Global existence of wave maps and some generalizations on ...

GLOBAL EXISTENCE OF WAVE MAPS AND SOME GENERALIZATIONS ON EXPANDING SPACETIMES ¨ VOLKER BRANDING AND KLAUS KRONCKE

arXiv:1709.06520v1 [math.DG] 19 Sep 2017

Abstract. We prove the global existence of wave maps 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 Dirac-wave maps with curvature term under similar assumptions.

1. 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. 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. Most of the mathematical research on wave maps focuses on the case that the domain is Minkowski space, see for example [17] and [16] for an overview. There are a less articles that consider the case of a domain being a non-flat globally hyperbolic manifold, see for example [9, 10] and [11, 13] and references therein. Let us now introduce the wave map equation in more detail. Suppose that (M, h) is a globally hyperbolic manifold and (P, G) a Riemannian manifold. Consider a map φ : M → P , 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) := −hαβ ∇∂α dφ(∂β ). For the wave map equation, we will prove the following global existence result on an open class of globally hyperbolic manifolds that satisfies a suitable growth condition: Theorem 1.1. Let g˜t be a smooth family of complete Riemannian metrics on Σn−1 , N ∈ ˜ = (R × Σ, −N 2 dt2 + g˜t ) be a globally C ∞ (R × Σ) with 0 < A ≤ N ≤ B < ∞ and (M n , h) hyperbolic manifold Lorentzian manifold that satisfies the following condition: There exists a R∞ monotonically increasing smooth function s : R → R+ with 0 s−1 dt < ∞, such that for the conformal metric ˜ = −s−2 dt2 + gt , h = (N s)−2 h

the metrics gt admit a uniform Sobolev constant and kRgt kC k (gt ) + kN kC k (gt ) + k∇ν N kC k (gt ) + kIIkC k (gt ) ≤ C < ∞ for all k ∈ N. Here, ν is the future-directed unit normal of the hypersurfaces {t} × Σ 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 initial data (φ0 , φ1 ) for the wave map equation satisfies kφ0 kH r+1 (˜g0 ) + kφ1 kH r (˜g0 ) < ε, Date: September 20, 2017. 2010 Mathematics Subject Classification. 58J45; 53C27; 53C50; 35L71. Key words and phrases. wave maps; Dirac-wave maps with curvature term; global existence. 1

(1.1)

2

¨ VOLKER BRANDING AND KLAUS KRONCKE

the unique wave map with initial data φ|t=0 = φ0 , ∂t φ|t=0 = φ1 exists for all times t ∈ [0, ∞) and satisfies φ ∈ C 0 ([0, ∞), H r+1 (Σ, P )) ∩ C 1 ([0, ∞), H r (Σ, P )). Remark 1.2. Observe that we are able to control arbitrary high H r -regularity. To the best of our knowledge, this is the first result for wave maps where this has been achieved. In modern quantum field theory one considers extensions of the wave map system, one of them being the supersymmetric nonlinear σ-model [1]. Recently, there has been a lot of interested in this model from a mathematical perspective. In mathematical terms, the model leads to a geometric variational problem that couples a map between manifolds to spinor fields. Most of the mathematical research on this model so far is concerned with the case that both manifolds are Riemannian in which case one is lead to the notion of Dirac-harmonic maps [8] and Dirac-harmonic map with curvature term [4, 7], which represent semilinear elliptic problems. At present, many results regarding the geometric and analytic structure of Dirac-harmonic maps and Dirac-harmonic maps with curvature are known [5], but no existence result for these kind of equations could be achieved. In case that the domain manifold has a Lorentzian metric the critical points of the supersymmetric nonlinear σ-model lead to a system of the wave map equation coupled to spinor fields. For this system two existence results are available [12, 6] for the domain being two-dimensional Minkowski space. Extending our previous existence result for wave maps we also derive an existence result for Dirac-wave maps with curvature term from a large class of globally hyperbolic manifolds. In order to couple the wave-map equation to spinor fields we have to recall some concepts from spin geometry on globally hyperbolic manifolds. We have to make the additional assumption that the manifold M is spin, which guarantees the existence of the spinor bundle SM . Sections in the spinor bundle are called spinors. Moreover, we fix a spin structure. The spinor bundle is a vector bundle on which we choose a metric connection compatible with the hermitian scalar product denoted by h·, ·iΣM . On the spinor bundle we have the Clifford multiplication of spinors with tangent vectors, which satisfies hψ, X · ξiSM = hX · ψ, ξiSM for all X ∈ T M, ψ, ξ ∈ SM . In addition, the Clifford relations X · Y + Y · X = −2h(X, Y ) hold for all X, Y ∈ T M , where h represents the metric on M . The natural operator acting on spinors is the Dirac operator, which is defined as the composition of applying the covariant derivative first and Clifford multiplication in the second step. More precisely, the Dirac operator acting on spinors is given by ∂/ := hαβ ∂α · ∇∂β . The Dirac operator is a linear first order hyperbolic differential operator. For more background on spin geometry on globally hyperbolic manifolds we refer to [3, 2]. The Dirac operator itself is anti-self-adjoint with respect to the L2 -norm. However, the combination i∂/ yields a self-adjoint operator, that is Z Z hξ, i∂/ψi dVh . hi∂/ξ, ψi dVh = M

M

The square of the Dirac operator satisfies the Schroedinger-Lichnerowicz formula 2 ∂/ = +

scalM , 4 2

where scalM denotes the scalar curvature of the manifold M . Note that ∂/ 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

GLOBAL EXISTENCE OF WAVE MAPS AND SOME GENERALIZATIONS ON EXPANDING SPACETIMES 3

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 ∇SM ⊗φ

∗T P

= ∇SM ⊗ 1φ

∗T P

+ 1SM ⊗ ∇φ

∗T P

.

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 / := hαβ ∂α · ∇∂SM ⊗φ D β

∗T N

.

/ is also self-adjoint We assume that the connection on φ∗ T P is metric and thus the operator iD with respect to the L2 -norm. After these considerations we are ready to present the energy functional for Dirac-wave maps with curvature term Z 1 ˜/ ψi ˜ − 1 hψ, ˜ iD ˜ RP (ψ, ˜ ψ) ˜ ψi) ˜ dV˜ . ˜ ˜ (|dφ|2 + hψ, (1.2) S(φ, ψ, h) = h 2 M 6 In the last term the indices are contracted as follows ˜ RP (ψ, ˜ ψ) ˜ ψi ˜ SM ⊗φ∗ T P = RIJKL hψ˜I , ψ˜K iSM hψ˜J , ψ˜L iSM , hψ, 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 (1.2). However, we want to employ methods from the geometric calculus of variation and due to this reason we are employing standard spinors. The critical points of (1.2) can easily be calculated as 1 1 ˜ i∂α · ψ)dφ(∂ ˜ ˜ ψ) ˜ ψ, ˜ ψi, ˜ τ (φ) = RP (ψ, h(∇RP )♯ (ψ, (1.3) α) − 2 12 ˜/ ψ˜ = 1 RP (ψ, ˜ ψ) ˜ ψ. ˜ (1.4) iD 3 The critical points of (1.3), (1.4) are called Dirac-wave maps with curvature term. Our main theorem on Dirac-wave maps with curvature term is as follows: Theorem 1.3. Let g˜t be a smooth family of complete Riemannian metrics on Σn−1 , N ∈ ˜ = (R × Σ, −N 2 dt2 + g˜t ) be a globally C ∞ (R × Σ) with 0 < A ≤ N ≤ B < ∞ and (M n , h) hyperbolic manifold Lorentzian spin manifold that satisfies the following R ∞ −1 condition: There exists a monotonically increasing smooth function s : R → R+ with 0 s dt < ∞, such that the conformal metric ˜ = −s−2 dt2 + gt h = (N s)−2 h has bounded geometry, the metrics gt have a uniform Sobolev constant and kN kC k (gt ) + k∇ν N kC k (gt ) + kIIkC k (gt ) ≤ C < ∞ for all k ∈ N. Here, ν is the future-directed unit normal of the hypersurfaces {t} × Σ 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 initial data (φ0 , φ1 , ψ0 ) for the system (1.3), (1.4) satisfies kφ0 kH r+1 (˜g0 ) + kφ1 kH r (˜g0 ) + kψ0 kH r+1 (˜g0 ) < ε,

(1.5)

the unique solution of the system (1.3), (1.4) with initial data φ|t=0 = φ0 , ∂t φ|t=0 = φ1 , ψ|t=0 = ψ0 exists for all times t ∈ [0, ∞) and satisfies φ ∈ C 0 ([0, ∞), H r+1 (Σ, P )) ∩ C 1 ([0, ∞), H r (Σ, P )), ψ ∈ C 0 ([0, ∞), H r+1 (M, ΣM ⊗ φ∗ T P )) ∩ C 1 ([0, ∞), H r (M, ΣM ⊗ φ∗ T P )). Remark 1.4. In contrast to Theorem 1.1, one also has to control the R0ij0 -components of the curvature tensor in this case, which appear when computing the curvature of the spinor bundle. This explains the stronger curvature assumption above.


4

Remark 1.5. It seems that the class of Lorentzian manifold 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 Theorem 1.1 and Theorem 1.3. The simplest class is the class of Robertson-Walker spacetimes −dt2 + s2 (t)g with s−1 ∈ L1 ([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 Λ > 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. 2. Conformal Euler-Lagrange equations In the following we calculate how the energy functional (1.2) transforms under a certain conformal transformation of the metric on M . ˜ = (N s)2 h and the vector spinors via Lemma 2.1. If we transform the metric h 1−n ψ˜ := (N s) 2 ψ

the energy functional (1.2) acquires the form Z 1 1 / − (N s)2−n hψ, RP (ψ, ψ)ψi) dVh . ((N s)n−2 |dφ|2 + hψ, iDψi S(φ, ψ, h) = 2 M 6

(2.1)

¯ = (N s)2 h the volume elements transform as Proof. Under a conformal change of the metric h n dVh¯ = (N s) dVh . The Dirac operators transform as n−1 ¯/ ¯ = (N s)− n+1 2 Dψ. / D((N s)− 2 ψ)

/ transforms in the same way as the standard Dirac Note that the twisted Dirac operator D ∗ / operator ∂ since the twist bundle φ T P does not depend on the metric on M . Inserting our choice for ψ˜ we get ˜/ ψi ˜ iD ˜ = (N s)−n hψ, iDψi, / hψ, which completes the proof.

For the sake of completeness we calculate the critical points of (2.1). Lemma 2.2 (Critical points). The critical points of (2.1) are given by 1 h φ =(n − 2)(N S)−1 ∇∇(N s) φ − (N s)2−n hαβ RP (ψ, i∂α · ψ)dφ(∂β ) 2 1 4−2n P ♯ h(∇R ) (ψ, ψ)ψ, ψi, + (N s) 12 1 / =(N s)2−n RP (ψ, ψ)ψ. iDψ 3 Here, := −hαβ ∇∂α dφ(∂β ) denotes the wave map operator.

(2.2)


Proof. First, we vary the vector spinors ψ keeping the map φ fixed. More precisely, we consider λ a variation of ψ : (−ε, ε) × M → ΣM ⊗ φ∗ T P denoted by ψλ satisfying ∇ψ ∂λ λ=0 = ξ. We calculate Z d 1 1 / λ i − (N s)2−n hψλ , RP (ψλ , ψλ )ψλ i) dVh (hψλ , iDψ λ=0 dλ 2 M 6 Z 1 4 / + hψ, iDξi / − (N s)2−n hξ, RP (ψ, ψ)ψi) dVh = (hξ, iDψi 2 M 6 Z 1 / − (N s)2−n hξ, RP (ψ, ψ)ψi) dVh , (Rehξ, iDψi = 3 M yielding the equation for the vector spinor. Afterwards, we keep the vector spinors ψ fixed and consider a variation of φ : (−ε, ε) × M → P λ denoted by φλ satisfying ∂φ ∂λ t=0 = η. We calculate Z Z 1 ∂φλ d n−2 2 (N s)n−2 h∇ (N s) |dφλ | dVh = , dφλ i dVh λ=0 λ=0 dλ 2 M ∂λ ZM (N s)n−2 hη, φi − (n − 2)(N s)n−1 ∇(N s)hη, ∇φi dVh . = M

Moreover, we have Z Z d 1 1 2−n P / − (N s) hαβ hψ, iRP (η, dφ(∂α ))∂β · ψi dVh , hψ, iDψi hψ, R (ψ, ψ)ψi) dVh = dλ λ=0 2 M 6 M

where we used the equation for ψ. Moreover, we have

hαβ hψ, iRP (η, dφ(∂α ))∂β · ψi = hαβ hη, RP (ψ, i∂α · ψ)dφ(∂β )i. Finally, we calculate Z Z

d 4−2n P ♯ 4−2n P (N s) h(∇R ) (ψ, ψ)ψ, ψi, η dVh , (N s) hψ, R (ψ, ψ)ψi) dV = h dλ λ=0 M M

which completes the proof.

We want to turn the system (2.2) into a system of two wave-type equations. To this end we / recall the following Weitzenboeck formula for the twisted Dirac operator D. / satisfies Lemma 2.3. The square of the twisted Dirac operator D /2 = + D

scalM 1 + hαγ hβδ ∂α · ∂β · RP (dφ(∂γ ), dφ(∂δ ))ψ. 4 2

Proof. For a proof the we refer to [14, Theorem 8.17].

(2.3)

Making use of (2.3) we obtain the following rescaled Euler-Lagrange equations 1 h φ =(n − 2)(N s)−1 ∇∇(N s) φ − (N s)2−n hαβ RP (ψ, i∂α · ψ)dφ(∂β ) (2.4) 2 1 + (N s)4−2n h(∇RP )♯ (ψ, ψ)ψ, ψi, 12 scalM 1 1 h ψ = − ψ − hαγ hβδ ∂α · ∂β · RP (dφ(∂γ ), dφ(∂δ ))ψ + ∇(RP (ψ, ψ)(N s)2−n ) · ψ (2.5) 4 2 3 1 4−2n P P R (ψ, ψ)R (ψ, ψ)ψ, + (N s) 9 where h denotes the corresponding wave-operator.


6

In terms of local coordinates this system acquires the form (I = 1, . . . , dim P ) 1 ˙ t φI + D ∗ DφI s2 ∂t2 φI + ss∂ ˙ t φI + s2 tr g∂ 2 = − (n − 2)ss∂ ˙ t φI − (n − 2)s2 N −1 ∂t N ∂t φI + (n − 2)N −1 D∂i N D∂i φI 1 I − (N s)2−n RJKL hαβ hψ K , i∂α · ψ L i∂β φJ 2 1 + (N s)4−2n GIJ ∇J RKLM N hψ K , ψ M ihψ L , ψ N i 12 1 2 2 I s ∇t ψ + ss∇ ˙ t ψ I + s2 tr g∇ ˙ t ψ I + D ∗ Dψ I , 2 scalM I 1 αγ βδ I =− ψ − h h RJKL ∂γ φK ∂δ φL ∂α · ∂β · ψ J 4 2 1 I hψ J , ψ L i∂yI ∂α · ψ K + ∇∂α (N s)2−n RJKL 3 1 I K J L M S R + (N s)4−2n RJKL RM RS hψ , ψ ihψ , ψ iψ . 9 ˜ is equivalent to the Remark 2.4. Note that the system (1.3), (1.4) on the manifold (M, h) system (2.2) on the conformally transformed manifold (M, h). From now on we will use the system (2.4), (2.5) which follows from (2.2). 3. The energy estimate In this section we first develop several formulas that are useful in the study of energy estimates for sections in arbitrary vector bundles. Later on, we will apply these techniques to the cases of wave maps and Dirac-wave maps with curvature term. 3.1. Energies on general vector bundles. Let Σn be a manifold, s : R → R+ smooth and gt , t ∈ R be a smooth family of Riemannian metrics on Σ. We consider the (globally hyperbolic) Lorentzian manifold (M, h) = (R × Σ, −s(t)−2 dt2 + gt ). The non-vanishing Christoffel symbols of this metric are given by 1 1 s˙ Γ(h)0ij = s2 g˙ ij , Γ(h)ji0 = Γ(h)j0i = gjk g˙ ik , Γ(h)kij = Γ(g)kij . Γ(h)000 = − , s 2 2 Let V be a Riemannian vector bundle over M which is equipped with a metric connection ∇. As usual, iterating yields a map ∇k : Γ(V ) → Γ(T ∗ M ⊗k ⊗ V ). We define the associated wave operator : Γ(V ) → Γ(V ) by the sign convention such that ξ = −hαβ ∇2αβ ξ for ξ ∈ Γ(V ). The covariant derivative ∇ restricts for each t ∈ R to the spatial covariant derivative D which yields a map D : Γ(V ) → Γ(π ∗ (T ∗ Σ) ⊗ V ), where π : M → Σ is the canonical projection. In the following, we will write T ∗ Σ instead of π ∗ (T ∗ Σ) for notational convenience. The covariant derivative naturally extends as a map D : Γ(T ∗ Σ⊗k ⊗ V ) → Γ(T ∗ Σ⊗k+1 ⊗ V ) by defining (DX ξ)(X1 , . . . , Xk ) = V ∇X (ξ(X1 , . . . , Xk )) −

k X i=1

ξ(X1 , . . . , DX Xi , . . . , Xk ),


where for each t ∈ R, DX Xi denotes the covariant derivative of gt . Furthermore, we define its formal adjoint D ∗ : Γ(T ∗ Σ⊗k+1 ⊗ V ) → Γ(T ∗ Σ⊗k ⊗ V ) by D ∗ ξ(X1 , . . . , Xk ) = −g ij D∂i ξ(∂j , X1 , . . . , Xk ). Finally, we define a covariant time-derivative ∇t : Γ(T ∗ Σ⊗k ⊗ V ) → Γ(T ∗ Σ⊗k ⊗ V ) by (∇t ξ)(X1 , . . . , Xk ) = ∇V∂t (ξ(X1 , . . . , Xk )) −

k X

ξ(X1 , . . . , ∇∂t Xi , . . . , Xk ).

i=1

Observe that this definition makes sense as ∇∂t Xi ∈ Γ(T M ) is always tangential to Σ due to the structure of the metric h. A quick computation shows that the wave operator can be written as 1 ξ = s2 ∇t ∇t ξ + ss∇ ˙ t ξ + s2 trg g˙ · ∇t ξ + D ∗ Dξ. 2

(3.1)

Lemma 3.1. We use the notations from above. Let h., .i be the natural t-dependent scalar product induced on Γ(T ∗ Σ⊗k ⊗ V ). Let ξ, η ∈ Γ(V ). Then we have the product rule ∂t hD k φ, D k ηi = h∇t D k φ, D k ηi + hD k φ, ∇t D k ηi. Proof. Let t0 be a fixed time and let {e1 , . . . , en } be an orthonormal basis of (Σ, gt0 ) at x ∈ Σ. Then the scalar product can be also written as hD k φ, D k ηi(t0 , x) =

n X

hDeki

1

i1 ,...,ik =1

k ,...,eik ξ, Dei1 ,...,eik ηi(t0 , x),

where the scalar products on the right hand side are the ones on V . Now think of {e1 , . . . , en } as an orthonormal system in T(t0 ,x) M . Parallel transport along the curve t 7→ (t, x) yields orthonormal systems {e1 , . . . , en } for each T(t,x) M . Because ∇∂t ∂t = − ss˙ ∂t , h(∂t , ei ) ≡ 0 for each i ∈ {1, . . . , n} so that we in fact obtain orthonormal bases of Tx Σ with respect to the metric gt for each t ∈ R. Therefore we get k

k

∂t hD φ, D ηi =

n X k k hDeki ,...,ei ξ, ∇t (Deki ,...,ei η)i h∇t (Dei ,...,ei ξ), Dei ,...,ei ηi + 1 1 1 1 k k k k i1 ,...,ik =1 i1 ,...,ik =1 n n X X hDeki ,...,ei ξ, ∇t Deki ,...,ei ηi h∇t Deki ,...,ei ξ, Deki ,...,ei ηi + 1 1 1 1 k k k k i1 ,...,ik =1 i1 ,...,ik =1 n X

= h∇t D k φ, D k ηi + hD k φ, ∇t D k ηi.

Remark 3.2. In the above lemma, the special structure of the metric h is essential. For a section ξ ∈ Γ(V ), we now define the kth energy density ek (ξ) = s2 |D k ∇t ξ|2 + |D k+1 ξ|2 , which is a nonnegative function on M and the kth energy as Z ek (ξ) dVgt , Ek (ξ)(t) = Σ

where dVgt is the volume element of the metric gt and the integral has to be understood as an integral over {t} × Σ.


8

Proposition 3.3. Let ξ ∈ Γ(V ) be spacelike compactly supported. Then its kth energy satisfies the evolution equation Z Z 1 d k k Ek (ξ) = 2 hD ξ, D ∇t ξi dVgt + (|D k+1 ξ|2 − s2 |D k ∇t ξ|2 ) trg g˙ dVgt dt 2 Σ Σ Z Z 2 k k + 2s hD ∇t ξ, [∇t , D ]∇t ξi dVgt + 2 hD k ∇t ξ, [D ∗ D, D k ]ξi dVgt Σ Z Σ + 2 hD k+1 ξ, [∇t , D k+1 ]ξi dVgt . Σ

Proof. We compute Z Z d 1 k 2 |D ∇t ξ| dVgt + Ek (ξ) = 2ss ˙ ek (ξ) trg g˙ dVgt dt 2 Σ Σ Z Z 2 k k + 2s h∇t D ∇t ξ, D ∇t ξi dVgt + 2 h∇t D k+1 ξ, D k+1 ξi dVgt Σ Σ Z Z 1 k 2 ek (ξ) trg g˙ dVgt |D ∇t ξ| dVgt + = 2ss ˙ 2 Σ Σ Z Z 2 k k 2 + 2s hD ∇t ∇t ξ, D ∇t ξi dVgt + 2s h[∇t , D k ]∇t ξ, D k ∇t ξi dVgt Σ Σ Z Z k+1 k+1 + 2 hD ∇t ξ, D ξi dVgt + 2 h[∇t , D k+1 ]ξ, D k+1 ξi dVgt . Σ

Σ

Moreover, we have Z Z k+1 k+1 2 hD ∇t ξ, D ξi dVgt = 2 hD k ∇t ξ, D ∗ D k+1 ξi dVgt Σ Z ZΣ k ∗ k = 2 hD ∇t ξ, D D Dξi dVgt + 2 hD k ∇t ξ, [D ∗ D, D k ]ξi dVgt . Σ

Σ

The statement follows by combining both equalities and using (3.1).

Lemma 3.4. Let ξ ∈ Γ(T ∗ Σ⊗k ⊗ V ). Then we have the identity 1 ([∇t , D]ξ)(X, X1 , . . . , Xk ) = R∂Vt ,X (ξ(X1 , . . . , Xk )) − Dg(X) ξ(X1 , . . . , Xk ) 2 ˙ k 1X ξ(X1 , . . . , DX g(X ˙ i ), . . . , Xk ). − 2 i=1

Proof. At first, we compute

(∇t Dξ)(X, X1 , . . . , Xk ) = ∇V∂t (Dξ(X, X1 , . . . , Xk )) − (Dξ)(∇∂t X, X1 , . . . , Xk ) −

k X (Dξ)(X, X1 , . . . , ∇∂t Xi , . . . , Xk ) i=1

= ∇V∂t (∇VX (ξ(X1 , . . . , Xk ) −

k X

ξ(X1 , . . . , DX Xi , . . . Xk ))

i=1

− (D∇∂t X ξ)(X1 , . . . , Xk ) −

k X

(DX ξ)(X1 , . . . , ∇∂t Xi , . . . , Xk )

i=1

= ∇V∂t (∇VX (ξ(X1 , . . . , Xk ))) −

k X

(∇t ξ)(X1 , . . . , DX Xi , . . . Xk )

i=1

−

k X i=1

ξ(X1 , . . . , ∇∂t DX Xi , . . . Xk )) −

n X

i,j=1 i6=j

ξ(X1 , . . . , DX Xi , . . . , ∇∂t Xj , . . . Xk ))


− (D∇∂t X ξ)(X1 , . . . , Xk ) −

k X

(DX ξ)(X1 , . . . , ∇∂t Xi , . . . , Xk ).

i=1

Similarly we find,

(D(∇t ξ))(X, X1 , . . . , Xk ) =

∇VX ((∇t ξ)(X1 , . . . , Xk ))

k X (∇t ξ)(X1 , . . . , DX Xi , . . . , Xk ) − i=1

= ∇VX (∇V∂t (ξ(X1 , . . . , Xk )) −

k X

ξ(X1 , . . . , ∇∂t Xi , . . . , Xk ))

i=1

k X (∇t ξ)(X1 , . . . , DX Xi , . . . , Xk ) − i=1

=

∇VX (∇V∂t (ξ(X1 , . . . , Xk )))

−

k X

(DX ξ)(X1 , . . . , ∇∂t Xi , . . . Xk )

i=1

−

k X

ξ(X1 , . . . , DX ∇∂t Xi , . . . Xk )) −

n X

ξ(X1 , . . . , DX Xi , . . . , G ∇∂t Xj , . . . Xk ))

i,j=1 i6=j

i=1

k X (∇t ξ)(X1 , . . . , DX Xi , . . . , Xk ). − i=1

Summing up, we obtain ([∇t , D]ξ)(X, X1 , . . . , Xk ) = ∇V∂t (∇VX (ξ(X1 , . . . , Xk ))) − ∇VX (∇V∂t (ξ(X1 , . . . , Xk ))) −

k X

ξ(X1 , . . . , ∇∂t DX Xi − DX ∇∂t Xi , . . . Xk ))

i=1

− (D∇∂t X ξ)(X1 , . . . , Xk ). As this expression is tensorial in X and the Xi , we may assume that the components of these vector fields with respect to a chart of Σ are independent of time. By raising an index with respect to gt we can think of g˙ t as an endomorphism on T ∗ Σ. Then the formulas for the Christoffel symbols imply ∇∂t X = ∇X ∂t = 12 g(X) ˙ and the same holds for Xi . Moreover, we have [∂t , X] = [∂t , Xi ] = 0 and ∇∂t DX Xi − DX ∇∂t Xi =

1 1 (g(D ˙ X Xi ) − DX (g(X ˙ i )) = DX g(X ˙ i ). 2 2

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 ), [∇t , D]φ ∈ Γ(T ∗ Σ ⊗ φ∗ T P ) and we get 1 ˙ ([∇t , D]φ)(X) = − dφ(g(X)). 2


10

More generally, if E = φ∗ T P in the Lemma above, we get 1 ([∇t , D]ξ)(X, X1 , . . . , Xk ) = RP (dφ(∂t ), dφ(X))(ξ(X1 , . . . , Xk )) − Dg(X) ξ(X1 , . . . , Xk ) 2 ˙ k 1X ξ(X1 , . . . , DX g(X ˙ i ), . . . , Xk ) − 2 i=1

1 = RP (∇t φ, DX φ)(ξ(X1 , . . . , Xk )) − Dg(X) ξ(X1 , . . . , Xk ) 2 ˙ k 1X − ξ(X1 , . . . , DX g(X ˙ i ), . . . , Xk ). 2 i=1

By an iteration argument, we find [∇t , D k ]∇t φ =

P

li +

+

X

P

G

mj =k−1

k−1 X

∇l1 RP ⋆ D m1 +1 φ ⋆ . . . ⋆ D ml1 +1 φ ⋆D l2 ∇t φ ⋆ D l3 +1 φ ⋆ D l4 ∇t φ {z } | l1 −times

D k−l g˙ ⋆ D l ∇t φ,

l=0

[∇t , D

k+1

]φ =

P

li +

+

X

P

G

mj =k−1

k X

∇l1 RN ⋆ D m1 +1 φ ⋆ . . . ⋆ D ml1 +1 φ ⋆D l2 ∇t φ ⋆ D l3 +1 φ ⋆ D l4 +1 φ {z } | l1 −times

D k−l g˙ ⋆ D l+1 φ.

l=0

In the case of a vector spinor ψ ∈ Γ(SM ⊗ φ∗ T P ), the formulas are 1 ˙ ([∇t , D]ψ)(X) = RSM (∂t , X) · ψ + RP (∇t φ, DX φ)ψ − Dφ(g(X)) 2 and ([∇t , D]ξ)(X, X1 , . . . , Xk ) = RSM (∂t , X) · ξ + RP (∇t φ, DX φ)(ξ(X1 , . . . , Xk )) k

1 1X − Dg(X) ξ(X1 , . . . , DX g(X ˙ i ), . . . , Xk ). ξ(X1 , . . . , Xk ) − ˙ 2 2 i=1

By iterating the formula from above, we get [∇t , D k ]∇t ψ =

k−1 X

D l (RSM (∂t , .)) ⋆ D k−1−l ∇t ψ +

l=0

+

P

[∇t , D

]ψ =

li +

k X

G

P

mj =k−1

l

SM

D (R

l1

P

∇ R ⋆D |

(∂t , .)) ⋆ D

k−l

P

m1 +1

ψ+

l=0

+

D k−l g˙ ⋆ D l+1 ∇t ψ

l=0

X

and k+1

k−1 X

φ ⋆ . . . ⋆ D ml1 +1 φ ⋆D l2 ∇t φ ⋆ D l3 +1 φ ⋆ D l4 ∇t ψ {z } l1 −times

k X

D k−l+1 g˙ ⋆ D l+1 ψ

l=0

X

li +

P

mj =k

G

l1

P

∇ R ⋆D |

m1 +1

φ ⋆ . . . ⋆ D ml1 +1 φ ⋆D l2 ∇t φ ⋆ D l3 +1 φ ⋆ D l4 ψ. {z } l1 −times


Lemma 3.5. Let ξ ∈ Γ(T ∗ Σ⊗k ⊗ V ). Then we have the identity ([D ∗ D, D]ξ)(X, X1 , . . . , Xk ) = −DRic(X) ξ(X1 , . . . , Xk ) − 2

k X

Dei ξ(X1 , . . . RX,ei Xj , . . . , Xk )

j=1

+

V Dei RX,e (ξ(X1 , . . . , Xk )) i

−

k X

V + 2RX,e (Dei ξ(X1 , . . . , Xk )) i

ξ(X1 , . . . Dei RX,ei Xj , . . . , Xk ),

j=1

where {ei }1≤i≤n−1 is a local orthonormal frame. Here and throughout the proof, we sum over i. Proof. A direct computation yields 3 ([D ∗ D, D]ξ)(X, X1 , . . . , Xk ) = DX,e ξ(X1 , . . . , Xk ) − De3i ,ei ,X ξ(X1 , . . . , Xk ) i ,ei 3 = DX,e ξ(X1 , . . . , Xk ) − De3i ,X,ei ξ(X1 , . . . , Xk ) i ,ei

+ De3i ,X,ei ξ(X1 , . . . , Xk ) − De3i ,ei ,X ξ(X1 , . . . , Xk ) = RX,ei Dξ(ei , X1 , . . . , Xk ) + Dei RX,ei ξ(X1 , . . . , Xk ) = −DRic(X) ξ(X1 , . . . , Xk ) −

k X

Dei ξ(X1 , . . . RX,ei Xj , . . . , Xk )

j=1

+ RX,ei (Dei ξ(X1 , . . . , Xk )) V

+ Dei (R ◦ ξ)(X, ei , X1 , . . . , Xk ) − Dei

k X

ξ(X1 , . . . RX,ei Xj , . . . , Xk )

j=1

= −DRic(X) ξ(X1 , . . . , Xk ) − 2

k X

Dei ξ(X1 , . . . RX,ei Xj , . . . , Xk )

j=1

V (ξ(X1 , . . . , Xk )) + 2RX,ei (Dei ξ(X1 , . . . , Xk )) + Dei RX,e i

−

k X

ξ(X1 , . . . Dei RX,ei Xj , . . . , Xk ).

j=1

Here,

RV

◦ξ ∈

Γ(T ∗ Σ⊗k+2

⊗ V ) is defined as

V (RV ◦ ξ)(X, Y, X1 , . . . , Xk ) = RX,Y (ξ(X1 , . . . , Xk ))

and DRV is the covariant derivative of the curvature endomorphism on V restricted to vectors tangential to Σ. In other words, V V V V V DX RY,Z ξ := DX (RY,Z ξ) − RD ξ − RY,D ξ − RY,Z (∇X ξ). X Y,Z X ,Z

In the case φ ∈ C ∞ (M, P ), we obtain ([D ∗ D, D]φ)(X) = −Dφ(RicM (X)) + RP (Dφ(X), Dφ(ei ))(Dφ(ei )) and for ξ ∈ Γ(T ∗ Σ⊗k ⊗ φ∗ T P ), we have ([D ∗ D, D]ξ)(X, X1 , . . . Xk ) = 2

k X

Dei ξ(X1 , . . . , RM (ei , X)Xl , . . . Xk )

l=1

+

k X

ξ(X1 , . . . , Dei RM (ei , X)Xl , . . . Xk ) − DRic(X) ξ(X1 , . . . , Xk )

l=1

+ DDφ(ei ) RP (Dφ(X), Dφ(ei ))(ξ(X1 , . . . Xk )) + RP (D 2 φ(ei , X), Dφ(ei ))(ξ(X1 , . . . Xk )) + RP (Dφ(X), D 2 φ(ei , ei ))(ξ(X1 , . . . Xk )) + 2RP (Dφ(X), Dφ(ei ))(Dei ξ(X1 , . . . , Xk )).


12

Thus by iteration we get [D ∗ D, D k ]φ =

k−1 X

D l RM ⋆ D k−l φ

l=0

+

P

li +

X

P

G

mj =k−1

∇l1 RP ⋆ D m1 +1 φ ⋆ . . . ⋆ D ml1 +1 φ ⋆D l2 +1 φ ⋆ D l3 +1 φ ⋆ D l4 +1 φ. | {z } l1 −times

In the case of vector spinors, that is ψ ∈ Γ(SM ⊗ φ∗ T P ), we obtain

SM SM ([D ∗ D, D]ψ)(X) = −DRic(X) ψ + 2RP (Dφ(X), Dφ(ei ))(Dψ(ei )) + 2RX,e · Dei ψ + Dei RX,e ·ψ i i

+ ∇Dφ(ei ) RP (Dφ(X), Dφ(ei ))ψ + RP (D 2 φ(ei , X), Dφ(ei ))ψ + RP (Dφ(X), D 2 φ(ei , ei ))ψ and for ξ ∈ Γ(T ∗ Σ⊗k ⊗ SM ⊗ φ∗ T P ), we have ([D ∗ D, D]ξ)(X, X1 , . . . , Xk ) = −DRic(X) ξ(X1 , . . . , Xk ) + 2RP (Dφ(X), Dφ(ei ))(Dei ψ(X1 , . . . , Xk )) SM SM · Dei ψ(X1 , . . . , Xk ) + Dei RX,e · ψ(X1 , . . . , Xk ) + 2RX,e i i

+ ∇Dφ(ei ) RP (Dφ(X), Dφ(ei ))(ψ(X1 , . . . , Xk )) + RP (D 2 φ(ei , X), Dφ(ei ))(ψ(X1 , . . . , Xk )) + RP (Dφ(X), D 2 φ(ei , ei ))(ψ(X1 , . . . , Xk )) +2

k X

Dei ξ(X1 , . . . , RM (ei , X)Xl , . . . Xk )

l=1

+

k X

ξ(X1 , . . . , Dei RM (ei , X)Xl , . . . Xk ).

l=1

By iteration, we then obtain ∗

k

[D D, D ]ψ =

k−1 X

l

DR

M

⋆D

k−l

ψ+

l=0

+

k X

D l RSM ⋆ D k−l ψ

l=0

P

X

li +

P

G

mj =k

l1

∇ RN ⋆ D m1 +1 φ ⋆ . . . ⋆ D ml1 +1 φ ⋆D l2 +1 φ ⋆ D l3 +1 φ ⋆ D l4 ψ. | {z } l1 −times

We conclude this section with a very important lemma, which we will frequently make use of when deriving energy estimates. Lemma 3.6. Let M m be an m-dimensional Riemannian manifold, r a natural number satisfying r> m connection, k ≥ 2, ξ1 , . . . ξk ∈ H r (E) and 2 , E → M a Riemannian vector bundle with a P li i = 1, . . . k natural numbers satisfying li ≤ r and i li ≤ 2r. Then the following inequality holds Z IIki=1 |∇li ξi | dV ≤ CSob · IIki=1 kξi kH r . M

l i

∇ i ξi p ≤ C kξi k r so that Proof. Choose pi ∈ [1, ∞], i = 1, . . . , lk such that p1i > 21 − r−l H n L i by Sobolev embedding. Furthermore, ! k k k X 1 1 X r k kr 1 X 1 r − li li ) = 1 + (k − 2) − + ( − li − 2r < 1 = − + 2 n 2 n n 2 n n i=1

i=1

due to our assumptions. Therefore, we can choose the pi such that of the H¨older inequality finishes the proof of the lemma.

i=1

Pk

1 i=1 pi

= 1. An application


3.2. Energy of the map. We define the kth energy density of the map part as ek (φ) = s2 |D k ∂t φ|2 + |D k Dφ|2

(3.2)

and the kth energy as Ek (φ) =

Z

ek (φ) dVgt .

(3.3)

Σ

The kth total energy is given by Fk (φ) =

k X

El (φ) =

Z X k Σ l=0

l=0

el (φ) dVgt = s2 k∇t φk2H k + kDφk2H k ,

(3.4)

where the Sobolev norms are taken with respect to the metric gt . In (3.8) below, we also define the total energy Fk (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 φ ∈ C 0 ([0, T ), H r+1 (Σ, P )) ∩ C 1 ([0, T ), H r (Σ, P )), ψ ∈ C 0 ([0, T ), H r+1 (M, SM ⊗ φ∗ T P )) ∩ C 1 ([0, T ), H r (M, SM ⊗ φ∗ T P )). Then the kth energy of the map satisfies the following inequality d Ek (φ) ≤ C1 (k) kgk ˙ C k−1 + k∂t log N kC k + s−1 kD log N kC k + s−1 kRΣ kC k−1 Fk (φ) dt Z |D k ∇t φ|2 dVgt

− (n − 2)ss˙

Σ

+ C2 (k, n, g)s

−1

P

kR kC k−1

k−1 X

Fr (φ)2+l/2

l=1

+ C3 (k, n, g)s2−n kN 2−n kC k kRP kC k k∂t kC k

k X

Fr (φ)1+l/2 Fr (ψ)

(3.5)

l=0

+ C4 (k, n, g)s1−n kN 2−n kC k kRP kC k

k X

Fr (φ)1+l/2 Fr (ψ)

l=0

+ C5 (k, n, g)s

3−2n

kN

4−2n

P

kC k kR kC k+1

k X

Fr (φ)1/2+l/2 Fr (ψ)2 ,

l=0

where the positive constants Ci , i ∈ {1, . . . , 5} depend on n, k and the Sobolev constant of the metric gt . 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 Z Z d 1 Ek (φ) =2 hD k h φ, D k ∇t φi dVgt + (|D k Dφ| − s2 |D k ∇t φ|2 ) trg g˙ dVgt dt 2 Σ Σ Z Z 2 k k + 2s h[∇t , D ]∇t φ, D ∇t φi dVgt + 2 h[D ∗ D, D k ]φ, D k ∇t φi dVgt Σ Z Σ + 2 h[∇t , D k+1 ]φ, D k+1 φi dVgt . Σ

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 explicitly. We have to estimate all terms on the right hand side and


14

start by estimating the commutator terms Z 2 s hD k ∇t φ, [∇t , D k ]∇t φi dVgt Σ Z X 2 ∇l1 RP ⋆ D m1 +1 φ ⋆ . . . ⋆ D ml1 +1 φ ⋆D l2 ∂t φ ⋆ D l3 +1 φ ⋆ D l4 ∂t φ ⋆ D k ∂t φ dVgt =s | {z } P P Σ li +

+ s2

mj =k−1

k−1 Z X

l1 −times

D k−l g˙ ⋆ D l+1 ∂t φ ⋆ D k ∂t φ dVgt

Σ

l=0

≤ C(k)s2 kRP kC k−1

k−1 X

3 2 ˙ C k−1 k∂t φk2H k kDφkl+1 H r k∂t φkH r + C(k)s kgk

l=0 k−1 X

≤ C(k)s−1 kRP kC k−1

(Fr (φ))2+l/2 + C(k)kgk ˙ C k−1 Fk (φ).

l=0

The second commutator can be controlled as follows Z hD k+1 φ, [∇t , D k+1 φ]i dVgt Σ Z X ∇l1 RP ⋆ D m1 +1 φ ⋆ . . . ⋆ D ml1 +1 φ ⋆D l2 ∂t φ ⋆ D l3 Dφ ⋆ D l4 Dφ ⋆ D k Dφ dVgt = {z } | P P Σ li +

+

mj =k−1

k Z X

l1 −times

D k−l g˙ ⋆ D l Dφ ⋆ D k Dφ dVgt

Σ

l=0

≤ C(k)kRP kC k−1

k−1 X

r ˙ C k−1 kDφk2H k kDφkl+3 H r k∂t φkH + C(k)kgk

l=0

≤ C(k)s

−1

k−1 X kR kC k−1 (Fr (φ))2+l/2 + C(k)kgk ˙ C k−1 Fk (φ). P

l=0

The third commutator can be estimated as follows Z hD k ∂t φ, [D ∗ D, D k ]φi dVgt Σ

=

k−1 Z X l=0

+

P

D l RΣ ⋆ D k−l φ ⋆ D k ∂t φ dVgt

Σ

li +

X

P

mj =k−1

Z

∇l1 RP ⋆ D m1 +1 φ ⋆ . . . ⋆ D ml1 +1 φ ⋆D l2 Dφ ⋆ D l3 Dφ ⋆ D l4 Dφ ⋆ D k ∇t φ dVgt | {z } Σ l1 −times

≤ C(k)kRΣ kC k−1 kDφkH k−1 k∂t φkH k + C(k)kRP kC k−1

k−1 X

kDφkl+3 H r k∂t φkH r

l=0

≤ C(k)s−1 kRΣ kC k−1 Fk (φ) + C(k)s−1 kRP kC k−1

k−1 X

(Fr (φ))2+l/2 .

l=0

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 (N s)−1 ∇∇(N s) φ = −ss∇ ˙ t φ − s2 N −1 ∂t N ∇t φ + N −1 hDN, Dφig .


This allows us to derive the following estimate Z D k h (N s)−1 ∇∇(N s) φ , D k ∇t φi dVgt Σ

= − ss˙

Z

|D k ∇t φ|2 dVgt +

Σ

+ s2

l=0

k Z X l

≤ − ss˙

Zi

Σ

k Z X

Dl Σ

DN ⋆ D k−l Dφ ⋆ D k ∇t φ dVgt N

∂t N ⋆ D l2 ∇t φ ⋆ D k ∇t φ dVgt N

D l1 Σ

|D k ∇t φ|2 dVgt + C(k)s−1 kD log N kC k Fk (φ) + C(k)k∂t log N kC k Fk (φ).

In order to treat the second term on the right hand side of (2.4) we have to consider spatial and time derivatives separately D k RP (ψ, i∂t · ψ)dφ(∂t ) X ∇l1 RP ⋆ D m1 +1 φ ⋆ . . . ⋆ D ml1 +1 φ ⋆D l2 ∂t ⋆ D l3 ψ ⋆ D l4 ψ ⋆ D l5 ∇t φ, = {z } | P P li +

mj =k

l1 −times

D R (ψ, i∂i · ψ)dφ(∂i ) X = ∇l1 RP ⋆ D m1 +1 φ ⋆ . . . ⋆ D ml1 +1 φ ⋆D l2 ψ ⋆ D l3 ψ ⋆ D l4 Dφ {z } | P P k

P

li +

mj =k

l1 −times

such that we find

D k N 2−n RP (ψ, i∂α · ψ)dφ(∂α ) X = D l1 N 2−n ⋆ s2 ∇l2 RP ⋆ D m1 +1 φ ⋆ . . . ⋆ D ml2 +1 φ ⋆D l3 ∂t ⋆ D l4 ψ ⋆ D l5 ψ ⋆ D l6 ∇t φ | {z } P P li +

mj =k

l2

P

+∇ R ⋆D |

m1 +1

φ ⋆ ... ⋆ D {z l2 −times

ml2 +1

l2 −times l4

φ ⋆D ψ ⋆ D ψ ⋆ D l5 Dφ . } l3

This allows us to derive the following estimate: Z 1 hD k − hαβ (N s)2−n RP (ψ, i∂α · ψ)dφ(∂β ) , D k ∇t φi dVgt 2 Σ Z 2−n X s D l1 N 2−n =− 2 P P Σ li +

2

l2

mj =k

P

⋆ s ∇ R ⋆ D m1 +1 φ ⋆ . . . ⋆ D ml2 +1 φ ⋆D l3 ∂t ⋆ D l4 ψ ⋆ D l5 ψ ⋆ D l6 ∇t φ ⋆ D k ∇t φ | {z } l2 −times

l2

P

+D R ⋆D |

m1 +1

φ ⋆ . . . ⋆ D ml2 +1 φ ⋆D l3 ψ ⋆ D l4 ψ ⋆ D l5 Dφ ⋆ D k ∇t φ dVgt {z } l2 −times

≤ C(k, n)s4−n kN 2−n kC k kRP kC k k∂t kC k

k X

kDφklH r kψk2H r k∇t φk2H r

l=0

+ C(k, n)s2−n kN 2−n kC k kRP kC k

k X

2 kDφkl+1 H r kψkH r k∇t φkH r

l=0

≤ C(k, n)s

2−n

kN

2−n

P

kC k kR kC k k∂t kC k

k X

Fr (φ)1+l/2 Fr (ψ)

l=0

+ C(k, n)s1−n kN 2−n kC k kRP kC k

k X l=0

Fr (φ)1+l/2 Fr (ψ).


16

The third term on the right hand side of (2.4) can be computed as D k N 4−2n h(∇RP )♯ (ψ, ψ)ψ, ψi X = D l1 N 4−2n ⋆ ∇l2 +1 RP ⋆ D m1 +1 φ ⋆ . . . ⋆ D ml2 +1 φ ⋆D l3 ψ ⋆ D l4 ψ ⋆ D l5 ψ ⋆ D l6 ψ. | {z } P P li +

mj =k

l2 −times

This allows us to derive the following estimate Z

hD k ((N s)4−2n h(∇RP )♯ (ψ, ψ)ψ, ψi), D k ∂t φi dVgt Σ Z X D l1 N 4−2n ⋆ ∇l2 +1 RP =s4−2n P

li +

P

mj =k

Σ

⋆ D m1 +1 φ ⋆ . . . ⋆ D ml2 +1 φ ⋆D l3 ψ ⋆ D l4 ψ ⋆ D l5 ψ ⋆ D l6 ψ ⋆ D k ∂t φ dVgt | {z } l2 −times

≤s4−2n C(k)kN 4−2n kC k kRP kC k+1

k X

kDφklH r kψk4H r k∂t φkH r

l=0

≤s

3−2n

C(k)kN

4−2n

P

kC k kR kC k+1

k X

Fr (φ)1/2+l/2 Fr (ψ)2 .

l=0

Adding up the different contributions concludes the proof.

3.3. Energy of the spinor. We define the kth energy density of the spinor part as ek (ψ) = s2 |D k ∇t ψ|2 + |D k Dψ|2

(3.6)

and the kth energy as Ek (ψ) =

Z

ek (ψ) dVgt .

(3.7)

Σ

The kth total energy is given by

Fk (ψ) =

k X l=0

El (ψ) + kψk2L2 =

Z X k Σ l=0

el (ψ) dVgt + kψk2L2 = s2 k∇t ψk2H k + kψk2H k+1 .

(3.8)

Note that d 1 1 kψk2L2 ≤ 2k∇t ψkL2 kψkL2 + k tr gk ˙ L∞ kψk2L2 ≤ (s−1 + k tr gk ˙ L∞ )F1 (ψ). dt 2 2

(3.9)

Proposition 3.8. 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 φ ∈ C 0 ([0, T ), H r+1 (Σ, P )) ∩ C 1 ([0, T ), H r (Σ, P )), ψ ∈ C 0 ([0, T ), H r+1 (M, SM ⊗ φ∗ T P )) ∩ C 1 ([0, T ), H r (M, SM ⊗ φ∗ T P )).


Then the energy of the spinor satisfies the following inequality d ˙ C k )Fk (ψ) Ek (ψ) ≤ C1 (k, n)(kRSM (∂t , ·)kC k−1 + kgk dt + C2 (n, k) · s−1 (kRΣ kC k−1 + kRSM kC k + kscalM kC k )Fk (ψ) + C3 (n, k, g)(s−1 + k∂t kC k )kRP kC k+1

k X

Fr (φ)l/2+1 Fr (ψ)

l=0

+ C4 (n, k, g)s2−n kN 2−n kC k kRP kC k+1 k∂t kC k

k+1 X

Fr (φ)l/2 Fr (ψ)2

l=0

+ (n − 2) · C5 (n, k, g)s

3−n

kN

2−n

P

∂t log N kC k kR kC k k∂t kC k

k X

Fr (φ)l/2 Fr (ψ)2

l=0

+ (n − 2) · C6 (n, k, g)ss ˙ 2−n kN 2−n kC k kRP kC k k∂t kC k

k X

Fr (φ)l/2 Fr (ψ)2

l=0

+ C7 (n, k, g)s1−n kN 2−n kC k+1 kRP kC k+1

k+1 X

Fr (φ)l/2 Fr (ψ)2

l=0

+ C8 (n, k, g)s1−n kN 2−n kC k kRP k2C k

2k X

Fr (φ)l/2 Fr (ψ)3 ,

l=0

(3.10)

where the positive constants Ci , i ∈ {3, . . . , 8} depend on n, k and the Sobolev constant of the metric gt . Remark 3.9. The curvature quantities coming from the spinor bundle above have to be understood as the following sections: RSM (∂t , ·) ∈ Γ(T ∗ Σ ⊗ End(S)), RSM = RSM (., .) ∈ Γ(Λ2 T Σ ⊗ End(S)), both by restricting to vectors tangent to Σ. Moreover ∂t ∈ Γ(End(S)) acts by Clifford multiplication. The C k norms with respect to gt are then defined in a canonical way. Proof. Using (3.3) the energy of the spinor ψ satisfies Z Z 1 d Ek (ψ) =2 hD k h ψ, D k ∇t ψi dVgt + (|D k Dψ| − s2 |D k ∇t ψ|2 ) trg g˙ dVgt dt 2 Σ Σ Z Z 2 k k + 2s h[∇t , D ]∇t ψ, D ∇t ψi dVgt + 2 h[D ∗ D, D k ]ψ, D k ∇t ψi dVgt Σ Z Σ + 2 h[∇t , D k+1 ]ψ, D k+1 ψi dVgt . Σ

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: Z 2 s hD k ∇t ψ, [∇t , D k ]ψi dVgt Σ

=s

2

k−1 Z X l=0

+ s2

D l (RSM (∂t , ·)) ⋆ D k−l−1 ∇t ψ ⋆ D k ∇t ψ dVgt Σ

k−1 Z X l=0

+s

2 P

li +

D k−l g˙ ⋆ D l+1 ∇t ψ ⋆ D k ∇t ψ dVgt

Σ

X

P

Z

mj =k−1 Σ

∇l1 RP


18

≤s

2

⋆ D m1 +1 φ ⋆ . . . ⋆ D ml1 +1 φ ⋆D l2 ∇t φ ⋆ D l3 Dφ ⋆ D l4 ∇t ψ ⋆ D k ∇t ψ dVgt | {z } l1 −times SM C(k)kR (∂t , ·)kC k−1 k∇t ψk2H k 2

P

+ s C(k)kR kC k−1

k−1 X

+ s2 C(k)kgk ˙ C k k∇t ψk2H k

2 kDφkl+1 H r k∂t φkH r k∇t ψkH r

l=0

≤ C(k)kR

SM

˙ C k Fk (ψ) (∂t , ·)kC k−1 Fk (ψ) + C(k)kgk

+ s−1 C(k)kRP kC k−1

k−1 X

Fr (φ)l/2+1 Fr (ψ).

l=0

The second commutator term can be estimated as follows: Z h[D ∗ D, D k ]ψ, D k ∇t ψi dVgt Σ

=

k−1 Z X

Σ

l=0

+

D l RM ⋆ D k−l ψ ⋆ D k ∇t ψ dVgt +

P

X

li +

P

mj =k

Z

k Z X

D l RSM ⋆ D k−l ψ ⋆ D k ∇t ψ dVgt

Σ

l=0

∇l1 RP ⋆ D m1 +1 φ ⋆ . . . ⋆ D ml1 +1 φ ⋆D l2 Dφ + D l3 Dφ ⋆ D l4 ψ ⋆ D k ∇t ψ dVgt {z } | Σ l1 −times

≤ C(k)kR kC k−1 kψkH k k∇t ψkH k + C(k)kRSM kC k kψkH k k∇t ψkH k M

+ C(k)kRP kC k

k X

kDφkl+2 H r kψkH r k∇t ψkH r

l=0

≤s

−1

M

C(k)(kR kC k−1 + kR

SM

kC k )Fk (ψ) + s

−1

P

C(k)kR kC k

k X

Fr (φ)l/2+1 Fr (ψ).

l=0

The third commutator can be controlled as follows: Z h[∇t , D k+1 ]ψ, D k+1 ψi dVgt Σ

=

k Z X l=0

+

P

l

D (R

SM

(∂t , ·)) ⋆ D

k−l

ψ⋆D

k+1

ψ dVgt +

Σ

li +

X

P

mj =k−1

k Z X l=0

Z

D k−l−1 g˙ ⋆ D l+1 ψ ⋆ D k+1 ψ dVgt Σ

∇l1 RP ⋆ D m1 +1 φ ⋆ . . . ⋆ D ml1 +1 φ ⋆D l2 ∂t φ ⋆ D l3 Dφ ⋆ D l4 ψ ⋆ D k+1 ψ dVgt | {z } Σ

l1 −times 2 ˙ C k kψk2H k+1 ≤ C(k)kR (∂t , ·)kC k kψkH k+1 + C(k)kgk k X 2 r kDφkl+1 + C(k)kRP kC k H r k∇t φkH kψkH r+1 l=0 SM

≤ C(k)kR

SM

˙ C k Fk (ψ) + s (∂t , ·)kC k Fk (ψ) + C(k)kgk

−1

P

C(k)kR kC k

k X

Fr (φ)l/2+1 Fr (ψ).

l=0

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 Z

k

1X scalM ψ), D k ∇t ψi dVgt = hD ( 4 4 Σ k

l=0

Z

D l scalM ⋆ D k−l ψ ⋆ D k ∇t ψ dVgt

Σ

≤C(k)kscalM kC k s−1 Fk (ψ).


Regarding the second term on the right hand side of (2.5) we again expand space and time contributions hαγ hβδ ∂α · ∂β · RP (dφ(∂γ ), dφ(∂δ ))ψ = − 2s2 gij ∂t · ∂i · RP (dφ(∂t ), dφ(∂j ))ψ + g ij gkl ∂i · ∂k · RP (dφ(∂j ), dφ(∂l ))ψ. Note that we do not get a term proportional to |dφ(∂t )|2 due to symmetry reasons. We then find D k hαγ hβδ ∂α · ∂β · RP (dφ(∂γ ), dφ(∂δ ))ψ) X D l1 RP ⋆ D m1 +1 φ ⋆ . . . ⋆ D ml1 +1 φ ⋆D l2 Dφ ⋆ D l3 Dφ ⋆ D l4 ψ = {z } | P P li +

+s

mj =k

P

l1 −times

X

2

li +

P

D ∂t ⋆ D RP ⋆ D m1 +1 φ ⋆ . . . ⋆ D ml2 +1 φ ⋆D l3 dφ(∂t ) ⋆ D l4 dφ ⋆ D l5 ψ . | {z } l1

mj =k

l2

l2 −times

This allows us to derive the following estimate Z hD k hαγ hβδ ∂α · ∂β · RP (dφ(∂γ ), dφ(∂δ ))ψ , D k ∇t ψi dVgt Σ Z X D l1 RP ⋆ D m1 +1 φ ⋆ . . . ⋆ D ml1 +1 φ ⋆D l2 dφ ⋆ D l3 dφ ⋆ D l4 ψ ⋆ D k ∇t ψ dVgt = | {z } P P Σ li +

+ s2

mj =k

P

l1 −times

X

li +

P

mj =k

Z

D l1 ∂t ⋆ D l2 RP ⋆ D m1 +1 φ ⋆ . . . ⋆ D ml2 +1 φ {z } | Σ

l3

l2 −times

l4

l5

k

⋆ D dφ(∂t ) ⋆ D dφ ⋆ D ψ ⋆ D ∇t ψ dVgt P

≤ C(k)kR kC k

k X

kDφkl+2 H r kψkH r+1 k∇t ψkH r

l=0

+ s2 C(k)k∂t kC k kRP kC k

k X

r r r kDφkl+1 H r k∇t φkH kψkH k∇t ψkH

l=0

≤ s−1 C(k)kRP kC k

k X

Fr (φ)l/2+1 Fr (ψ) + C(k)k∂t kC k kRP kC k

l=0

k X

Fr (φ)l/2+1 Fr (ψ).

l=0

To manipulate the third term on the right hand side of (2.5) we first consider the derivative with respect to t. We find ∇t (RP (ψ, ψ)(N s)2−n )∂t · ψ =(∇dφ(∂t ) RP )(ψ, ψ)(N s)2−n ∂t · ψ + 2RP (∇t ψ, ψ)(N s)2−n ∂t · ψ + (2 − n)RP (ψ, ψ)(N s)2−n ∂t log N ∂t · ψ + (2 − n)ss ˙ 1−n N 2−n RP (ψ, ψ)∂t · ψ. As a next step we calculate the k-th spatial derivative of this expression k

P

D (∇dφ(∂t ) R )(ψ, ψ)N ⋆D |

m1 +1

2−n

φ ⋆ ... ⋆ D {z

∂t · ψ =

ml2 +1 +1

l2 +1−times

D k RP (∇t ψ, ψ)N 2−n ∂t · ψ = ⋆D |

m1 +1

φ ⋆ ... ⋆ D {z l2 −times

ml2 +1

P

P

k X

li +

l3

P

D l1 N 2−n ⋆ D l2 +1 RP

mj =k

φ ⋆D ψ ⋆ D l4 ψ ⋆ D l5 ∂t ⋆ D l6 ψ ⋆ D l7 dφ(∂t ), } k X

li + l3

P

D l1 N 2−n ⋆ D l2 RP

mj =k

φ ⋆D ∇t ψ ⋆ D l4 ψ ⋆ D l5 ∂t ⋆ D l6 ψ, }


20

k

P

D R (ψ, ψ)N ⋆D |

m1 +1

2−n

∂t log N ∂t · ψ) =

φ ⋆ ... ⋆ D {z

ml2 +1

l2 −times

D k N 2−n RP (ψ, ψ)∂t · ψ = ⋆D |

m1 +1

φ ⋆ ... ⋆ D {z l2 −times

li +

l3

P

D l1 N 2−n ∂t log N ⋆ D l2 RP

mj =k

φ ⋆D ψ ⋆ D ψ ⋆ D l5 ∂t ⋆ D l6 ψ, }

P

ml2 +1

P

k X l4

k X

li +

P

D l1 N 2−n ⋆ D l2 RP

mj =k

l3

φ ⋆D ψ ⋆ D l4 ψ ⋆ D l5 ∂t ⋆ D l6 ψ. }

These manipulations allow us to derive the following estimates: s2

Z

hD k ∇t (RP (ψ, ψ)(N s)2−n )∂t · ψ , D k ∇t ψi dVgt Σ Z X 4−n D l1 N 2−n ⋆ D l2 +1 RP ⋆ D m1 +1 φ ⋆ . . . ⋆ D ml2 +1 +1 φ =s {z } | P P Σ li +

mj =k

l3

l2 +1−times

l4

l5

l6

l7

k

⋆ D ψ ⋆ D ψ ⋆ D ∂t ⋆ D ψ ⋆ D dφ(∂t ) ⋆ D ∇t ψ dVgt Z k X 4−n D l1 N 2−n ⋆ D l2 RP ⋆ D m1 +1 φ ⋆ . . . ⋆ D ml2 +1 φ +s | {z } P P Σ li +

mj =k

l3

l2 −times

l4

l5

l6

k

⋆ D ∇t ψ ⋆ D ψ ⋆ D ∂t ⋆ D ψ ⋆ D ∇t ψ dVgt Z k X 3−n D l1 N 2−n ⋆ D l2 RP ⋆ D m1 +1 φ ⋆ . . . ⋆ D ml2 +1 φ + (2 − n)ss ˙ {z } | P P Σ li +

l3

l4

l3

l4

mj =k

l5

l2 −times

l6

k

l6

k

⋆ D ψ ⋆ D ψ ⋆ D ∂t ⋆ D ψ ⋆ D ∇t ψ dVgt Z k X 4−n D l1 N 2−n ∂t log N ⋆ D l2 RP ⋆ D m1 +1 φ ⋆ . . . ⋆ D ml2 +1 φ + (2 − n)s | {z } P P Σ li +

mj =k

l5

l2 −times

⋆ D ψ ⋆ D ψ ⋆ D ∂t ⋆ D ψ ⋆ D ∇t ψ dVgt

≤C(k)s

4−n

kN

2−n

P

kC k kR kC k+1 k∂t kC k

k X

kDφklH r kψk3H r k∇t φkH r k∇t ψkH r

l=0

+ C(k)s4−n kN 2−n kC k kRP kC k k∂t kC k

k X

kDφklH r kψk2H r k∇t ψk2H r

l=0

+ C(k, n)(n − 2)ss ˙ 3−n kN 2−n kC k kRP kC k k∂t kC k

k X

kDφklH r kψk3H r k∇t ψkH r

l=0

+ C(k, n)(n − 2)s4−n kN 2−n ∂t log N kC k kRP kC k k∂t kC k

k X


l=0

≤C(k)s2−n kN 2−n kC k kRP kC k+1 k∂t kC k

k X

Fr (φ)l/2+1/2 Fr (ψ)2

l=0

+ C(k)s2−n kN 2−n kC k kRP kC k k∂t kC k

k X l=0

Fr (φ)l/2 Fr (ψ)2


+ C(k, n)(n − 2)ss ˙

2−n

kN

2−n

P

kC k kR kC k k∂t kC k

k X

Fr (φ)l/2 Fr (ψ)2

l=0

+ C(k, n)(n − 2)s3−n kN 2−n ∂t log N kC k kRP kC k k∂t kC k

k X

Fr (φ)l/2 Fr (ψ)2 .

l=0

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 D(RP (ψ, ψ)(N s)2−n ) · ψ =(Ddφ RP )(ψ, ψ)(N s)2−n · ψ + 2RP (Dψ, ψ)(N s)2−n ) · ψ + s2−n RP (ψ, ψ)D(N )2−n ) · ψ. The kth spatial derivative of these terms acquire the forms D k Ddφ RP )(ψ, ψ)(N s)2−n · ψ X D l1 +1 RP ⋆ D m1 +1 φ ⋆ . . . ⋆ D ml1 +1 +1 φ ⋆D l2 ψ ⋆ D l3 ψ ⋆ D l4 ψ ⋆ D l5 N 2−n , = s2−n | {z } P P li +

k

mj =k

P

2−n

D R (Dψ, ψ)(N s) X = s2−n P

k

2−n

D (s)

li +

P

(l1 +1)−times

)·ψ

mj =k

P

D l1 RP ⋆ D m1 +1 φ ⋆ . . . ⋆ D ml1 +1 φ ⋆D l2 Dψ ⋆ D l3 ψ ⋆ D l4 ψ ⋆ D l5 N 2−n , {z } | l1 −times

2−n

R (ψ, ψ)D(N ) ) · ψ) X l1 P D R ⋆ D m1 +1 φ ⋆ . . . ⋆ D ml1 +1 φ ⋆D l2 ψ ⋆ D l3 ψ ⋆ D l4 ψ ⋆ D l5 DN 2−n . =s | {z } P P 2−n

li +

mj =k

l1 −times

These manipulations allow us to derive the following estimates: Z hD k D(RP (ψ, ψ)(N s)2−n )∂i · ψ , D k ∇t ψi dVgt Σ Z X 2−n D l1 N 2−n ⋆ D l2 +1 RP ⋆ D m1 +1 φ ⋆ . . . ⋆ D ml2 +1 +1 φ =s | {z } P P Σ li +

l3

mj =k

(l2 +1)−times

l4

l5

li +

mj =k

k

⋆ D ψ ⋆ D ψ ⋆ D ψ ⋆ D ∇t ψ dVgt Z X D l1 N 2−n ⋆ D l2 RP ⋆ D m1 +1 φ ⋆ . . . ⋆ D ml2 +1 φ + s2−n {z } | P P Σ l3

l4

l2 −times

l5

k

⋆ D Dψ ⋆ D ψ ⋆ D ψ ⋆ D ∇t ψ dVgt Z X 2−n D l1 D(N 2−n ) ⋆ D l2 RP ⋆ D m1 +1 φ ⋆ . . . ⋆ D ml2 +1 φ +s | {z } P P Σ l3

li +

mj =k

l4

l5

l2 −times

k

⋆ D ψ ⋆ D ψ ⋆ D ψ ⋆ D ∇t ψ dVgt

≤ C(k)s2−n kN 2−n kC k kRP kC k+1

k X

3 kDφkl+1 H r kψkH r k∇t ψkH r

l=0

+ C(k)s2−n kN 2−n kC k kRP kC k

k X


l=0

+ C(k)s

2−n

kN

2−n

P

kC k+1 kR kC k

k X


l=0

≤ C(k)s1−n kN 2−n kC k+1 kRP kC k+1

k+1 X l=0

Fr (φ)l/2 Fr (ψ)2 .


22

To control the last term from the right hand side of (2.5) we calculate D k (N s)2−n RP (ψ, ψ)RP (ψ, ψ)ψ X D l1 N 2−n ⋆ D l2 RP ⋆ D m1 +1 φ ⋆ . . . ⋆ D ml2 +1 +1 φ =s2−n {z } | P P P l3

li1 +

mi2 +

P

q1 +1

⋆D R ⋆D |

qi3 =k

l2 −times

φ ⋆ ... ⋆ D {z

ql3 +1 +1

l3 −times

l4

l5

l6

φ ⋆D ψ ⋆ D ψ ⋆ D ψ ⋆ D l7 ψ ⋆ D l8 ψ. }

Consequently, we obtain the following estimate Z hD k ((N s)2−n RP (ψ, ψ)RP (ψ, ψ)ψ), D k ∇t ψi dVgt Σ

≤s

2−n

C(k)kN

2−n

kC k kRP k2C k

2k X


l=0

≤s

1−n

C(k)kN

2−n

kC k kRP k2C k

2k X

Fr (φ)l/2 Fr (ψ)3 .

l=0

Adding up the different contributions yields the claim.

Remark 3.10. Note that all terms on the right hand side of (3.10) have a similar analytic structure except the terms proportional to C3 and C8 . 3.4. 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 ∈ Γ(T ∗ Σ⊙2 ) of a hypersurface ({t} × Σ, gt ) ⊂ (M, h) is given by s 1 ˙ Y ) · ∂t , νih = − g(X, ˙ Y ), II(X, Y ) = h∇X Y − DX Y, νih = hs2 g(X, 2 2 where ν is the future-directed unit normal of the hypersurface. Lemma 3.11. We have the estimates kgk ˙ C k ≤ 2s−1 kIIkC k , k X

Σ

kR kC k ≤ C(k)

k∂t kC k ≤ s−1 (1 + kIIkC k−1 ),

kIIklC k kRM kC k + C(k) kIIk2C k ,

l=0

kRSM kC k ≤ C(k, n)

k X

kIIklC k kRM kC k ,

l=0

kRSM (∂t , ·)kC k ≤ C(k, n)s−1

k+1 X

kIIklC k kRM kC k .

l=0

Here, we defined the C k -norm of RM by taking the kth covariant derivative with respect to h but taking the norm with respect to the Riemannian reference metric s−2 dt2 + gt in order to get a nonnegative quantity. Proof. The first estimate follows from the definition of II and the second from the fact that ∇X ∂t = 12 g(X). ˙ Now we recall that for every tensor T ∈ Γ(T ∗ Σ⊗k ) on the manifold M , the difference between the covariant derivatives ∇ and D can be expressed as DT = ∇T + II ⋆ T, which by induction yields Dk T =

X

l1 +l2 +

P

mi =k

m1 . . ⋆ D ml1 II} ⋆∇l2 T. |D II ⋆ .{z l1 −times


This formula in combination with the Gauß equation RΣ = RM + II ⋆ II yields the third inequality. To prove the last two formulas, we recall that R

SM

n−1 1 X M (X, Y )ψ = R (X, Y, ∂α , ∂β )∂α · ∂β · ψ 4 α,β=0

for X, Y ∈ Γ(T M ), where {∂α } is a local pseudo-orthonormal frame. Therefore, we get in the above situation X m1 (D k RSM )ψ = D II ⋆ .{z . . ⋆ D ml1 II} ⋆∇l2 RM ∗ ψ, | P l1 +l2 +

mi =k

l1 −times

which yields the fourth inequality. Similarly, by using the product rule, we obtain X m1 (D k RSM (∂t , ·))ψ = D l1 ∂t ⋆ D II ⋆ .{z . . ⋆ D ml2 II} ⋆∇l3 RM ⋆ ψ | P

li +

P

mj =k

l2 −times

and by using the second inequality we obtain the last one.

At this point we are ready to control the total energy of (φ, ψ), which we define by Fr (φ, ψ) = Fr (φ) + Fr (ψ) = =s

2

k∂t φk2H r

+

r X

Ek (φ) +

k=0 kDφk2H r

+s

2

r X

Ek (ψ) + kψk2L2

k=0 k∇t ψk2H r

+ kDψk2H r + kψk2L2 .

Proposition 3.12. 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 φ ∈ C 0 ([0, T ), H r+1 (Σ, P )) ∩ C 1 ([0, T ), H r (Σ, P )), ψ ∈ C 0 ([0, T ), H r+1 (M, SM ⊗ φ∗ T P )) ∩ C 1 ([0, T ), H r (M, SM ⊗ φ∗ T P )). Suppose that s˙ ≥ 0 and that the following uniform bounds hold: 0 < C2 ≤ N ≤ C3 ,

kN kC r+1 < C4 ,

kRM kC r < C7 ,

kIIkC r < C6 ,

k∇ν N kC r+1 < C5 ,

kRP kC r+1 < C8

for some positive constants Ci , i = 1, . . . , 8. Suppose finally that there is a uniform bound on all the Sobolev constants of gt . Then the total energy satisfies the following inequality r 2r+4 X X d Fr (φ, ψ)l/2+2 , Fr (φ, ψ)l/2+1 + C(n − 2)ss ˙ 1−n Fr (φ, ψ) ≤ C · s−1 dt l=0

(3.11)

l=0

where C > 0 depends on the above bounds.

Proof. The estimate is a direct consequence of the evolution inequalities (3.5), (3.9), (3.10), Lemma 3.11 and elementary estimates. Proof of Theorem 1.3. The existence of a local solution to the system (1.3), (1.4) can be obtained by standard methods, see for example [15, 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 [15, Lemma 9.14] if suffices to obtain a uniform bound on Fr (φ, ψ) for all times t ∈ [0, ∞). We set f (t) = (n − 2)s1−n s. ˙ 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 Z ∞ Z ∞ d 2−n 1−n (s )dt = − s2−n |t=∞ − s2−n |t=0 = s(0)2−n < ∞. ss ˙ dt = − (n − 2) dt 0 0

24


As long as Fr (φ, ψ) ≤ 1 we have the differential inequality d Fr (φ, ψ) ≤ C s−1 (t) + f (t) Fr (φ, ψ), dt which can easily be integrated as Z T Fr (φ, ψ)|t=T ≤ Fr (φ, ψ)|t=0 exp (s−1 (t) + f (t))dt . 0 R ∞ −1 Now, we set Φ := 0 (s (t) + f (t))dt < ∞ and choose ε > 0 small enough such that Fr (φ, ψ)|t=0 ≤ (2Φ)−1 . Suppose that T0 is the first time for which Fr (φ, ψ)|T0 = 1. However, the energy inequality from above gives 1 Fr (φ, ψ)|t=T0 ≤ ΦFr (φ, ψ)|t=0 = < 1, 2 which yields a contradiction. Therefore we can conclude that Fr (φ, ψ) < 1 < ∞ for all times t ∈ [0, ∞) completing the proof.

Proof of Theorem 1.1. 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. Acknowledgements: The first author gratefully acknowledges the support of the Austrian Science Fund (FWF) through the START-Project Y963-N35 of Michael Eichmair. The second author wants to thank the Fields Institute for Research in Mathematical Sciences for its hospitality during the program on Geometric Analysis. Both authors thank the Erwin Schr¨ odinger International Institute for Mathematics and Physics and the organizers of its “Geometry and Relativity” program, where part of this paper was written. References [1] Luis Alvarez-Gaum´e and Daniel Z. Freedman. Geometrical structure and ultraviolet finiteness in the supersymmetric σ-model. Comm. Math. Phys., 80(3):443–451, 1981. [2] Christian B¨ ar, Paul Gauduchon, and Andrei Moroianu. Generalized cylinders in semi-Riemannian and Spin geometry. Math. Z., 249(3):545–580, 2005. [3] Helga Baum. Spin-Strukturen und Dirac-Operatoren u ¨ber pseudoriemannschen Mannigfaltigkeiten, volume 41 of Teubner-Texte zur Mathematik [Teubner Texts in Mathematics]. BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1981. With English, French and Russian summaries. [4] Volker Branding. Some aspects of Dirac-harmonic maps with curvature term. Differential Geom. Appl., 40:1–13, 2015. [5] Volker Branding. Energy estimates for the supersymmetric nonlinear sigma model and applications. Potential Anal., 45(4):737–754, 2016. [6] Volker Branding. Energy methods for Dirac-type equations in two-dimensional Minkowski space. arXiv preprint arXiv:1706.05971, 2017. [7] Q. Chen, J. Jost, and G. Wang. Liouville theorems for Dirac-harmonic maps. J. Math. Phys., 48(11):113517, 13, 2007. [8] Qun Chen, J¨ urgen Jost, Jiayu Li, and Guofang Wang. Dirac-harmonic maps. Math. Z., 254(2):409–432, 2006. [9] Yvonne Choquet-Bruhat. Global wave maps on curved space times. In Mathematical and quantum aspects of relativity and cosmology (Pythagoreon, 1998), volume 537 of Lecture Notes in Phys., pages 1–29. Springer, Berlin, 2000. [10] Yvonne Choquet-Bruhat. Global wave maps on Robertson-Walker spacetimes. Nonlinear Dynam., 22(1):39– 47, 2000. Modern group analysis. [11] Piero D’Ancona and Qidi Zhang. Global existence of small equivariant wave maps on rotationally symmetric manifolds. Int. Math. Res. Not. IMRN, (4):978–1025, 2016. [12] Xiaoli Han. Dirac-wave maps. Calc. Var. Partial Differential Equations, 23(2):193–204, 2005. [13] Andrew Lawrie. The Cauchy problem for wave maps on a curved background. Calc. Var. Partial Differential Equations, 45(3-4):505–548, 2012.


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