frequently make use of when deriving energy estimates.
If in addition, we have ξ k+1 ∈ H r −1 (E) and k+1 i=1 l i ≤ r − 1, then Here, C sob = C sob (g, l 1 , . . . , l k , r ) depends on the constant from the Sobolev embedding on M and the numbers l 1 , . . . , l k , r.
Proof We prove the first inequality, the second one is shown very similarly. Choose p i ∈ [1, ∞], i = 1, . . . , l k such that 1 due to our assumptions. Therefore, we can choose the p i such that k i=1 1 p i = 1 2 . An application of the Hölder inequality finishes the proof of the inequality.

Energy of the map
We define the kth energy density of the map part as and the kth energy as The kth total energy is given by where the Sobolev norms are taken with respect to the metric g t . In (3.9) below, we also define the total energy F k (ψ) of the spinor part, which we already need in the following proposition: Proposition 3.7 Let T > 0, r ∈ N, r > (n − 1)/2, k ∈ {1, . . . , r } and (φ, ψ) be a solution of (2.4), (2.5) such that Then the kth energy of the map satisfies the following inequality where the positive constants C i , i ∈ {1, . . . , 5} depend on n, k and the Sobolev constant of the metric g t .
Proof Assume for the moment that the initial data is compactly supported such that the solution is spacelike compactly supported. Using the general formula (3.3) we find Due to finite speed of propagation and an exhaustion procedure, this equality also holds generally for solutions in the above space. Note also that we will use Lemma 3.6 frequently in the proof without mentioning it explicitly. We have to estimate all terms on the right hand side and start by estimating the commutator terms The second commutator can be controlled as follows The third commutator can be estimated as follows As a second step we estimate the terms that arise when inserting the equation (2.4) for φ into (3.3). In order to estimate the first term we calculate This allows us to derive the following estimate In order to treat the second term on the right hand side of (2.4) we have to consider spatial and time derivatives separately This allows us to derive the following estimate: The third term on the right hand side of (2.4) can be computed as This allows us to derive the following estimate Adding up the different contributions concludes the proof.

Energy of the spinor
We define the kth energy density of the spinor part as and the kth energy as The kth total energy is given by Then the kth energy of the spinor satisfies the following inequality (3.11) where the positive constants C i , i ∈ {3, . . . , 8} depend on n, k and the Sobolev constant of the metric g t .

Remark 3.9
The curvature quantities coming from the spinor bundle above have to be understood as the following sections: R SM (∂ t , ·) ∈ (T * ⊗ End(SM)), R SM = R SM (., .) ∈ ( 2 T ⊗ End(SM)), both by restricting to vectors tangent to . Moreover ∂ t ∈ (End(SM)) acts by Clifford multiplication. The C k norms with respect to g t are then defined in a canonical way.
Proof In order to derive an energy estimate for the spinor we make use of (3.2). As before, we may at first assume that the solution is spacelike compactly supported before we obtain this equality for the general case by an exhaustion procedure. We will also frequently use Lemma 3.6 in this proof. Again, we have to estimate all terms on the right hand side and start by estimating the commutator terms: The second commutator term can be estimated as follows: The third commutator can be controlled as follows: As a next step we estimate the terms that arise when inserting (2.5) into (3.3). The first term can easily be controlled as Regarding the second term on the right hand side of (2.5) we again expand space and time contributions Note that we do not get a term proportional to |dφ(∂ t )| 2 due to symmetry reasons. We then find This allows us to derive the following estimate As a next step we take care of the spatial derivatives in the third term on the right hand side of (2.5). These can be computed as The kth spatial derivative of these terms acquire the forms These manipulations allow us to derive the following estimates: To control the last term from the right hand side of (2.5) we calculate Consequently, we obtain the following estimate The last term of (3.2) can be estimated as Adding up the different contributions yields the claim.

Remark 3.10
Note that all terms on the right hand side of (3.11) have a similar analytic structure except the terms proportional to C 3 and C 8 , which contain higher powers of the H r -norms of φ and ψ.

Energy of the coupled pair
Before we prove an energy estimate for the coupled pair, we give more geometric interpretations of the terms appearing in the above estimates. At first, recall that the second fundamental form II ∈ (T * 2 ) of a hypersurface ({t} × , g t ) ⊂ (M, h) is given by where ν is the future-directed unit normal of the hypersurface.

Lemma 3.11
We have the estimates Here, we defined the C k -norm of R M by taking the kth covariant derivative with respect to h but taking the norm with respect to the Riemannian reference metric s −2 dt 2 + g t in order to get a nonnegative quantity.
Proof The first estimate follows from the definition of II, the second and third from the facts that ∇ X ∂ t = 1 2ġ (X ) and e 0 = s∂ t . Now we recall that for every tensor T ∈ (T * ⊗k ) on the manifold M, the difference between the covariant derivatives ∇ and D can be expressed as R M X , Y , ∂ α , ∂ β ∂ α · ∂ β · ψ for X , Y ∈ (T M), where {∂ α } is a local pseudo-orthonormal frame. Therefore, we get in the above situation D k R SM ψ = l 1 +l 2 + m i =k D m 1 II · · · D m l 1 II which yields the fifth inequality. Similarly, by using the product rule, we obtain and by using the third inequality we obtain the last one.
At this point we are ready to control the total energy of (φ, ψ), which we define by Suppose thatṡ ≥ 0 and that the following uniform bounds II C r < C 6 , R M C r < C 7 , R P C r+1 < C 8 , II L ∞ < C 9 s −1 hold for some positive constants C i , i = 1, . . . , 9. Suppose finally that there is a uniform bound on all the Sobolev constants of g t . Then the total energy satisfies the following inequality d dt F r (φ, ψ) ≤ C · s −1 2r +4 l=0 F r (φ, ψ) l/2+1 + C(n − 2)ṡs 1−n r l=0 F r (φ, ψ) l/2+2 , (3.12) where C > 0 depends on the above bounds.
Proof of Theorem 1. 1 The existence of a local solution to the system (1.2), (1.3) can be obtained by standard methods, see for example [21,Proposition 9.12]. In order to prove long-time existence, it suffices to prove longtime existence of the solution (φ, ψ) of the equivalent system (2.4), (2.5) on the conformal manifold (M, h). Moreover, by the continuation criterion for hyperbolic partial differential equations [21,Lemma 9.14] if suffices to obtain a uniform bound on F r (φ, ψ) for all times t ∈ [0, ∞).
We set f (t) = (n − 2)s 1−nṡ . Note that f (t) is integrable with respect to t. For n = 2, this is trivial. For n > 2, we get due to the assumptions on s that As long as F r (φ, ψ) ≤ 1 we have the differential inequality which can easily be integrated as Now, we set := ∞ 0 (s −1 (t) + f (t))dt < ∞ and choose ε > 0 small enough such that F r (φ, ψ)| t=0 ≤ (2 ) −1 . Suppose that T 0 is the first time for which F r (φ, ψ)| T 0 = 1. However, the energy inequality from above gives which yields a contradiction. Therefore we can conclude that F r (φ, ψ) < 1 < ∞ for all times t ∈ [0, ∞) completing the proof.

Proof of Theorem 1.3
The proof is as above but we additionally assume that ψ ≡ 0. In this case, we just need the assumptions in Proposition 3.7 and not the slightly stronger ones in Proposition 3.8. As the spinor is not involved, we obviously can also remove the spin condition.