Future complete U(1) symmetric Einsteinian spacetimes, the unpolarized case.

Yvonne Choquet-Bruhat

February 20, 2003

1 Introduction.

In this paper I generalize the non linear stability theorem obtained in collaboration with V. Moncrief (CB-M1, CB-M2) for vacuum Einsteinian 4 - manifolds (V,^{(4)} g) where V = M × R with M a circle bundle over a compact, orientable surface Σ of genus greater than 1. The Lorentzian metric (4)g admits a Killing symmetry along the (spacelike) circular fibers. I remove the so-called polarization condition, i.e. the orthogonality of the fibers to quotient 3-manifolds. The reduced field equations take now the form of a wave map equation, instead of a linear wave equation in the polarized case, coupled to 2+1 gravity. I use results on wave maps from curved manifolds obtained in CB1, CB2. Like in CB-M2 we do not restrict the conformal geometry of Σ to avoid those regions of Teichmüller space for which the lowest positive eigenvalues of the scalar Laplacian lie, in our normalization, in the gap (0, \frac{1}{8}]. A consequence is that the asymptotic behaviour of the wave map field does not exhibit a universal rate of decay but instead develops a decay rate which depends upon the asymptotic values of the lowest eigenvalue.

Under the Kaluza-Klein reduction which one carries out in the presence of the assumed spacelike Killing field one is first led to field equations of the type of an Einstein - Maxwell - Jordan system on the 3 - manifold Σ × R. To transform this to a more convenient Einstein - wave map system one needs a further topological restriction on the fields allowed. The need for this arises from considering the constraint equation for the effective 2+1 dimensional electric type field density ẽ = e^{a} \frac{∂}{∂x^{a}} which reads e_{,a}^{a} = 0. On a higher genus surface Σ the general solution of this equation (which results from a Hodge decomposition of one - forms on Σ) takes the form e^{a} = ε^{ab}(ω,_{b} +h_{b}) where h_{b}dx^{b} is a harmonic one - form on Σ. A consistent simplification results from setting this harmonic contribution to zero so that e^{a} \frac{∂}{∂x^{a}} becomes expressible purely in terms of the so - called twist potential ω. Taken together with the norm of the (U(1) generating) Killing field Ỹ, conveniently expessed via Ỹ .Ỹ = e^{2γ}, the twist potential ω and the function γ provide a map from Σ × R to R^{2}. When expressed in terms of the pair (γ, ω) Einstein’s equations take the form of a wave map from a 2+1 Lorentzian manifold (Σ × R,^{(3)} g) into the Poincaré plane with its standard metric 2dγ^{2} +\frac{1}{2}e^{−4γ}dω^{2}; the metric (3)g satisfies the 2+1 Einstein equations on Σ×R, with source the wave map. These 2+1 Einstein equations, supplemented by suitable coordinate conditions to fix the gauge, reduce to an elliptic system on each slice Σ_{t} of Σ × R for the lapse, the shift, and the conformal factor of a 2 dimensional metric, together with an ordinary differential system for the Teichmüller parameters with determine the conformal geometry. The wave maps fields and the Teichmüller parameters represent therefore the true propagating gravitational degrees of freedom of the original problem.

The basic methods we use to prove existence for an infinite proper time involve the construction of higher order energies to control the Sobolev norms of the wave map combined with an application of the ”Dirichlet energy” function in Teichmüller space to control the Teichmüller parameters degrees of freedom. A subtlety is that the most obvious definition of wave map energies does not lead to a well defined rate of decay so that suitable corrected energies must be developed which exploit information about the lowest eigenvalues of the spatial Laplacian which appears in the relevant wave operators. The eigenvalues vary with position in Teichmüller space and thus evolve along with Teichmuller parameters. If the lowest (non trivial) eigenvalue asymptotically avoids a well known gap in the spectrum ( the gap (0,\frac{1}{8}] in our normalization which has the more familiar form (0,\frac{1}{4}] if one instead normalizes the Gauss curvature on the higher genus surface) then we obtain a universal rate of decay for the energies asymptotically. If the lowest eigenvalue however drifts into this gap and remains there asymptotically then the rate of decay of the energies will depend upon the asymptotic value of this lowest eigenvalue and will no longer be universal. We need slightly different forms for the corrected energies to handle these different eventualities (universal versus non universal rates of decay). In all cases the conformal geometry of our circle bundles undergoes a kind of Cheeger - Gromov collapse in which the circular fibers (after a conformal rescaling needed to take out the overall expansion) collapse to zero length asymptotically while only the conformal 2 - geometry remains well behaved. In our set up the Sobolev constants depend only on the conformal 2 - geometry (i.e. upon the Teichmüller parameters) and, so long as the evolution remains in a compact subspace of Teichmüller space, these constants remain under control.

The sense in which our solutions are global in the expanding direction is that they exhaust the maximal range allowed for the mean curvature function on a manifold of negative Yamabe type, for which a zero mean curvature cannot be achieved but only asymptotically approached. In addition however our estimates prove that the normal trajectories to our spatial slices all have infinite future proper time length, and allow us to establish, using [CB-C], causal geodesic completeness in the expanding direction.

If the harmonic one - form discussed above were allowed to be non zero it would disturb the pure wave map character of the reduced field equations. On the other hand it seems plausible that energy arguments could still be made to work in the presence of h. Alternatively one might simply refrain from trying to force the reduced field equations into a wave map framework and instead develop energy arguments for the Einstein - Maxwell - Jordan type system itself which require no splitting of ẽ into twist potential and harmonic contributions. We shall not however pursue either of these possibilities here but leave them for further study.

I need as in CB-M1 and CB-M2 a smallness condition on suitably defined energies which control the norms of the evolving (herere wave map) field and for this reason I continue to restrict my attention to trivial S^{1} bundles over Σ (i.e., to those for which M = S^{1} × Σ). The reason for this is that the curvature of the U(1) connection and its assumed (U(1) - generating) Killing field has a quantized integral over Σ and, in the case of a non trivial bundle when this integral is not zero, cannot be adjusted to satisfy the smallness condition needed for the energy argument. It seems plausible that one could probably substract off this unavoidable topological contribution to curvature and work with suitable energies defined for the substracted fields to handle the case of non trivial bundles but I shall not attempt to do so here.

Another approach (suggested by V. Moncrief) to treating solutions on non trivial S^{1} bundles involves applying a well known action of SL(2, R) (the isometry goup of the Poincaré plane which plays the role of target for our wave map fields) to the fields defined on the base manifold Σ × R. In certain cases this group action can be used to transform solutions which lift to the trivial S^{1} bundle over Σ × R to other solutions which lift instead to another, non trivial bundle. There is an obstruction to obtaining such solutions in this way since a certain Casimir invariant (which is of course preserved under the group action) is necessarily positive for solutions which lift to the trivial bundle (it can be negative for a subset of solutions which lift to non trivial bundles). This formulation has so far only been developed for the case of circle bundles over S^{2} × R but can most likely be generalized to the cases of bundles over Σ×R where Σ is either a torus or a higher genus surface. That possibility is left for further study.

The small data future global existence theorem for solutions of Einstein’s equations of Andersson and Moncrief, this volume, makes no symmetry assumption whatsoever, but treats a different class of spatial 3 - manifolds which are taken to be compact hyperbolic. The results of their analysis show that the standard hyperbolic (i.e. constant negative curvature) metric on such a manifold serves as an attractor for the conformal geometry under the (future) Einstein flow. In other words the evolving conformal geometry has a well behaved limit in that problem. This fact plays a crucial role in their analysis since various Sobolev ”constants” (which are in fact functionals of the geometry) which are needed in the associated energy estimates are asymptotically under control since they are tending toward their (regular) limiting values for the hyperbolic metric. Thus the difficulty of degenerating Sobolev constants, avoided by the introduction of a conformal 2 metric in the case of our U(1) symmetry assumption, never arises in the Andersson - Moncrief work.

Besides the fact that the U(1) symmetric case is not included in the no symmetry case treated by Andersson and Moncrief, an interest of the U(1) case is that in our problem the number of effective spatial dimensions is two, and also that there is no known ”physical” reason why large data solutions should develop singularities in the direction of cosmological expansion. Black hole formation seems to be suppressed by the topological character of the assumed Killing symmetry (which is of translational rather than rotational type and excludes the appearance of an axis of symmetry) and the big bang singularity is avoided by considering the future evolution from an initially expanding Cauchy hypersurface. Any possible big crunch is excluded by our requirement that the spatial manifold M is of negative Yamabe type (which is true of all circle bundles over higher genus manifolds). Such manifolds are incompatible (in the vacuum and electrovacuum cases for example) with the development of a maximal hypersurface which would be a necessary prelude to the ”recollapse” of an expanding universe towards a hypothetical big crunch singularity. At a maximal hypersurface the scalar curvature of M would have to be everywhere positive - an impossibility on any manifold of negative Yamabe type. Thus it is conceivable that for large data future global existence holds for our problem. Up to now the only large data global results require simplifying assumptions so stringent that they effectively reduce the number of spatial dimensions to one (e.g., Gowdy models and their generalizations, plane symmetric gravitational vaves, spherically symmetric matter coupled with gravity) or zero (e.g. Bianchi models, 2+1 gravity). Unfortunately we have at present no way of proving this global existence, even in the polarized case for which the wave map equation reduces to a wave equation for a scalar function, because the reduced field equations are non local in character. The ”background” spacetime on which the scalar field evolves is not given a priori but is instead a certain functional (obtained by solution of elliptic equations) of the evolving field (and the Teichmüller parameters) itself. In the unpolarized case, the problem of global existence of strong solutions for wave maps on a fixed background in 2+1 dimensions is still unsolved. However there is a proof [M-S] of global existence of a weak solution (with no uniqueness) for wave maps from Minkowski spacetime. Any progress on the large data global existence, even of weak solutions, for the U(1) - symmetric problem would represent a ”quantum jump” forward in our understanding of long time existence problems for Einstein’s equations.

It is worth mentioning here that, again with suitable topological restrictions, using the reduction obtained by V. Moncrief (M2), an analogous Einstein - wave map form of the reduced field equations can be obtained even when one begins with the full Einstein - Maxwell system in 3+1 dimensions.

Some steps of the proof given here have been obtained independently, using other notations, by V. Moncrief. I thank him for communicating his manuscript to me, and for numerous conversations on the subject.

2 S^{1} invariant einsteinian universes.

2.1 Definition.

The spacetime manifold V is a principal fiber bundle with Lie group S^{1} and base Σ × R, with Σ a smooth orientable 2 dimensional manifold which we suppose here to be compact and of genus greater than one.

The spacetime metric ^{(4)}g is invariant under the action of S^{1}, the orbits are the fibers of V and are supposed to be space like. We write it in the form

(4)g = e^{−2γ(3)}g + e^{2γ}(θ)^{2},

where γ is a scalar function and ^{(3)}g a lorentzian metric on Σ × R which reads:

(3)g = −N^{2}dt^{2} + g_{ab}(dx^{a} + ν^{a}dt)(dx^{b} + ν^{b}dt)

equivalently, in terms of a moving frame

(3)g = −N^{2}(θ^{0})^{2} + g_{ab}θ^{a}θ^{b}, θ^{0} ≡ dt, θ^{a} ≡ dx^{a} + ν^{a}dt

N and ν are respectively the lapse and shift of ^{(3)}g,while

g = g_{ab}dx^{a}dx^{b}

is a riemannian metric on Σ, depending on t.

The 1-form θ is a connection on the fiber bundle V, represented in coordinates (x^{3}, x^{α}) adapted to a local trivialization of the bundle by

θ = dx^{3} + A_{α}dx^{α}

with x^{3} a coordinate on S^{1} (i.e. with x^{3} = 0 and x^{3} = 2π identified) and A ≡ A_{α}dx^{α} a locally defined 1-form on Σ × R.

2.2 Twist potential.

2.2.1 Definition.

The curvature of the connection locally represented by A is a 2-form F on Σ × R, given by, if the equations ^{(4)}R_{α3}=0 are satisfied,

F_{αβ} = (1/2)e^{−4γ}η_{αβλ}E^{λ}

with η the volume form of the metric ^{(3)}g, and E an arbitrary closed 1-form . Hence if Σ is compact

E = dω + H

where ω is a scalar function on V, called the twist potential, and H a representative of the 1-cohomology class of Σ×R, for instance defined by a 1-form on Σ, harmonic for some given riemannian metric m. For simplicity we take H = 0.

2.2.2 Construction of A.

The connection 1- form θ can be constructed, when F is known if the following integrability condition is satisfied (see [M1], [CB-M3])

∫_{Σ_{t}} F = \frac{1}{2} ∫_{Σ_{t}} F_{ab}dx^{a} ∧ dx^{b} = − ∫_{Σ_{t}} e^{−4γ}N^{−1}∂_{0}ωμ_{g} = 2πn.  (2.1)

where n is the Chern number of the bundle over Σ × R.

We will suppose here that this bundle is trivial, i.e. n = 0; this value of n is the only one compatible with the smallness assumptions on the energy that we will make. The 1 - form A ≡ A_{α}dx^{α} is then defined globally on Σ × R. It satisfies the equation

dA = F  (2.2)

We denote by Ã and F̃ the t dependent 1 - form and 2 - form on Σ given by

Ã ≡ A_{a}dx^{a}, F̃ ≡ \frac{1}{2}F_{ab}dx^{a} ∧ dx^{b}.  (2.3)

The equation 2.2 splits into

dÃ = F̃, i.e. ∂_{a}A_{b} − ∂_{b}A_{a} = F_{ab}  (2.4)

and, denoting by F_{(t)} the 1-form F_{ta}dx^{a} :

∂_{t}A_{a} − ∂_{a}A_{t} = F_{ta}, i.e. ∂_{t}Ã − dA_{t} = F_{(t)}.  (2.5)

We solve 2.4 by introducing a smooth m metric on Σ. We denote by δ_{m} and Δ̃_{m} ≡ δ_{m}d+dδ_{m} the codifferential and the de Rham Laplace operator in this metric. If we suppose that Ã satisfies the Coulomb gauge condition:

δ_{m}Ã = 0.  (2.6)

The equations 2.4 and 2.6 imply that

Δ̃_{m}Ã = δ_{m}F̃.  (2.7)

The general solution of this equation is the sum of the unique solution Â which is L^{2}(m)− orthogonal to the elements H_{(i)} of a basis of harmonic 1-forms, and an arbitrary harmonic 1-form, that is:

Ã = Â + ∑_{i} c_{i}H_{(i)}, (Â, H_{(i)})_{L^{2}(m)} = 0  (2.8)

where the c_{i} are t-dependent numbers. The solution Â satisfies a Sobolev inequality

||Â||_{H_{2}(m)} ≤ C_{m}||δ_{m}F̃||_{L^{2}(m)}, ||δ_{m}F̃||_{L^{2}(m)} ≤ C_{m}||F̃||_{H_{1}(m)}  (2.9)

A solution Ã of 2.7 satisfies 2.4 and 2.6 because 2.7 implies

dΔ̃_{m}Ã ≡ Δ̃_{m}dÃ = dδ_{m}F̃ ≡ Δ̃_{m}F̃  (2.10)

since F̃ is closed, and

δ_{m}Δ̃_{m}Ã ≡ Δ̃_{m}δ_{m}Ã = 0.  (2.11)

2.10 implies that dÃ − F̃ is a harmonic 2-form on Σ, it is zero because its period, i.e. its integral on the unique 2 cycle Σ, is zero (equation 2.1). 2.11 implies that the scalar function δ_{m}Ã is harmonic, but its integral on Σ is zero, since it is a divergence, therefore δ_{m}Ã = 0.

The equation 2.5 can be satisfied by choice of A_{t}, a t dependent function on Σ, if the 1-form ∂_{t}A − F_{(t)} is an exact differential. The commutation of partial derivatives and the closure of F show that a solution Ã of 2.4 satisfies the equation

∂_{a}∂_{t}A_{b} − ∂_{b}∂_{t}A_{a} = ∂_{t}F_{ab} = ∂_{a}F_{tb} − ∂_{b}F_{ta}, i.e. d(∂_{t}Ã − F_{(t)}) = 0. (2.12)

Since the form ∂_{t}Ã−F_{(t)} is closed it will be the differential of a function A_{t} if and only if it is L^{2}(m) orthogonal to the harmonic 1-forms, that is, because ∂_{t}Â is like Â L^{2}(m) orthogonal to the H_{(i)} which do not depend on t :

(∑_{j} \frac{dc_{j}}{dt} H_{(j)} − F_{(t)}, H_{(i)})_{L^{2}(m)} = 0.  (2.13)

Choosing the H′_{(i)}s to be L^{2}(m) orthonormal, this equation reduces to:

\frac{dc_{i}}{dt} = (F_{(t)}, H_{(i)})_{L^{2}(m)}.  (2.14)

These equations determine c_{i} by integration on t through its its initial value c_{i}(t_{0})

We complete the determination of A_{t} by remarking that for a scalar function the equation 2.5 implies, using 2.6,

Δ_{m}A_{t} = −δ_{m}F_{(t)},  (2.15)

an equation which determines uniquely A_{t} if we impose that its integral on Σ_{t} is zero. It satisfies then the inequality

||A_{t}||_{H_{2}(m)} ≤ C_{m}||δ_{m}F_{(t)}||_{L^{2}(m)}, ||δ_{m}F_{(t)}||_{L^{2}(m)} ≤ C_{m}||F_{t}||_{H_{1}(m)}. (2.16)

Remark 2.1 We can, instead of the Coulomb gauge, determine Ã in temporal gauge, i.e. impose A_{t} = 0. We determine the 1 - form Ã_{0}, value of Ã for t = t_{0} by the relation 2.4 through the value of F̃_{0} as above. The equation 2.5 is, when A_{t} and F_{(t)} are known, an ordinary differential equation for Ã which can be solved by integration on t :

Ã = Ã_{0} + ∫_{t_{0}}^{t} F_{(t)}.  (2.17)

When 2.5 is satisfied it implies, whatever be A_{t}, using the commutation of partial derivatives and the closure of F the equation

∂_{t}(∂_{a}A_{b} − ∂_{b}A_{a}) = ∂_{a}F_{tb} − ∂_{b}F_{ta} = ∂_{t}F_{ab}  (2.18)

i.e.

∂_{t}(dÃ − F̃) = 0  (2.19)

hence dÃ − F̃ = 0 for all t if it is so for t = t_{0}.

The disavantage of the temporal gauge is that it gives only H_{1} estimates for Ã.

2.3 Wave map equation.

The fact that F is a closed form together with the equation ^{(4)}R_{33} = 0 imply (with the choice H = 0 in the definition of ω) that the pair u ≡ (γ, ω) satisfies a wave map equation from (Σ × R,^{(3)} g) into the Poincaré plane (R^{2}, G),

G = 2(dγ)^{2} + (1/2)e^{−4γ}(dω)^{2},

It is a system of hyperbolic type when^{(3)}g is a known lorentzian metric which reads, denoting by ^{(3)}∇_{α} the components of covariant derivatives of tensors on Σ × R in the metric ^{(3)}g in the moving frame (θ^{a}, dt):

(3)∇^{α}∂_{α}γ + \frac{1}{2}e^{−4γ}g^{αβ}∂_{α}ω∂_{β}ω = 0

(3)∇^{α}∂_{α}ω − 4g^{αβ}∂_{α}ω∂_{β}γ = 0

The integral 2.1 is independent of t if F is closed, hence if the wave map equation is satisfied.

Remark 2.2 The non zero Christoffel symbols of the metric G are

G_{22}^{1} ≡ G_{ωω}^{γ} = \frac{1}{2}e^{−4γ}, G_{12}^{2} = G_{21}^{2} = G_{γω}^{ω} = −2

The scalar and Riemann curvature are:

R_{12,12} = −2e^{−4γ}, R = −4

2.4 3-dimensional Einstein equations

When ^{(4)}R_{3α} = 0 and ^{(4)}R_{33} = 0 the Einstein equations ^{(4)}R_{αβ} = 0 are equivalent to Einstein equations on the 3-manifold Σ × R for the metric ^{(3)}g with source the stress energy tensor of the wave map:

(3)R_{αβ} = ρ_{αβ} ≡ ∂_{α}u.∂_{β}u

where a dot denotes a scalar product in the metric of the Poincaré plane:

∂_{α}u.∂_{β}u ≡ 2∂_{α}γ∂_{β}γ + \frac{1}{2}e^{−4γ}∂_{α}ω∂_{β}ω

In dimension 3 the Einstein equations are non dynamical, except for the conformal class of g determined by Teichmüller parameters. They decompose into:

a. Constraints.

b. Equations for lapse and shift to be satisfied on each Σ_{t}. These equations, as well as the constraints, are of elliptic type.

c. Evolution equations for the Teichmüller parameters, ordinary differential equations.

2.4.1 Constraints on Σ_{t}.

One denotes by k the extrinsic curvature of Σ_{t} as submanifold of (Σ×R,^{(3)} g). Then, with ∇ the covariant derivative in the metric g,

k_{ab} ≡ (2N)^{−1}(−∂_{t}g_{ab} + ∇_{a}ν_{b} + ∇_{b}ν_{a}),

the equations (momentum constraint)

(3)R_{0a} ≡ N(−∇_{b}k_{a}^{b} + ∂_{a}τ) = ∂_{0}u.∂_{a}u  (2.20)

and (hamiltonian constraint)

2N^{−2(3)}S_{00} ≡ R(g) − k_{b}^{a}k_{a}^{b} + τ^{2} = N^{−2}∂_{0}u.∂_{0}u + g^{ab}∂_{a}u.∂_{b}u  (2.21)

do not contain second derivatives transversal to Σ_{t} of g or u. They are the constraints. To transform the constraints into an elliptic system one uses the conformal method. We set

g_{ab} = e^{2λ}σ_{ab},

where σ is a riemannian metric on Σ, depending on t, on which we will comment later, and

k_{ab} = h_{ab} + \frac{1}{2}g_{ab}τ

where τ is the g-trace of k, hence h is traceless.

We denote by D a covariant derivation in the metric σ. We set

u′ = N^{−1}∂_{0}u

with ∂_{0} the Pfaff derivative of u, namely

∂_{0} = \frac{∂}{∂t} − ν^{a}∂_{a} with ∂_{a} = \frac{∂}{∂x^{a}}

and

u^{.} = e^{2λ}u′

The momentum constraint on Σ_{t} reads if τ is constant in space, a choice which we will make

D_{b}h_{a}^{b} = L_{a}, L_{a} ≡ −D_{a}u.u^{.}  (2.22)

This is a linear equation for h, with left hand side independent of λ. The general solution is the sum of a transverse traceless tensor h_{T T} ≡ q (see 2.28 below) and a conformal Lie derivative r. Such tensors are L^{2}−orthogonal on (Σ, σ).

The hamiltonian constraint reads as the semilinear elliptic equation in λ :

Δλ = f(x, λ) ≡ p_{1}e^{2λ} − p_{2}e^{−2λ} + p_{3},  (2.23)

with Δ ≡ Δ_{σ} the Laplacian in the metric σ and:

p_{1} ≡ \frac{1}{4}τ^{2}, p_{2} ≡ \frac{1}{2}(| u^{.} |^{2} + | h |^{2}), p_{3} ≡ \frac{1}{2}(R(σ) − |Du|^{2})

2.4.2 Equations for lapse and shift.

Lapse and shift are gauge parameters for which we obtain elliptic equations on each Σ_{t} as follows.

We impose that the Σ′_{t}s have constant (in space) mean curvature, namely that τ is a given increasing function of t. The lapse N satisfies then the linear elliptic equation

ΔN − αN = −e^{2λ} \frac{∂τ}{∂t}  (2.24)

with (|.|pointwise norm in the metric σ)

α ≡ e^{−2λ}(| h |^{2} + | u^{.} |^{2}) + \frac{1}{2}e^{2λ}τ^{2}  (2.25)

The equation to be satisfied by the shift ν results from the the expression for h deduced from the definition of k

h_{ab} ≡ (2N)^{−1}[−(∂_{t}g_{ab} − \frac{1}{2}g_{ab}g^{cd}∂_{t}g_{cd}) + ∇_{a}ν_{b} + ∇_{b}ν_{a} − g_{ab}∇_{c}ν^{c}]

which implies, if g_{ab} ≡ e^{2λ}σ_{ab} and if n_{a} ≡ e^{−2λ}ν_{a} denotes the covariant components of the shift vector ν in the metric σ (thus n^{a} = ν^{a})

h_{ab} ≡ (2N)^{−1}e^{2λ}[−(∂_{t}σ_{ab} − \frac{1}{2}σ_{ab}σ^{cd}∂_{t}σ_{cd}) + D_{a}n_{b} + D_{b}n_{a} − σ_{ab}D_{c}n^{c}]

The equation is therefore:

(L_{σ}n)_{ab} ≡ D_{a}n_{b} + D_{b}n_{a} − σ_{ab}D_{c}n^{c} = f_{ab}  (2.26)
f_{ab} ≡ 2Ne^{−2λ}h_{ab} + ∂_{t}σ_{ab} − \frac{1}{2}σ_{ab}σ^{cd}∂_{t}σ_{cd}  (2.27)

The homogeneous associated operator, the conformal Killing operator L_{σ}, has injective symbol, and it has a kernel zero, since manifolds of genus greater than 1 admit no conformal Killing fields.

The kernel of the dual of L_{σ} is the space of transverse traceless symmetric 2-tensors, i.e. symmetric 2-tensors T such that

σ^{ab}T_{ab} = 0, D^{a}T_{ab} = 0.  (2.28)

These tensors are usually called TT tensors. The spaces of TT tensors are the same for two conformal metrics.

2.5 Teichmüller parameters.

On a compact 2-dimensional manifold of genus G ≥ 2 the space T_{eich} of conformally inequivalent riemannian metrics, called Teichmüller space, can be identified (cf. Fisher and Tromba) with M_{−1}/D_{0}, the quotient of the space of metrics with scalar curvature −1 by the group of diffeomorphisms homotopic to the identity. M_{−1}→T_{eich} is a trivial fiber bundle whose base can be endowed with the structure of the manifold R^{n}, with n = 6G − 6.

We require the metric σ_{t} to be in some chosen cross section Q → ψ(Q) of the above fiber bundle. Let Q^{I}, I = 1, ..., n be coordinates in T_{eich}, then ∂ψ/∂Q^{I} is a known tangent vector to M_{−1} at ψ(Q), that is a symmetric 2-tensor field on Σ, the sum of a transverse traceless tensor field X_{I}(Q) and of the Lie derivative of a vector field on the manifold (Σ, ψ(Q)). The tensor fields X_{I}(Q), I = 1, ...n span the space of transverse traceless tensor fields on (Σ, ψ(Q)). The matrix with elements

∫_{Σ} X_{I}^{ab}X_{Jab}μ_{ψ(Q)}

is invertible.

We have found in [CB-M1] an ordinary differential system satisfied by t ↦ Q(t) by using on the one hand the solvability condition for the shift equation which determines dQ^{I}/dt in terms of h_{t} which reads

∫_{Σ_{t}} f_{ab}X_{J}^{ab}μ_{σ_{t}} = 0, J = 1, ...6G − 6,  (2.29)

and on the other hand the necessary and sufficient conditions for the previous equations to imply also the remaining equations ^{(3)}R_{ab} − ρ_{ab} = 0, that is:

∫_{Σ_{t}} N(^{(3)}R_{ab} − ρ_{ab})X_{J}^{ab}μ_{σ_{t}} = 0, for J = 1, 2, ...6G − 6.  (2.30)

We have used the expression

∂_{t}σ_{ab} = \frac{dQ^{I}}{dt} X_{I,ab} + C_{ab}

where C_{ab} is a Lie derivative, L^{2} orthogonal to TT tensors, together with the decomposition h = q + r, with r a tensor in the range of the conformal Killing operator and q a TT tensor. This last tensor can be written with the use of the basis X_{I} of such tensors, the coefficients P^{I} depending only on t:

q_{ab} = P^{I}(t)X_{I,ab}

The orthogonality condition 2.29 reads, using the expression 2.27 of f_{ab} and the fact that the transverse tensors X_{I} are orthogonal to Lie derivatives and are traceless:

∫_{Σ_{t}}[2Ne^{−2λ}(r_{ab} + P^{I}X_{I,ab}) + (dQ^{I}/dt)X_{I,ab}]X_{J}^{ab}μ_{σ} = 0

The tangent vector dQ^{I}/dt to the curve t → Q(t) and the tangent vector P^{I}(t) to T_{eich} are therefore linked by the linear system

X_{IJ} \frac{dQ^{I}}{dt} + Y_{IJ}P^{I} + Z_{J} = 0

with

X_{IJ} ≡ ∫_{Σ_{t}} X_{I}^{ab}X_{J,ab}μ_{σ},  (2.31)

Y_{IJ} ≡ ∫_{Σ_{t}} 2Ne^{−2λ}X_{I}^{ab}X_{J,ab}μ_{σ}, Z_{J} ≡ ∫_{Σ_{t}} 2Ne^{−2λ}r_{ab}X_{J}^{ab}μ_{σ}  (2.32)

While, using

(3)R_{ab} ≡ R_{ab} − N^{−1}∂_{0}k_{ab} − 2k_{ac}k_{b}^{c} + τk_{ab} − N^{−1}∇_{a}∂_{b}N  (2.33)

where

R_{ab} ≡ \frac{1}{2}Rg_{ab}, ρ_{ab} ≡ ∂_{a}u.∂_{b}u

k_{ab} ≡ P^{I}X_{I,ab} + r_{ab} + \frac{1}{2}g_{ab}τ

and ∂_{0} is an operator on time dependent space tensors defined by, with L_{ν} the Lie derivative in the direction of ν,

∂_{0} ≡ ∂_{t} − L_{ν}

gives^{1} for 2.30 the expression:

∫_{Σ_{t}}(−∂_{0}k_{ab} − 2Ne^{2λ}h_{ac}h_{b}^{c} + τNh_{ab} − ∇_{a}∂_{b}N − ∂_{a}u.∂_{b}u)X_{J}^{ab}μ_{σ_{t}} = 0 (2.34)

We have thus obtained an ordinary differential system of the form

X_{IJ} \frac{dP^{I}}{dt} + Φ_{J}(P, \frac{dQ}{dt}) = 0

where Φ is a polynomial of degree 2 in P and dQ/dt with coefficients depending smoothly on Q and directly but continuously on t through the other unknowns, namely:

Φ_{J} ≡ A_{JIK}P^{I}P^{K} + B_{JIK}P^{I} \frac{dQ^{K}}{dt} + C_{JI}P^{I} + D_{J}

with

A_{JIK} ≡ ∫_{Σ_{t}} 2Ne^{2λ}X_{I,a}^{c}X_{K,bc}X_{J}^{ab}μ_{σ_{t}}

B_{JIK} ≡ ∫_{Σ_{t}} \frac{∂X_{I,ab}}{∂Q^{K}} X_{J}^{ab}μ_{σ_{t}}

C_{JI} ≡ ∫_{Σ_{t}}[(−L_{ν}X_{I})_{ab} + 4Ne^{−2λ}r_{b}^{c}X_{I,ac} − τNX_{I,ab}]X_{J}^{ab}μ_{σ_{t}}

and, using integration by parts and the transverse property of the X_{I} to eliminate second derivatives of N (recall that ∇_{a}∂_{b}N ≡ D_{a}∂_{b}N − 2∂_{a}λ∂_{b}N)

D_{J} ≡ ∫_{Σ_{t}}(−∂_{0}r_{ab} − 2Ne^{−2λ}r_{ac}r_{b}^{c} + τNr_{ab} + 2∂_{a}λ∂_{b}N − ∂_{a}u.∂_{b}u)X_{J}^{ab}μ_{σ_{t}}.

3 Cauchy problem.

3.1 Cauchy data.

The Cauchy data on Σ_{t_{0}} are:

1. A C^{∞} riemannian metric σ_{0} which projects onto a point Q(t_{0}) of T_{eich} and a C^{∞} tensor q_{0} which is TT in the metric σ_{0}.

1In the formula 2.33 indices are raised with g, in 2.34 they are raised with σ.

2. Cauchy data for u and u̇ on Σ_{t_{0}}, i.e.

u(t_{0}, .) = u_{0}, u̇(t_{0}, .) = u̇_{0}.

We say that a pair of scalar functions, u ≡ (γ, ω) or u̇ ≡ (γ̇, ω̇) belongs to W_{s}^{p} if it is so of each of the scalars; W_{s}^{p} and H_{s} ≡ W_{s}^{2} are the usual Sobolev spaces of scalar functions on the riemannian manifold (Σ, σ_{0}). We suppose that

u_{0} ∈ H_{2}, u̇_{0} ∈ H_{1}.

¿From these data one determines the values on Σ_{0} of the auxiliary unkown, h_{0} ∈ W_{2}^{p}, 1 < p < 2, the conformal factor lapse and shift λ_{0}, N_{0}, ν_{0} ∈ W_{3}^{p}.

One deduces then the usual Cauchy data for the wave map by

(∂_{t}u)_{0} = e^{−2λ_{0}}N_{0}u̇_{0} + ν_{0}^{a}∂_{a}u_{0}  (3.1)

It holds that

(∂_{t}u)_{0} ∈ H_{1}.  (3.2)

We suppose that the initial data satisfy the integrability condition 2.1 and we deduce from them an admissible Ã_{0}.

3.2 Local in time existence theorem.

The following theorem is a consequence of previous results (see CB-M2, CB1, CB-M1).

Theorem 3.1 The Cauchy problem with the above data for the Einstein equations with S^{1} isometry group has, if T − t_{0} is small enough, a solution with u ∈ C^{0}([t_{0}, T), H_{2}) u̇ ∈ C^{1}([t_{0}, T), H_{1}); λ, N, ν ∈ C^{0}([t_{0}, T), W_{3}^{p}) ∩C^{1}([t_{0}, T), W_{2}^{p}), 1 < p < 2 and N > 0 while σ ∈ C^{1}([t_{0}, T), C^{∞}) with σ_{t} uniformly equivalent to σ_{0}. This solution is unique up to t parametrization of τ, choice of A_{t}, and choice of a cross section of M_{−1} over T_{eich}.

3.3 Scheme for global existence.

If the universe is expanding the mean curvature τ starts negative and increases, the universe attains a moment of maximum expansion if it exists up to τ = 0. We choose the time parameter t by requiring that

t = −\frac{1}{τ}.  (3.3)

Then t increases from t_{0} > 0 to infinity when τ increases from τ_{0} < 0 to zero.

It results from the local existence theorem and a standard argument that the solution of the Einstein equations exists on [t_{0}, ∞) if the curve t ↦ Q(t) remains in a compact subset of T_{eich} and the norms ||γ(t, .), ω(t, .)||_{H_{2}}, ||∂_{t}γ(t, .), ∂_{t}ω(t, .)||_{H_{1}}do not blow up for any finite t.

It will result from the following sections that these norms do not blow up if it is so of the energies that we will now define. However this non blow up will be proved only for small initial data and the proof of the boundedness of Q will require the consideration of corrected energies, analogous to the corrected energies introduced in [CB-M1], but linked with the wave structure and more complicated to estimate.

4 Energies.

4.1 First energy.

4.1.1 Definition.

We denote by |.| a norm in the metric G and |.|_{g} a norm in the metrics g and G, in particular:

|u′|^{2} ≡ 2(γ′)^{2} + \frac{1}{2}e^{−4γ}(ω′)^{2} , |Du|_{g}^{2} ≡ g^{ab}(2D_{a}γD_{b}γ + \frac{1}{2}e^{−4γ}D_{a}ωD_{b}ω) (4.1)

The 2+1 dimensional Einstein equations with source the stress energy tensor of the wave map u contain the following equation (hamiltonian constraint)

2N^{−2}(T_{00} −^{(3)} S_{00}) = |u′|^{2} + |Du|_{g}^{2} + |k|_{g}^{2} − R(g) − τ^{2} = 0  (4.2)

The splitting of the covariant 2-tensor k into a trace and a traceless part:

k_{ab} = h_{ab} + \frac{1}{2}g_{ab}τ  (4.3)

gives that:

|k|_{g}^{2} = g^{ac}g^{bd}k_{ab}k_{cd} = |h|_{g}^{2} + \frac{1}{2}τ^{2}  (4.4)

and the hamiltonian constraint equation reads

|u′|^{2} + |Du|_{g}^{2} + |h|_{g}^{2} = R(g) + \frac{1}{2}τ^{2}  (4.5)

Inspired by this equation, we define the first energy by the following formula

E(t) ≡ ∫_{Σ_{t}}(I_{0} + I_{1} + \frac{1}{2}|h|_{g}^{2})μ_{g}

with

I_{0} ≡ \frac{1}{2} | u′ |^{2}≡ (γ′)^{2} + \frac{1}{4}e^{−4γ}(ω′)^{2},
I_{1} ≡ \frac{1}{2} | Du |_{g}^{2}≡| Dγ |_{g}^{2} +\frac{1}{4}e^{−4γ} | Dω |_{g}^{2}

that is:

E(t) ≡ \frac{1}{2}\{∥ u′ ∥_{g}^{2} + ∥ Du ∥_{g}^{2} + ∥ h ∥_{g}^{2}\}  (4.6)

with ∥ . ∥_{g}^{2}the square of the integral in the metric g of |.|_{g}^{2}. This energy is the first energy of the wave map u completed by the square of the L^{2}(g) norm of h.

4.2 Bound of the first energy.

The integration of the hamiltonian constraint on (Σ_{t,}g) using the constancy of τ and the Gauss Bonnet theorem which reads, with χ the Euler characteristic of Σ

∫_{Σ_{t}} R(g)μ_{g} = 4πχ  (4.7)

shows that

E(t) = \frac{τ^{2}}{4} V ol_{g}(Σ_{t}) + 2πχ, V ol_{g}(Σ_{t}) = ∫_{Σ_{t}} μ_{g}.

Recall that on a compact manifold

\frac{dV ol_{g}Σ_{t}}{dt} = \frac{1}{2} ∫_{Σ_{t}} g^{ab} \frac{∂g_{ab}}{∂t} μ_{g} = −τ ∫_{Σ_{t}} Nμ_{g}.

We use the equation

N^{−1(3)}R_{00} ≡ Δ_{g}N − N|k|_{g}^{2} + ∂_{t}τ = N|u′|^{2}

together with the splitting of k to write after integration, since τ is constant in space,

\frac{1}{2}τ^{2} ∫_{Σ_{t}} Nμ_{g} = \frac{dτ}{dt} V ol_{g}(Σ_{t}) − ∫_{Σ_{t}} N(|h|_{g}^{2} + |u′|^{2})μ_{g}

We then find as in [CB-M1] that it simplifies to:

\frac{dE(t)}{dt} = \frac{1}{2}τ ∫_{t}(|h|_{g}^{2} + |u′|^{2})Nμ_{g}.  (4.8)

We see that E(t) is a non increasing^{2} function of t if τ is negative. We remark that, due to the use of the constraints DN does not appear in 4.8, as it would have if we had used only the wave map energy.

We set

ε ≡ \{E(t)\}^{\frac{1}{2}} , ε_{0} ≡ \{E(t_{0})\}^{\frac{1}{2}} ,  (4.9)

we have proved that if τ ≤ 0 then

ε ≤ ε_{0}.  (4.10)

4.3 Second energy.

4.3.1 notations.

We denote by ˆ∇ a covariant derivative in the metrics g and G, for t dependent sections of the fiber bundle E^{p,q} with base Σ and fiber ⊗^{p}T_{x}^{∗}Σ⊗^{q} T_{u(x)}P, with P the Poincaré plane. That is we set, for ∂_{c}u^{A}, a section of E^{1,1},

ˆ∇_{b}∂_{c}u^{A} ≡ ∂_{b}∂_{c}u^{A} − Γ_{bc}^{a}∂_{a}^{A} + G_{BC}^{A}∂_{b}u^{B}∂_{c}u^{C}  (4.11)

where Γ_{bc}^{a} and G_{BC}^{A} denote respectively the connection coefficients of the metric g, and of G given in the remark 2.2. For u^{′A}, section of E^{0,1}, we have:

ˆ∇_{a}u^{′A} = ∂_{a}u^{′A} + G_{BC}^{A}∂_{a}u^{B}u^{′C}.  (4.12)

while for G_{AB}, section of E^{0,2} it holds that

ˆ∇_{a}G_{AB} = 0.  (4.13)

2The absence of the term |Du|^{2} prevents the use of this equality to obtain a decay estimate.

On the other hand we define by ˆ∂_{0} a differential operator mapping a t dependent section of a bundle E^{p,q} into another such section by the formula:

ˆ∂_{0} ˆ∇^{p}u^{A} = ¯∂_{0} ˆ∇^{p}u^{A} + G_{BC}^{A}∂_{0}u^{B} ˆ∇^{p}u^{C}  (4.14)

with ^{3}

¯∂_{0} ≡ ∂_{0} − L_{ν}  (4.15)

where L_{ν} denotes the Lie derivative with respect to the shift ν. In particular:

ˆ∂_{0}u^{′A} = ∂_{0}u^{′A} + G_{BC}^{A}∂_{0}u^{B}u^{′C}  (4.16)

and

ˆ∂_{0}G^{AB} = 0.  (4.17)

With these notations the wave map equation reads:

−ˆ∂_{0}u^{′A} + ˆ∇^{a}(N∂_{a}u^{A}) + Nτu′ = 0.  (4.18)

We will use the following lemma, which can be foreseen, and also checked^{4} by direct computation.

Lemma 4.1 The following commutation relations are satisfied:

ˆ∂_{0}∂_{a}u^{A} = ˆ∇_{a}∂_{0}u^{A},  (4.19)
ˆ∂_{0} ˆ∇_{a}∂_{0}u^{A} − ˆ∇_{a} ˆ∂_{0}∂_{0}u^{A} = R_{CB}^{A}_{D}∂_{0}u^{C}∂_{a}u^{B}∂_{0}u^{D},  (4.20)
ˆ∂_{0} ˆ∇_{a}∂_{b}u^{A} − ˆ∇_{a} ˆ∂_{0}∂_{b}u^{A} = R_{CB}^{A}_{D}∂_{0}u^{C}∂_{a}u^{B}∂_{b}u^{D} − ∂_{c}u^{A} ˆ∂_{0}Γ_{ab}^{c}.  (4.21)

We recall the identities

ˆ∂_{0}g^{ab} = ¯∂_{0}g^{ab} = 2Nk^{ab},  (4.22)
ˆ∂_{0}Γ_{ab}^{c} = ¯∂_{0}Γ_{ab}^{c} = ∇^{c}(Nk_{ab}) − ∇_{a}(Nk_{b}^{c}) − ∇_{b}(Nk_{a}^{c}).  (4.23)

3Operator on tensors denoted ˆ∂_{0} in [CB-Y].

4Used in the case of tensor fields in [CB-Y].

4.3.2 Definition.

We define the second energy by the following formula

E^{(1)}(t) ≡ ∫_{Σ_{t}}(J_{0} + J_{1})μ_{g}  (4.24)

with

J_{1} = \frac{1}{2} | Δ̂_{g}u |^{2}≡ \frac{1}{2}\{(2(Δ̂_{g}γ)^{2} + \frac{1}{2}e^{−4γ}(Δ̂_{g}ω)^{2}\}  (4.25)
J_{0} = \frac{1}{2} | ˆ∇u′ |_{g,}^{2}≡ \frac{1}{2}\{2| ˆ∇γ′|_{g}^{2} + \frac{1}{2}e^{−4γ}| ˆ∇ω′|_{g}^{2}\}.  (4.26)

4.3.3 Estimate.

We postpone the computation and estimate of the derivative of E^{(1)}(t) until after the estimates of h, λ, N and ν. We set:

E(t) ≡ ε^{2}, E^{(1)}(t) ≡ τ^{2}ε_{1}^{2}  (4.27)

4.4 Norms.

We suppose chosen a smooth cross section Q → ψ(Q) of M_{−1} over the Teichmuller space T_{eich}, together with a C^{1} curve t → Q(t). We are then given by lift to M_{−1} a regular metric σ_{t} for t ∈ [t_{0}, T], with scalar curvature -1.

Definition 4.1 Hypothesis H_{σ} : the curve is contained in a compact subset of T_{eich}.

Under the hypothesis H_{σ} the metric σ_{t} is uniformly equivalent to the metric σ_{0} ≡ σ_{t_{0}}. A t− dependent Sobolev constant on (Σ, σ_{t}) is uniformly equivalent to a number. We denote by C_{σ} any such number, which depends only on the considered compact subset of T_{eich}.

The spaces W_{s}^{p}(σ_{t}) are the usual Sobolev spaces of tensor fields on the riemannian manifold (Σ, σ_{t}). By the hypothesis on σ_{t} their norms are uniformly equivalent for t ∈ [t_{0,}, T] to the norm in W_{s}^{p}(σ_{0}) denoted simply W_{s}^{p}. We set W_{s}^{2} = H_{s}.

We have denoted by |.| a pointwise norm in the σ and G metrics; ∥ . ∥ and ∥ . ∥_{p} denote L^{2} and L^{p} norms in the σ metric.

We denote by D̂ a covariant derivative relative to the metrics σ and G. A lower case index m or M denotes respectively the lower or upper bound

of a scalar function on Σ_{t}. It may depend on t.

Lemma 4.2 It holds that:

1.

||Du||^{2} ≡ ||Du||_{g}^{2} ≤ 2ε^{2}, ||u′||^{2} ≤ e^{−2λ_{m}}||u′||_{g}^{2} ≤ 2e^{−2λ_{m}}ε^{2}.  (4.28)

2.

||D̂Du||^{2} ≤ 2e^{2λ_{M}} τ^{2}ε_{1}^{2} + ε^{2}.  (4.29)

Proof. 1. results directly from the definitions.

2. By definition

||D̂Du||^{2} = ∫_{Σ} D̂^{a}D^{b}u.D̂_{a}D_{b}uμ_{σ}  (4.30)

= ∫_{Σ}{D̂^{a}(D^{b}u.D̂_{a}D_{b}u) − D^{b}u.D̂^{a}D̂_{a}D_{b}u}μ_{σ} = − ∫_{Σ_{t}} D^{b}u.D̂^{a}D̂_{a}D_{b}uμ_{σ}

(4.31)

(since D^{b}u. ˆ∇_{a}D_{b}u is an ordinary covariant vector on Σ its divergence integrates to zero).

The Ricci commutation formula gives that, with ρ_{ab} = −\frac{1}{2}σ_{ab} the Ricci curvature of the metric σ:

D̂^{a}D̂_{a}D_{b}u^{C} = D̂^{a}D̂_{b}D_{a}u^{C} = D̂_{b}Δ̂u^{C} + ρ_{b}^{c}D_{c}u^{C} + D^{a}u^{A}D_{b}u^{B}R_{AB,}^{C}_{D}D_{a}u^{D}.

(4.32)

By another integration by parts 4.31 gives then

∫_{Σ} −D^{b}u.D̂^{a}D̂_{a}D_{b}uμ_{σ} = ∫_{Σ}{|Δ̂u|^{2}−D^{b}u.(ρ_{b}^{c}D_{c}u^{C}+D^{a}u^{A}D_{b}u^{B}R_{AB,}^{.}_{D}D_{a}u^{D})}μ_{σ}

On a 2 dimensional manifold the Riemann curvature is given by:

R_{AB,}^{C}_{D} = \frac{1}{2}R(G)\{δ_{A}^{C}G_{BD} − δ_{B}^{C}G_{AD}\}  (4.33)

A straightforward computation gives therefore

D^{b}u.D^{a}u^{A}D_{b}u^{B}R_{AB,}^{.}_{D}D_{a}u^{D} =  (4.34)
\frac{1}{2}R(G)D^{b}u^{E}D^{a}u^{A}D_{b}u^{B}D_{a}u^{D}\{G_{AE}G_{BD} − G_{BE}G_{AD}\} =  (4.35)
\frac{1}{2}R(G)\{(D^{b}u.D^{a}u)(D_{b}u.D_{a}u) − (D^{b}u.D_{b}u)(D^{a}u.D_{a}u)\} =  (4.36)

\frac{1}{2}R(G)\{|Du.Du|^{2} − | |Du|^{2}|^{2}\}  (4.37)

In the case of the Poincaré plane R(G) = −4, hence

−D^{b}u.D^{a}u^{A}D_{b}u^{B}R_{AB,}^{.}_{D}D_{a}u^{D} = 2{|Du.Du|^{2} − | |Du|^{2}|^{2}} ≤ 0, (4.38)

because

|Du.Du| ≤ |Du|^{2}.  (4.39)

5 First elliptic estimates.

The equations for h, λ, N, and ν are elliptic equations on (Σ_{t}, σ_{t}), identical with those written in [CB-M1], except that in the coefficients Du.u̇, |Du|^{2}, |u̇|^{2} which appear in these equations u is now a wave map and not a scalar function. The estimates obtained in [1] in terms of ε and ε_{1} will be valid if the new coefficients satisfy the same estimates in terms of our new ε and ε_{1}.

5.1 Basic bounds on N and λ.

The generalized maximum principle^{5} applied to the equations 2.24 and 2.23 satisfied respectively by N and λ shows that, with our choice of the time parameter

t = −τ^{−1},  (5.1)

it holds that

0 ≤ N_{m} ≤ N ≤ N_{M} ≤ 2,  (5.2)
e^{−2λ_{M}} ≤ e^{−2λ} ≤ e^{−2λ_{m}} ≤ \frac{1}{2}τ^{2}.  (5.3)

Definition 5.1 We say that the hypothesis^{6} H_{λ} is satisfied if there exists a number c_{λ} > 1, independent of t, such that

\frac{1}{\sqrt{2}}e^{λ_{M}} |τ| ≤ c_{λ}.  (5.4)

and we denote by C_{λ} any positive continuous function of c_{λ} ∈ R^{+}.

5The coefficients in these equations belong to the same functional spaces as in [CB-M1], as will be proved in the next subsection which will also estimate them.

6This hypothesis replaces the hypothesis H_{c} made on v in [CB-M1].

5.2 L^{2} estimates of || |Du|^{2}|| and || |u̇|^{2}||.

Under the hypothesis H_{σ} and H_{λ} there exist numbers C_{σ}, C_{λ} such that u satisfies the same inequalities as in the polarized case, that is:

Lemma 5.1

|| |Du|^{2}|| ≤ C_{σ}C_{λ}{ε^{2} + εε_{1}}.  (5.5)
∥ | u′ |^{2} ∥ ≤ C_{σ}C_{λ}τ^{2}ε(ε + ε_{1}).  (5.6)
∥ | u̇ |^{2} ∥ ≤ C_{σ}C_{λ}τ^{−2}ε(ε + ε_{1}).  (5.7)

Proof. A Sobolev embedding theorem applied to the scalar function |u′|^{2} gives that:^{7}

∥ | u′ |^{2} ∥ ≤ C_{σ}(∥|u′|^{2}∥_{1} + ∥D|u′|^{2}∥_{1}).

It holds that

D|u′|^{2} ≡ 2u′.D̂u′

therefore, since D̂u′ ≡ ˆ∇u′,

∥D|u′|^{2}∥_{1} ≤ 2||u′|| ||D̂u′|| ≤ 2e^{−λ_{m}}||u′||_{g} || ˆ∇u′||_{g}

Hence, since ∥|u′|^{2}∥_{1} ≡ ||u′||^{2}, and using the bound 5.3 of λ

∥ | u′ |^{2} ∥ ≤ C_{σ}e^{−λ_{m}}||u′||_{g}(e^{−λ_{m}}||u′||_{g} + ||D̂u′||_{g}) ≤ C_{σ}|τ|^{2}ε(ε + ε_{1})

By the definition of u̇ it holds that.

∥ |u̇|^{2} ∥≤ e^{4λ_{M}} ∥ |u′|^{2} ∥,

hence

∥ | u̇ |^{2} ∥ ≤ C_{σ}e^{4λ_{M}} τ^{2}ε(ε + e^{λ_{M}} |τ|ε_{1}) ≤ C_{σ}C_{λ}τ^{−2}ε(ε + ε_{1}).

On the other hand, since

D|Du|^{2} ≡ 2Du.D̂Du,  (5.8)

it holds that, using again the Sobolev embedding theorem

∥ | Du |^{2} ∥ ≤ C_{σ}∥Du∥(∥Du∥ + ∥D̂Du∥)

7See C.B 98 p. 160., here case n=2.

which gives using lemma 4.2

∥ | Du |^{2} ∥ ≤ C_{σ}ε(ε + e^{λ_{M}} |τ|ε_{1})

hence under the hypothesis H_{λ:} ∥ | Du |^{2} ∥ ≤ C_{σ}C_{λ}ε(ε + ε_{1}).

Lemma 5.2

|| |Du|^{2}||_{g} ≤ C_{σ}C_{λ}|τ|ε{ε + ε_{1}}.  (5.9)
∥ | u′ |^{2} ∥_{g} ≤ C_{σ}C_{λ}|τ|ε(ε + ε_{1}).  (5.10)

Proof. The inequalities 5.5 and 5.6 imply that

∥ | u′ |^{2} ∥_{g} ≡ (∫_{Σ_{t}} |u′|^{4}μ_{g})^{\frac{1}{2}} ≤ e^{λ_{M}} ∥ | u′ |^{2} ∥ ≤ C_{λ}C_{σ}|τ|ε(ε + ε_{1}) and, using the lower bound of λ,

∥ | Du |^{2} ∥_{g} ≡ (∫_{Σ_{t}} |Du|_{g}^{4}μ_{g})^{\frac{1}{2}} ≤ e^{−λ_{m}}∥ | Du |^{2} ∥ ≤ C_{σ}C_{λ}|τ|ε(ε + ε_{1}).

5.3 Estimate of h in H_{1}.

We have defined the auxiliary unknown h by

h_{ab} ≡ k_{ab} − \frac{1}{2}g_{ab}τ

5.3.1 Estimate of ||h||. The L^{2} norm of h on (Σ, σ) is bounded in terms of the first energy and an upper bound λ_{M} of the conformal factor since we have

∥ h ∥^{2}= ∫_{Σ_{t}} σ^{ac}σ^{bd}h_{ab}h_{cd}μ_{σ} ≤ e^{2λ_{M}} ∥ h ∥_{L^{2}(g)}^{2}≤ 2e^{2λ_{M}} ε^{2}.

5.3.2 Estimate of ∥ Dh ∥ .

The tensor h satisfies the equations

D_{a}h_{b}^{a} = L_{b} ≡ −∂_{a}u.u^{.}

It is the sum of a TT (transverse, traceless) tensor h_{T T} ≡ q and a conformal Lie derivative r:

h ≡ q + r

It results from elliptic theory that on each Σ_{t} the tensor r satisfies the estimate

∥ r ∥_{H_{1}}≤ C_{σ} ∥ Du.u^{.} ∥≤ C_{σ} ∥ |Du|^{2} ∥^{\frac{1}{2}} ∥ |u^{.}|^{2} ∥^{\frac{1}{2}} .

It results from the inequalities of the theorem 5.2 that

∥ r ∥_{H_{1}}≤ C_{σ}C_{λ}e^{2λ_{M}} |τ|ε\{(ε + ε_{1})(ε + ε_{1}e^{λ_{M}} |τ|)\}^{\frac{1}{2}}

We recall that for the transverse part h_{T T} = q it holds that

∥ Dq ∥=∥ q ∥≤∥ h ∥ + ∥ r ∥

therefore

∥ Dh ∥≤ e^{λ_{M}} ε\{\sqrt{2} + C_{σ}e^{λ_{M}} |τ|(ε + ε_{1}e^{λ_{M}} |τ|)^{\frac{1}{2}} (ε + ε_{1})^{\frac{1}{2}} \}.

Definition 5.2 We say that the hypothesis H_{E} is satisfied if there exists a positive number c_{E} such that ε + ε_{1} ≤ c_{E}. We denote by C_{E} any continuous and positive function of c_{E} ∈ R^{+}.

If the hypothesis H_{σ}, H_{λ} and H_{E} are satisfied, h verifies the inequality:

∥ Dh ∥≤ C_{λ}|τ|^{−1}ε(1 + C_{σ}C_{λ}C_{E}).

5.4 Estimates for the conformal factor λ.

The conformal factor λ satisfies on each Σ_{t} the equation

Δλ = f(λ) ≡ p_{1}e^{2λ} − p_{2}e^{−2λ} + p_{3}

where the coefficients p_{i} are given by

p_{1} = \frac{1}{4}τ^{2}, p_{2} = \frac{1}{2}(| h |^{2} + | u̇ |^{2}), p_{3} = −\frac{1}{2}(1+ | Du |^{2})

The equation admits the subsolution λ_{−} given by

e^{−2λ_{−}} = \frac{1}{2}τ^{2}

and it holds that

λ_{−} ≤ λ_{m} ≤ λ ≤ λ_{M} ≤ λ_{+}

with λ_{+} a supersolution, for example:

λ_{+} = θ + v − minv

where v is the solution with mean value zero on Σ_{t} of the linear equation

Δv = f(θ) ≡ p_{1}e^{2θ} − p_{2}e^{−2θ} + p_{3}

with^{8} e^{2θ} a t− dependent number, positive solution of the equation

p̄_{1}e^{4θ} + p̄_{3}e^{2θ} − p̄_{2} = 0,

where f̄ denotes the mean value on (Σ, σ) of a function f :

f̄ ≡ \frac{1}{V_{σ}} ∫_{Σ} fμ_{σ}, V_{σ} ≡ ∫_{Σ} μ_{σ} = −4πχ.

Using the expressions of p̄_{2}, p̄_{3} and p̄_{1} = \frac{1}{4}τ^{2}, together with

∥ u^{.} ∥^{2}≤ e^{2λ_{M}} ∥ u′ ∥_{g}^{2}, and ∥ h ∥^{2}≤ e^{2λ_{M}} ∥ h ∥_{g}^{2}

and the expression of ε^{2} ≡ E(t) we have found in [CB-M1], section 8.1 (recall that V_{σ} = −4πχ, a constant) that (we have renamed θ the t− dependent number ω of CB-M1):

0 ≤ \frac{1}{2}τ^{2}e^{2θ} − 1 ≤ V_{σ}^{−1}(1 + \frac{τ^{2}}{2} e^{2λ_{M}})ε^{2} ≤ C_{λ}ε^{2}  (5.11)

Lemma 5.3 The following inequalities hold

1.

0 ≤ λ_{M} − θ ≤ 2 ∥ v ∥_{L^{∞}} .  (5.12)

2.

1 ≤ \frac{1}{2}τ^{2}e^{2λ_{M}} ≤ 1 + C_{λ}ε^{2} + C_{λ}C_{E} ∥ v ∥_{L^{∞}} e^{4∥v∥_{L^{∞}}}  (5.13)

8We have renamed θ the function called ω in [CB-M1].

Proof. 1. It holds that

λ_{M} ≤ sup λ_{+} = θ + maxv − minv,  (5.14)

from which results 5.12.

2. The inequality 5.12 implies by elementary calculus

e^{2(λ_{M} −θ)} ≤ 1 + 4||v||_{L^{∞}}e^{4||v||_{L^{∞}}}  (5.15)

.The inequalities 5.11 and 5.15 imply that

\frac{1}{2}τ^{2}e^{2λ_{M}} ≤ (1 + C_{λ}ε^{2})(1 + 4 ∥ v ∥_{L^{∞}} e^{4∥v∥_{L^{∞}}}),  (5.16)

from which the inequality 5.13 follows.

5.4.1 Estimate of v.

The equation satisfied by v implies

∫_{Σ} |Dv|^{2}μ_{σ} = − ∫_{Σ} f(θ)vμ_{σ}

hence

∥ Dv ∥^{2}≤∥ f(θ) ∥∥ v ∥

but the Poincaré inequality applied to the function v which has mean value 0 on Σ gives

∥ v ∥^{2}≤ [Λ_{σ}]^{−1} ∥ Dv ∥^{2}

where Λ_{σ} is the first (positive) eigenvalue of - Δ_{σ} for functions on Σ_{t} with mean value zero. Therefore on each Σ_{t}

∥ Dv ∥≤ [Λ_{σ}]^{−1/2} ∥ f(θ) ∥

We use Ricci identity and R(σ) = −1 to obtain that

∥ Δ_{σ}v ∥^{2}=∥ D^{2}v ∥^{2} −\frac{1}{2} ∥ Dv ∥^{2}

The equation satisfied by v implies then, as in [1],

∥D^{2}v∥^{2} = ∥f(θ)∥^{2} + \frac{1}{2}∥Dv∥^{2}

Assembling these various inequalities gives that:

∥ v ∥_{H_{2}}≤ [1 + 3/(2Λ_{σ}) + 1/Λ_{σ}^{2}]^{1/2} ∥ f(θ) ∥

The Sobolev inequality

∥ v ∥_{L^{∞}}≤ C_{σ} ∥ v ∥_{H_{2}}

gives then a bound on the L^{∞} norm of v on Σ_{t} in terms of the L^{2} norm of f(θ), a Sobolev constant C_{σ} and the lowest eigenvalue Λ_{σ} of −Δ_{σ}, which is itself a number C_{σ}.

We now estimate the L^{2} norm of f(θ).

f(θ) ≡ f ≡ p_{1}e^{2θ} − p_{2}e^{−2θ} + p_{3}.

By the isoperimetric inequality, and since f̄ = 0, there exists a constant C_{σ} such that:

∥f∥ ≤ C_{σ}∥Df∥_{1}

We want to bound the right hand side in terms of the first and second energies of the wave map. We have by the definition of f and the expression of the p′s that:

∥Df∥_{1} ≤ \frac{1}{2}\{∥D|Du|^{2}∥_{1} + e^{−2θ}(∥D|h|^{2}∥_{1} + ∥D|u̇|^{2}∥_{1})\}

Lemma 5.4 The following estimate holds under the hypothesis H_{λ}:

\frac{1}{2} ∥ D|Du|^{2} ∥_{1}≤ C_{λ}(ε^{2} + εε_{1})

Proof. We have:

D|Du|^{2} = 2Du.D̂^{2}u

hence

∥D|Du|^{2}∥_{1} ≤ 2∥Du∥∥D̂^{2}u∥

We have seen that

∥Du∥ =∥ Du ∥_{g} , ∥D̂^{2}u∥ ≤ e^{λ_{M}} ∥Δ̂_{g}u∥_{g} + (1/\sqrt{2})∥Du∥_{g}  (5.17)

which implies the given result under the hypothesis H_{λ} and the definitions of ε and ε_{1}.

Lemma 5.5 The following estimates hold under the hypothesis H_{σ}, H_{E} and H_{λ} :

1.

\frac{1}{2}e^{−2θ} ∥ D|h|^{2} ∥_{1}≤ C_{σ}C_{E}C_{λ}ε^{2}

2.

\frac{1}{2}e^{−2θ}∥D|u̇|^{2}∥_{1} ≤ C_{E}C_{λ}C_{σ}(ε^{2} + εε_{1}).  (5.18)

Proof. 1. We have:

∥ D|h^{2}| ∥_{1}≤ 2 ∥ h ∥∥ Dh ∥

Using the inequalities of sections 5.3.1 and 5.3.2 we find that

∥ D|h^{2}| ∥_{1}≤ C_{E}C_{λ}τ^{−2}ε^{2}

The given result follows from the bound 5.11 of e^{−2θ}.

2. We recall the following estimates:

∥ | u′ |^{2} ∥ ≤ C_{λ}C_{σ}τ^{2}{ε^{2} + εε_{1}}  (5.19)

and

∥ | Du |^{2} ∥ ≤ C_{λ}C_{σ}{ε^{2} + εε_{1}}  (5.20)

and, by the Sobolev and Cauchy Schwarz inequalities

∥ |h|^{2} ∥≤ C_{σ}{∥ h ∥^{2} +||h|| ∥ Dh ∥}

which gives, using previous results

∥ |h|^{2} ∥≤ C_{σ}C_{λ}C_{E}τ^{−2}ε^{2}.  (5.21)

The estimate given in [CB-M1], lemma 21, when u is a scalar function holds when u is a wave map, , with the same proof which, as far as u is concerned, contains only norms. It gives the announced inequality.

We denote by C_{E,λ,σ} a number depending only on c_{E}, c_{λ} and the considered compact domain of T_{eich}.

Lemma 5.6 There exists a number C_{E,λ,σ} such that the L^{∞} norm of v is bounded by the following inequality

∥ v ∥_{∞}≤ C_{E,λ,σ}(ε^{2} + εε_{1})  (5.22)

Proof. Recall that there exists a Sobolev constant C_{σ} such that

∥v∥_{∞} ≤ C_{σ}{∥D|Du|^{2}∥_{1} + e^{−2ω}(∥D|h|^{2}∥_{1} + ∥D|u̇|^{2}∥_{1})}

The three terms in the sum have been evaluated in the lemma 5.4.

Theorem 5.7 1.It holds that:

1 ≤ \frac{1}{2}τ^{2}e^{2λ_{M}} ≤ 1 + C_{E,λ,σ}(ε + ε_{1})^{2},  (5.23)

2. There exists a number η_{1} > 0 such that the hypothesis H_{λ} is satisfied, i.e.:

1 ≤ \frac{1}{\sqrt{2}}|τ|e^{λ_{M}} ≤ c_{λ}, c_{λ} > 1,  (5.24)

as soon as

ε + ε_{1} ≤ η_{1}.  (5.25)

Proof. 1. The lemmas 5.3 and 5.6.

2. By 5.23 it holds that, with C_{E,λ,σ} the number of that inequality,

1 ≤ \frac{1}{\sqrt{2}}|τ|e^{λ_{M}} < c_{λ} if (ε + ε_{1})^{2} < \frac{c_{λ}^{2} − 1}{C_{E,λ,σ}}.  (5.26)

The result follows from a continuity argument.

5.4.2 Bound of λ in H_{1}.

The following theorem holds, with the same proof as the theorem 23 of [CB-M1].

Theorem 5.8 The following estimate holds

||Dλ||_{H_{1}} ≤ C_{E,λ,σ}(ε^{2} + εε_{1}).  (5.27)

6 Estimates in W_{s}^{p}.

6.1 Estimates for h in W_{2}^{p}.

The estimates of h in W_{2}^{p}, with 1 < p < 2 (for definiteness we will choose p = \frac{4}{3}) will be obtained using estimates for the conformal factor λ which have been obtained by using the H_{1} norm of h.

Theorem 6.1 Under the H hypotheses there exists a positive number C_{E,λ,σ} such that the W_{2}^{p} norm of h, choosing to be specific p = \frac{4}{3}, is bounded by

∥ h ∥_{W_{2}^{p}}≤ C_{E,λ,σ}|τ|^{−1}(ε + ε_{1})

Corollary 6.2 It holds that

|τ| ∥ h ∥_{∞}≤ C_{E,λ,σ}(ε + ε_{1})

and that

∥ h ∥_{L^{∞}(g)}≤ C_{E,λ,σ}|τ|(ε + ε_{1}).

Proof. The inequalities satisfied by ||h||_{W_{2}^{p}} in [CB-M1], with p = \frac{4}{3}, are still valid when u is a wave map, with the same proof, in particular because ∥ Du.u^{.} ∥_{\frac{4}{3}} and ∥ D(Du.u^{.}) ∥_{\frac{4}{3}} satisfy estimates of the same type as in [CB-M1]; indeed, using section 5.2:

∥ Du.u^{.} ∥_{\frac{4}{3}} ≤∥ Du ∥∥ u^{.} ∥_{4}≤ C_{E,λ,σ}|τ|^{−1}ε^{\frac{3}{2}} (ε + ε_{1})^{\frac{1}{2}} .

On the other hand a straightforward calculation gives

D(Du.u^{.}) ≡ D̂(Du).u̇ + Du.D̂u̇  (6.1)

hence

∥ D(Du.u^{.}) ∥_{\frac{4}{3}} ≤∥ D̂^{2}u ∥∥ u^{.} ∥_{4} + ∥ |Du| ∥_{4}∥ D̂u^{.} ∥

which gives, using previous estimates

∥ D(Du.u^{.}) ∥_{\frac{4}{3}} ≤ C_{λ}C_{σ}e^{λ_{M}} \{ε^{\frac{1}{2}} (ε + ε_{1})^{\frac{3}{2}} + ε^{\frac{3}{2}} (ε + ε_{1})^{\frac{1}{2}} \}

The result of the theorem follows from the bound of ε by ε + ε_{1}. The corollary is a consequence of the Sobolev embedding theorem,

∥ h ∥_{∞}≤ C_{σ} ∥ h ∥_{W_{2}^{p}} if p > 1,

and the estimate .

∥ h ∥_{L^{∞}(g)}= Sup_{Σ}\{g^{ac}g^{bd}h_{ab}h_{cd}\}^{\frac{1}{2}} ≤ e^{−2λ_{m}} ∥ h ∥_{∞}≤ \frac{1}{2}τ^{2} ∥ h ∥_{∞}

6.2 W_{3}^{p} estimates for N.

6.2.1 H_{2} estimates of N.

Theorem 6.3 There exists a number C_{E,λ,σ} such that the H_{2}norm of N satisfies the inequality

∥ 2 − N ∥_{H_{2}}≤ C_{E,λ,σ}(ε^{2} + εε_{1})

Corollary 6.4 a. It holds that:

∥ 2 − N ∥_{L^{∞}}≤ C_{E,λ,σ}(ε^{2} + εε_{1})  (6.2)

b. There exists η_{2} > 0 such that

ε + ε_{1} ≤ η_{2}  (6.3)

implies the existence of a positive number N_{m} (independent of t) such that

N ≥ N_{m} > 0.

Proof. We write, as in [CB-M1] the equation satisfied by N in the form

Δ(2 − N) − (2 − N) = β

with, having chosen the parameter t such that ∂_{t}τ = τ^{2},

β ≡ (2 − N)(e^{2λ} \frac{1}{2}τ^{2} − 1) − N(e^{2λ} | u′ |^{2} +e^{−2λ} | h |^{2})

The standard elliptic estimate applied to the form given to the lapse equation gives

∥ 2 − N ∥_{H_{2}}≤ C_{σ} ∥ β ∥

Since 0 < N ≤ 2 and e^{−2λ} ≤ \frac{1}{2}τ^{2} it holds that

∥ β ∥≤ 2(\frac{1}{2}e^{2λ_{M}} τ^{2} − 1)V_{σ}^{1/2} + 2(e^{2λ_{M}} ∥ |u′|^{2} ∥ +\frac{1}{2}τ^{2} ∥ |h|^{2} ∥)  (6.4)

The L^{4} norms of h and u′ as well as \frac{1}{2}e^{2λ_{M}} τ^{2} − 1 have been estimated in the section on the conformal factor estimate. We deduce from these estimates the bound

∥ β ∥≤ C_{E,λ,σ}(ε^{2} + εε_{1}).

which gives the result of the theorem.

The corollary a. is a consequence of the Sobolev embedding theorem, b. is a consequence of a.

6.2.2 L^{∞} estimate of DN.

Theorem 6.5 Under the hypotheses H there exist numbers C_{E}, C_{λ} and C_{σ} such that if 1 < p < 2, for instance p = \frac{4}{3}

∥ 2 − N ∥_{W_{3}^{p}}≤ C_{λ}C_{σ}C_{E}(ε^{2} + εε_{1}).  (6.5)

Corollary 6.6 The gradient of N satisfies the inequality:

∥ DN ∥_{L^{∞}(g)} | ≤ C_{λ}C_{σ}C_{E}|τ|(ε^{2} + εε_{1})  (6.6)

Proof. The proof, essentially the same as in [CB-M1] rests on the W_{1}^{p} estimate of β, since applying the standard elliptic estimate gives

∥ 2 − N ∥_{W_{3}^{p}}≤ C_{σ} ∥ β ∥_{W_{1}^{p}}  (6.7)

The estimate of

∥ β ∥_{p}≤ V_{σ}^{\frac{1}{p} − \frac{1}{2}} ∥ β ∥

is the same as in [1] theorem 28. In the estimate ∥ Dβ ∥_{p}the difference could be only in the estimate of the term ∥ D|u′|^{2} ∥_{p}. We have here

D|u′|^{2} ≡ 2u′.D̂u′  (6.8)

hence, with p = \frac{4}{3}

∥ D|u′|^{2} ∥_{p}≤ 2 ∥ u′ ∥_{4}∥ Du′ ∥ +2e^{4(γ_{M} −γ_{m})}||Dγ||_{4}||e^{−2γ}ω′||_{4}^{2}

which leads to the same estimate as in [CB-M1]:

||β||_{W_{1}^{p}} ≤ C_{E,λ,σ}(ε^{2} + εε_{1}).

The corollary is a consequence of the Sobolev embedding theorem and the relation between σ and g norms:

∥ DN ∥_{L^{∞}(g)}≤ e^{−λ_{m}} ∥ DN ∥_{∞}≤ e^{−λ_{m}}C_{σ} ∥ DN ∥_{W_{2}^{p}}≤ C_{E,λ,σ}|τ|(ε^{2} + εε_{1})

7 ∂_{t}σ and shift estimates.

7.1 ∂_{t}σ estimate.

We have chosen a section ψ of M_{−1} over Teichmuller space, denoted σ ≡ ψ(Q) and supposed (Hypothesis H_{σ}) that Q remains in a compact subset of T_{eich}.

We then have:

∂_{t}σ_{ab} = \frac{∂ψ_{ab}}{∂Q^{I}} \frac{dQ^{I}}{dt}  (7.1)

where \frac{∂ψ}{∂Q^{I}} is uniformly bounded. Hence it holds that

|∂_{t}σ| ≤ C_{σ}|\frac{dQ}{dt}|  (7.2)

We recall that Q satisfies the differential equation

X_{IJ} \frac{dQ^{I}}{dt} + Y_{IJ}P^{I} + Z_{J} = 0

where X_{IJ} ≡ ∫_{Σ_{t}} X_{I}^{ab}X_{J,ab}μ_{σ_{t}} is a matrix X with uniformly bounded inverse while Y_{IJ} and Z_{J} admit the following bounds, deduced from the basic estimates of N, λ and the L^{2} bound of r :

|Y_{IJ}| ≡ | ∫_{Σ_{t}} 2Ne^{−2λ}X_{I}^{ab}X_{J,ab}μ_{σ_{t}}| ≤ C_{σ}τ^{2}

|Z_{J}| ≡ | ∫_{Σ_{t}} 2Ne^{−2λ}r_{ab}X_{J}^{ab}μ_{σ_{t}}| ≤ C_{σ}τ^{2}||r|| ≤ C_{σ}C_{E}|τ|ε(ε + ε_{1}).

On the other hand we recall that

q_{ab} ≡ h_{ab}^{T T} ≡ X_{I,ab}P^{I}  (7.3)

hence

P^{I} ≡ (X^{−1})^{IJ} ∫_{Σ_{t}} X_{J}^{ab}q_{ab}μ_{σ}.  (7.4)

Therefore:

|P^{I}| ≤ C_{σ}||q|| ≤ C_{σ}(||h|| + ||r||) ≤ C_{σ}C_{E}|τ|^{−1}(ε + ε_{1})  (7.5)

hence

|Y_{IJ}P^{J}| ≤ C_{σ}C_{E}|τ|(ε + ε_{1})  (7.6)

We have obtained inequalities of the following type:

|\frac{dQ}{dt}| ≤ C_{σ}C_{E}|τ|(ε + ε_{1}), |∂_{t}σ| ≤ C_{σ}C_{E}|τ|(ε + ε_{1}).  (7.7)

the derivatives D^{k}∂_{t}σ satisfy inequalities of the same type.

7.2 Shift estimate.

The equation to be satisfied by the shift ν reads, with n_{a} the covariant components of the vector ν in the metric σ, i.e. n_{a} ≡ σ_{ab}ν^{b} ≡ e^{−2λ}g_{ab}ν^{b}:

(L_{σ}n)_{ab} ≡ D_{a}n_{b} + D_{b}n_{a} − σ_{ab}D_{c}n^{c} = f_{ab}  (7.8)

f_{ab} ≡ 2Ne^{−2λ}h_{ab} + ∂_{t}σ_{ab} − \frac{1}{2}σ_{ab}σ^{cd}∂_{t}σ_{cd}

The elliptic theory for this first order system gives the estimate

||n||_{W_{3}^{p}} ≡ ||ν||_{W_{3}^{p}} ≤ C_{σ}||f||_{W_{2}^{p}}.  (7.9)

with, if p > 1, using the bound 5.3 of e^{−2λ},

||f||_{W_{2}^{p}} ≤ C_{σ}C_{E}{τ^{2}||N||_{W_{2}^{p}}||λ||_{W_{2}^{p}}||h||_{W_{2}^{p}} + ||∂_{t}σ||_{W_{2}^{p}}}.  (7.10)

Hence, using previous estimates

||f||_{W_{2}^{p}} ≤ C_{σ}C_{E}|τ|(ε + ε_{1}).  (7.11)

8 Second energy estimate.

We have defined the energy E^{(1)}(t) of gradient u by the formula

τ^{2}ε_{1}^{2} ≡ E^{(1)}(t) ≡ ∫_{Σ_{t}}(J_{0} + J_{1})μ_{g}  (8.1)

with

J_{1} = \frac{1}{2} | Δ̂_{g}u |^{2}≡ \frac{1}{2}\{2(Δ̂_{g}γ)^{2} + \frac{1}{2}e^{−4γ}(Δ̂_{g}ω)^{2}\}  (8.2)
J_{0} = \frac{1}{2} | D̂u′ |_{g}^{2}≡ \frac{1}{2}\{2|D̂γ′|_{g}^{2} + \frac{1}{2}e^{−4γ}|D̂ω′|_{g}^{2}\}  (8.3)

8.1 Second energy equality.

We have:

\frac{d}{dt} ∫_{Σ_{t}}(J_{1} + J_{0})μ_{g} = ∫_{Σ_{t}}\{∂_{t}(J_{1} + J_{0}) − (Nτ − ∇_{a}ν^{a})(J_{1} + J_{0})\}μ_{g} (8.4)

On a compact manifold Σ, divergences integrate to zero, which leads to the following formula where the shift does not appear explicitly:

\frac{d}{dt} ∫_{Σ_{t}}(J_{1} + J_{0})μ_{g} = ∫_{Σ_{t}}\{∂_{0}(J_{1} + J_{0}) − Nτ(J_{1} + J_{0})\}μ_{g}  (8.5)

with, since ˆ∂_{0}G_{AB} = 0,

∂_{0}J_{1} = ˆ∂_{0}Δ̂_{g}u.Δ̂_{g}u  (8.6)

We deduce from the commutation relation of the lemma 4.1 that

ˆ∂_{0}Δ̂_{g}u^{A} = g^{ab}( ˆ∇_{a} ˆ∂_{0}∂_{b}u^{A} − ∂_{c}u^{A} ˆ∂_{0}Γ_{ab}^{c}) + ˆ∂_{0}g^{ab} ˆ∇_{a}∂_{b}u^{A} + F̂_{1}^{A}  (8.7)

with

F̂_{1}^{A} ≡ g^{ab}R_{CB}^{A}_{D}∂_{0}u^{C}∂_{a}u^{B}∂_{b}u^{D}.  (8.8)

Hence, using the identities 4.23:

ˆ∂_{0}Δ̂_{g}u^{A} = g^{ab} ˆ∇_{a} ˆ∂_{0}∂_{b}u^{A} + NτΔ̂_{g}u^{A} + F_{1}^{A} + F̂_{1}^{A},  (8.9)

with

F_{1}^{A} ≡ 2∂_{c}u^{A}(h_{g}^{ac}∂_{a}N + N∇_{a}k^{ac}) + 2Nh_{g}^{ab} ˆ∇_{a}∂_{b}u^{A}.  (8.10)

We have therefore, using Stokes formula, and ˆ∂_{0}∂_{b}u ≡ ˆ∇_{b}(Nu′)

∫_{Σ_{t}} ∂_{0}J_{1}μ_{g} = ∫_{Σ_{t}}{−N ˆ∇_{a}u′. ˆ∇^{a}Δ̂_{g}u+2NτJ_{1}−∂_{a}Nu′. ˆ∇^{a}Δ̂_{g}u+(F_{1}+F̂_{1}).Δ̂_{g}u}μ_{g}

(8.11)

On the other hand

∂_{0}J_{0.} = g^{ab} ˆ∂_{0} ˆ∇_{a}u′. ˆ∇_{b}u′ + N(h_{g}^{ab} + \frac{1}{2}g^{ab}τ)∂_{a}u′.∂_{b}u′

where we have used the identity, h_{g}^{ab} denoting the contravariant components of h_{ab} computed with the metric g,

ˆ∂_{0}g^{ab} = 2Nk^{ab} ≡ 2Nh_{g}^{ab} + Ng^{ab}τ.

The commutation relation 4.20 gives that:

g^{ab} ˆ∂_{0} ˆ∇_{a}u′. ˆ∇_{b}u′ ≡ ˆ∂_{0} ˆ∇_{a}u′. ˆ∇^{a}u′ = ˆ∇_{a} ˆ∂_{0}u′. ˆ∇^{a}u′ + F̂_{0},  (8.12)

with

F̂_{0} ≡ R_{AB,CD}u^{′D}∂_{0}u^{A}∂_{a}u^{B} ˆ∇^{a}u^{′C}  (8.13)

Therefore, using the wave map equation

−ˆ∂_{0}u′ + NΔ̂_{g} + ∂^{a}N∂_{a}u + Nτu′ = 0.  (8.14)

we find that

ˆ∂_{0} ˆ∇_{a}u′. ˆ∇^{a}u′ = ˆ∇_{a}[NΔ̂_{g}u + ∂^{c}N∂_{c}u) + Nτu′]. ˆ∇^{a}u′ + F̂_{0}  (8.15)

Therefore

∫_{Σ_{t}} ∂_{0}J_{0.}μ_{g} = ∫_{Σ_{t}}{N ˆ∇^{a}Δ̂_{g}u. ˆ∇^{a}u′ + 3NτJ_{0} + F_{0} + F̂_{0}}μ_{g}  (8.16)

with

F_{0} ≡ [∂^{a}NΔ̂_{g}u + ˆ∇^{a}(∂^{c}N∂_{c}u)]. ˆ∇_{a}u′ + τ∂^{a}Nu′. ˆ∇_{a}u′ + +Nh_{g}^{ab} ˆ∇_{a}u′. ˆ∇_{b}u′

(8.17)

We see that the third order terms in u disappear from the integral of ∂_{0}(J_{0.} + J_{1}) which reduces to

∫_{Σ_{t}} ∂_{0}(J_{0.} + J_{1})μ_{g} = ∫_{Σ_{t}}{3NτJ_{0} + 2NτJ_{1}}μ_{g} + Z_{1}  (8.18)

with

Z_{1} ≡ ∫_{Σ_{t}}{(F_{1} + F̂_{1}).Δ̂_{g}u + (F_{0} + F̂_{0}}μ_{g}  (8.19)

We have obtained

\frac{dE^{(1)}}{dt} = ∫_{Σ_{t}} Nτ(2J_{0} + J_{1})μ_{g} + Z_{1}  (8.20)

which we write

\frac{dE^{(1)}}{dt} − 2τE^{(1)} = τ ∫_{Σ_{t}} NJ_{0}μ_{g} + Z_{2} + Z_{1}  (8.21)

with

Z_{2} ≡ τ ∫_{Σ_{t}}(N − 2)(2J_{0} + J_{1})μ_{g}.  (8.22)

8.2 Second energy inequality.

Since τ is negative (and N positive) the equality 8.21 implies the inequality

\frac{dE^{(1)}}{dt} − 2τE^{(1)}(t) ≤ Z_{1} + Z_{2}.  (8.23)

We now estimate the various terms of Z_{1}, Z_{2}, called non linear terms because they are all homogeneous and cubic in h_{g}^{ab}, N −2, and the derivatives of N and u. These estimates are essentially the same as the ones given in [CB-M1], due to the estimates of the previous section. We first write, using the estimate of ||N − 2||_{L^{∞}(g)} and the definition of ε_{1} :

|Z_{2}| ≡ |τ ∫_{Σ_{t}}(N − 2)(2J_{0} + J_{1})μ_{g}| ≤ C_{λ}C_{σ}C_{E}|τ|^{3}(ε + ε_{1})^{4}.  (8.24)

We now estimate the different terms of Z_{1}, beginning with the terms X_{1} and X_{2} coming from F_{0} :

|X_{1}| ≡ | ∫_{Σ_{t}}{(∂^{a}NΔ̂_{g}u + ∂^{c}N ˆ∇^{a}∂_{c}u). ˆ∇_{b}u′}μ_{g}|

A proof analogous to the proof of the lemma 4.2 gives

| ˆ∇Du|_{g}^{2} ≤ |Δ̂_{g}u|_{g}^{2} − \frac{1}{2}R(g)|Du|_{g}^{2}.  (8.25)

The hamiltonian constraint 2.21 implies that

R(g) = |u′|^{2} + |Du|_{g}^{2} + |h|_{g}^{2} − \frac{1}{2}τ^{2} ≥ −\frac{1}{2}τ^{2}  (8.26)

therefore

|| ˆ∇Du||_{g}^{2} ≤ ||Δ̂_{g}u||_{g}^{2} + \frac{1}{4}τ^{2}||Du||_{g}^{2}  (8.27)

Using the estimate 6.6 of DN gives then

|X_{1}| ≤ C_{E,λ,σ}|τ|^{3}(ε + ε_{1})^{4}  (8.28)

The remaining, X_{2}, of the integral of F_{0} is estimated as follows

|X_{2}| = | ∫_{Σ_{t}}( ˆ∇_{a}∂^{c}N)∂_{c}u)]. ˆ∇_{b}u′μ_{g}| ≤ (||∇^{2}N||_{L^{4}(g)}|| |Du| ||_{L^{4}(g)}|| ˆ∇u′||_{g}

(8.29)

The estimates of ||∇^{2}N||_{L^{4}(g)} and || |Du| ||_{L^{4}(g)} given in [CB-M1], section 10.2.2, for the estimate of Y_{2} applies here, due to the lemma 5.1, and give

|X_{2}| ≤ C_{E,λ,σ}|τ|^{3}(ε + ε_{1})^{4}.  (8.30)

The terms in

X_{3} ≡ ∫_{Σ_{t}} F_{1}.Δ̂_{g}uμ_{g} ≡ ∫_{Σ_{t}}[2∂_{c}u(h_{g}^{ac}∂_{a}N + N∇_{a}k^{ac}) + 2Nh_{g}^{ab} ˆ∇_{a}∂_{b}u].Δ̂_{g}uμ_{g}

(8.31)

are analogous to terms found in [CB-M1] and can be estimated similarly, giving an inequality of the form

|X_{3}| ≤ C_{E,λ,σ}|τ|^{3}(ε + ε_{1})^{3}.  (8.32)

The new terms, in our unpolarized case, are

X_{4} ≡ ∫_{Σ_{t}} F̂_{1}.Δ̂_{g}uμ_{g} ≡ ∫_{Σ_{t}} g^{ab}R_{CB,AD}∂_{0}u^{C}∂_{a}u^{B}∂_{b}u^{D}Δ̂_{g}u^{A}μ_{g}  (8.33)

and

X_{5} ≡ ∫_{Σ_{t}} F̂_{0}μ_{g} ≡ ∫_{Σ_{t}} R_{AB,CD}u^{′D}∂_{0}u^{A}∂^{a}u^{B} ˆ∇_{a}u^{′C}μ_{g}  (8.34)

We have here |Riemann(G)| = 4, therefore, using ∂_{0}u ≡ Nu′ and 0 < N ≤ 2, and the Hölder inequality:

|X_{4}| ≤ 8 ∫_{Σ_{t}} |u′| |Du|_{g}^{2}|Δ̂_{g}u|_{g}μ_{g} ≤ 8|τ|ε_{1}||u′||_{L^{6}(g)}|| |Du|^{2} ||_{L^{3}(g)}. (8.35)

and

|X_{5}| ≤ 8||u′||_{L^{6}(g)}^{2}|| |Du|_{g} ||_{L^{6}(g)}|τ|ε_{1}  (8.36)

The L^{6}(g) norms can be estimated as follows. It results from the definitions that:

|| |Du|_{g} ||_{L^{6}(g)} = \{∫_{Σ_{t}} e^{−4λ}|Du|^{6}μ_{σ}\}^{\frac{1}{6}} ≤ e^{− \frac{2}{3} λ_{m}}|| |Du| ||_{6}  (8.37)

while, by the Sobolev embedding theorem

|| |Du| ||_{6} ≤ C_{σ}(||Du|| + || D|Du| ||  (8.38)

It holds that

D|Du| = \frac{D|Du|^{2}}{2|Du|} = \frac{D̂Du.Du}{|Du|}  (8.39)

hence

|D|Du| | ≤ |D̂Du|  (8.40)

and

|| |Du| ||_{6} ≤ C_{σ}(||Du|| + || D̂Du ||)  (8.41)

The inequalities of the lemma 5.1 and the lower bound of λ_{m} give then

|| |Du| ||_{L^{6}(g)} ≤ C_{σ}C_{λ}|τ|^{\frac{2}{3}} (ε + ε_{1}))  (8.42)

An analogous proof gives that

||u′||_{L^{6}(g)} ≤ e^{\frac{1}{3} λ_{M}} ||u′||_{6}  (8.43)

and

||u′||_{6} ≤ C_{σ}(||u′|| + ||D|u′| ||),  (8.44)

with

|D|u′| | ≤ |D̂u′|, ||D̂u′|| = ||D̂u′||_{g}  (8.45)

therefore, using again the lemma 5.1

||u′||_{L^{6}(g)} ≤ C_{σ}e^{\frac{1}{3} λ_{M}} |τ|(ε + ε_{1}) ≤ C_{σ}C_{λ}|τ|^{\frac{2}{3}} (ε + ε_{1}).  (8.46)

These estimates imply that

|X_{4}| ≤ C_{σ}C_{λ}|τ|^{3}(ε + ε_{1})^{3}ε_{1}  (8.47)

and the same inequality for X_{5}. We have proved the following theorem Theorem 8.1 The second energy satisfies an equality of the form

\frac{dE^{(1)}}{dt} − 2τE^{(1)}(t) ≤ |τ|^{3}B_{1}.  (8.48)

with

|B_{1}| ≤ C_{σ}C_{λ}C_{E}(ε + ε_{1})^{3}.

9 Corrected energies.

9.1 Corrected first energy

9.1.1 Definition and lower bound.

One defines as follows a corrected first energy where α is a constant, which we will choose positive:

E_{α}(t) = E(t) − ατE_{c}(t), E_{c}(t) ≡ ∫_{Σ_{t}}(u − u).u′μ_{g}  (9.1)

where we have set:

(u − u).u′ ≡ 2(γ − γ)γ′ + \frac{1}{2}e^{−4γ}(ω − ω)ω′  (9.2)

and denoted by f the mean value on (Σ_{t}, σ) of a scalar function f :

f = \frac{1}{V ol_{σ}(Σ_{t})} ∫_{Σ_{t}} fμ_{σ} ≡ −\frac{1}{4πχ} ∫_{Σ_{t}} fμ_{σ}

An estimate of E_{α} will involve second derivatives of u, it cannot alone gives a bound of the first energy E.

The Cauchy Schwarz inequality on (Σ, g) and the relation between g and σ imply that:

| ∫_{Σ_{t}}(γ − γ)γ′μ_{g}| ≤ ||γ − γ̄||_{g} ∥ γ′ ∥_{g}≤ e^{λ_{M}} ||γ − γ̄|| ||γ′||_{g}  (9.3)

Using the Poincaré inequality and recalling that Λ_{σ} denotes the first positive eigen value of the laplacian Δ_{σ} on functions with mean value zero gives the majoration (recall that ∥ Df ∥=∥ Df ∥_{g} if f is a scalar function):

| ∫_{Σ_{t}}(γ − γ)γ′μ_{g}| ≤ e^{λ_{M}} Λ_{σ}^{−1/2} ∥ Dγ ∥∥ γ′ ∥_{g}  (9.4)

an analogous procedure gives, with γ_{m} and γ_{M} the lower and upper bounds of γ :

| ∫_{Σ_{t}} e^{−4γ}(ω − ω)ω′μ_{g}| ≤ e^{2(γ_{M} −γ_{m})}e^{λ_{M}} Λ_{σ}^{−1/2} ∥ e^{−2γ}Dω ∥∥ e^{−2γ}ω′ ∥_{g} (9.5)

since

∥ Dω ∥≤ e^{2γ_{M}} ∥ e^{−2γ}Dω ∥ .

Using the definition of the G−norm we see that the inequalities 9.4, 9.5 imply:

| ∫_{Σ_{t}}(u − u).u′μ_{g}| ≤ e^{λ_{M}} Λ_{σ}^{− \frac{1}{2}} e^{2(γ_{M} −γ_{m})} ∥ Du ∥∥ u′ ∥_{g}  (9.6)

with (theorem 5.7)

e^{λ_{M}} ≤ |τ|^{−1}\{\sqrt{2} + C_{σ}C_{E}C_{λ}(ε + ε_{1})\}.  (9.7)

Lemma 9.1 It holds that:

γ_{M} − γ_{m} ≤ C_{σ}C_{E}C_{λ}{ε + ε_{1}}.

Proof. We have:

0 ≤ γ_{M} − γ_{m} ≤ 2 ∥ γ − γ ∥_{L^{∞}}

The Sobolev imbedding theorem gives therefore that:

γ_{M} − γ_{m} ≤ 2C_{σ} ∥ γ − γ ∥_{H_{2}}

hence, using again the Poincaré inequality to estimate ||γ − γ̄||,

γ_{M} − γ_{m} ≤ 2C_{σ}{(Λ_{σ}^{−1} + 1) ∥ Dγ ∥ + ∥ D^{2}γ ∥}.

It results from the definitions of the G norm and of ε that

||Dγ||^{2} ≤ \frac{1}{2}||Du||^{2} ≤ ε^{2}.  (9.8)

On the other hand since γ is a scalar function on a 2-manifold with constant scalar curvature −1, it holds that:

||D^{2}γ||^{2} = ||Δγ||^{2} + \frac{1}{2}||Dγ||^{2}  (9.9)

We have

Δγ = Δ̂γ − \frac{1}{2}e^{−4γ}|Dω|^{2}

hence

||Δγ|| ≤ ||Δ̂γ|| + \frac{1}{2}|| |e^{−2γ}Dω|^{2}||  (9.10)

with

||Δ̂γ||^{2} ≤ \frac{1}{2}||Δ̂u||^{2} ≤ \frac{1}{2}e^{2λ_{M}} ||Δ̂_{g}u||^{2} ≤ C_{λ}ε_{1}^{2},  (9.11)

and, using the lemma 5.1

|| |e^{−2γ}Dω|^{2}|| ≤ || |Du|^{2}|| ≤ C_{σ}C_{λ}{ε^{2} + εε_{1}}.  (9.12)

hence:

||D^{2}γ||^{2} ≤ C_{E}C_{λ}C_{σ}(ε_{1}^{2} + ε^{2})  (9.13)

Using the hypothesis H_{E} we deduce from all these inequalities the announced result.

We deduce from this lemma and the elementary calculus formula:

e^{2(γ_{M} −γ_{m})} ≤ 1 + 2(γ_{M} − γ_{m})e^{2(γ_{M} −γ_{m})}

that there exists an inequality of the form

e^{2(γ_{M} −γ_{m})} ≤ 1 + C_{E,λ,σ}(ε + ε_{1}).  (9.14)

We have proved that

|τ ∫_{Σ_{t}}(u − u).u′μ_{g}| ≤ \sqrt{2}Λ_{σ}^{− \frac{1}{2}} ∥ Du ∥∥ u′ ∥_{g} +A_{1}  (9.15)

with

|A_{1}| ≤ C_{E,λ,σ}ε^{2}(ε + ε_{1}).  (9.16)

Therefore:

E_{α}(t) ≥ Q_{α,Λ}(x_{0}, x_{1}) − C_{E,λ,σ}ε^{2}(ε + ε_{1}).  (9.17)

where ∥ Du ∥= x_{1}, ∥ u′ ∥_{g}= x_{0}, x = (x_{0}, x_{1}) and Q_{α,Λ} is the quadratic form

Q_{α,Λ}(x) ≡ \frac{1}{2}(x_{0}^{2} + x_{1}^{2}) − α\sqrt{2}Λ_{σ}^{− \frac{1}{2}} x_{0}x_{1}.  (9.18)

The right hand side of the inequality 9.17 can be positive only if this quadratic form is positive, that is if:

α < \frac{Λσ^{\frac{1}{2}}}{\sqrt{2}}.  (9.19)

There exists a number K > 0 such that:

Q_{α,Λ}(x) ≥ KE(t) ≡ \frac{1}{2}K(x_{0}^{2} + x_{1}^{2})  (9.20)

if the following quadratic form Q_{K} is positive definite

Q_{K}(x) ≡ (1 − K)(x_{0}^{2} + x_{1}^{2}) − 2α\sqrt{2}Λ_{σ}^{− \frac{1}{2}} x_{0}x_{1})  (9.21)

that is, under the condition 9.19 on α,

0 < K < 1 − α\sqrt{2}Λ_{σ}^{− \frac{1}{2}} .  (9.22)

There will then exist a number 0 < ℓ < K such that

E_{α}(t) ≥ ℓE(t)  (9.23)

as soon as

ε + ε_{1} ≤ η_{3}  (9.24)

with (C_{σ,E,λ} denotes the coefficient of this type in 9.17)

η_{3} = \frac{K − ℓ}{C_{σ,E,λ}} < \frac{1 − α\sqrt{2}Λ_{σ}^{− \frac{1}{2}}}{C_{σ,E,λ}} .  (9.25)

9.2 Time derivative of the corrected first energy.

We have (recall that terms involving the shift are exact divergences which integrate to zero, and we have set \frac{dτ}{dt} = τ^{2}):

\frac{dE_{α}}{dt} = \frac{dE}{dt} − ατR with R ≡\frac{dE_{c}}{dt} + τE_{c},

that is:

R ≡ ∫_{Σ_{t}}{ˆ∂_{0}u′.(u − u) + u′.ˆ∂_{0}(u − u) − Nτu′.(u − u) + τu′.(u − u)}μ_{g} (9.26)

The mapping u satisfies the wave map equation:

−ˆ∂_{0}u′ + ˆ∇^{a}(N∂_{a}u) + τNu′ = 0,  (9.27)

therefore, performing an integration by parts where we derivate u − ū as if it were a section of E^{0,1} we obtain that:

∫_{Σ_{t}}(ˆ∂_{0}u′ − Nτu′).(u − u)μ_{g} = ∫_{Σ_{t}} ˆ∇^{a}(N∂_{a}u).(u − ū)μ_{g}  (9.28)

= ∫_{Σ_{t}} −Ng^{ab}∂_{a}u.ˆ∂_{b}(u − ū)μ_{g}  (9.29)

where

ˆ∂_{b}(u^{B} − ū^{B}) ≡ ∂_{b}(u^{B} − ū^{B}) + G_{CD}^{B}∂_{b}u^{C}(u^{D} − ū^{D})  (9.30)

that is, due to the values of the coefficients G_{BC}^{A} (remark 2.2):

ˆ∂_{b}(γ − γ̄) ≡ ∂_{b}(γ − γ̄) + \frac{1}{2}e^{−4γ}∂_{b}ω(ω − ω̄),  (9.31)
ˆ∂_{b}(ω − ω̄) ≡ ∂_{b}(ω − ω̄) − 2e^{−4γ}[∂_{b}ω(γ − γ̄) + ∂_{b}γ(ω − ω̄)]  (9.32)

hence, using the expression of the metric G :

∫_{Σ_{t}} −Ng^{ab}∂_{a}u.ˆ∂_{b}(u − ū)μ_{g} = − ∫_{Σ_{t}} N{|Du|_{g}^{2} − e^{−4γ}g^{ab}∂_{a}ω∂_{b}ω(γ − γ̄)}μ_{g}

(9.33)

The non linear term can be estimated using previous results, namely (recall g^{ab}μ_{g} = σ^{ab}μ_{σ}):

| ∫_{Σ_{t}} Ne^{−4γ}g^{ab}∂_{a}ω∂_{b}ω(γ − γ̄)μ_{g}| ≤ 2||γ − γ̄|| || |e^{−2γ}Dω|^{2}||.  (9.34)

It holds that

||γ − γ̄|| ≤ Λ_{σ}^{− \frac{1}{2}} ||Dγ||, ||Dγ||^{2} ≤ ε^{2}  (9.35)

and, due to the definition of the norm in G and the lemma 5.1,

|| |e^{−2γ}Dω|^{2}|| ≤ 2|| |Du|^{2}|| ≤ C_{σ}C_{λ}(ε + ε_{1})^{2}  (9.36)

On the other hand:

ˆ∂_{0}(u − u)^{A} = ∂_{0}(u − ū)^{A} + G_{CD}^{A}∂_{0}u^{C}(u − ū)^{D}.  (9.37)

A straightforward computation using the values of the coefficients G_{CD}^{A} gives that

∫_{Σ_{t}} u′.ˆ∂_{0}(u − u)μ_{g} = ∫_{Σ_{t}}{N|u′|^{2} − Ne^{−4γ}ω^{′2}(γ − γ̄)}μ_{g}

−∂_{t}γ̄ ∫_{Σ_{t}} \frac{1}{2}γ′μ_{g} − ∂_{t}ω̄ ∫_{Σ_{t}} 2e^{−4γ}ω′μ_{g}.  (9.38)

The non linear term can be estimated as before:

∫_{Σ_{t}} Ne^{−4γ}ω^{′2}(γ − γ̄)μ_{g} ≤ 2e^{2λ_{M}} Λ_{σ}^{− \frac{1}{2}} ε|| |u′|^{2}|| ≤ C_{σ}C_{λ}ε^{2}(ε + ε_{1}) (9.39)

To bound the remaining terms we observe that for a scalar function f , since V_{σ} = −4πχ is a constant, it holds that

∂_{t}f̄ = \frac{1}{V_{σ}} ∂_{t} ∫_{Σ_{t}} fμ_{σ} = \frac{1}{V_{σ}} ∫_{Σ_{t}}\{∂_{0}f + ν^{a}∂_{a}f + \frac{1}{2}fσ^{ab}∂_{t}σ_{ab}\}μ_{σ}  (9.40)

with, for the considered metric σ

∫_{Σ_{t}} σ^{ab}∂_{t}σ_{ab}μ_{σ} = 0.  (9.41)

We write ∂_{t}f̄ under the form (recall that f′ ≡ N^{−1}∂_{0}f)

∂_{t}f̄ = \frac{1}{V_{σ}} ∫_{Σ_{t}}\{2f′ + (N − 2)f′ + ν^{a}∂_{a}f + \frac{1}{2}(f − f̄)σ^{ab}∂_{t}σ_{ab}\}μ_{σ} (9.42)

with

\frac{1}{V_{σ}} ∫_{Σ_{t}} 2f′μ_{σ} ≡ \frac{τ^{2}}{V_{σ}} ∫_{Σ_{t}} f′μ_{g} + \frac{1}{V_{σ}} ∫_{Σ_{t}}(2 − e^{2λ}τ^{2})f′μ_{σ}  (9.43)

We deduce from these equalities that

−∂_{t}γ̄ ∫_{Σ_{t}} γ′μ_{g} = −\frac{2}{V_{σ}} (∫_{Σ_{t}} γ′μ_{g})^{2} + \frac{1}{V_{σ}} X  (9.44)

with:

X ≡ \{∫_{Σ_{t}}[(2 − e^{2λ}τ^{2} + N − 2)γ′ + ν^{a}∂_{a}γ + \frac{1}{2}(γ − γ̄)σ^{ab}∂_{t}σ_{ab}]μ_{σ}\}\{∫_{Σ_{t}} γ′μ_{g}\}

(9.45)

The equality 9.44 implies the inequality:

−∂_{t}γ̄ ∫_{Σ_{t}} γ′μ_{g} ≤ \frac{1}{V_{σ}} X.  (9.46)

All the terms in X are non linear in the energies and can be estimated. Indeed:

| ∫_{Σ_{t}} γ′μ_{g}| ≤ Vg^{\frac{1}{2}} ||γ′||_{g} ≤ e^{λ_{M}} V_{σ}||γ′||_{g}  (9.47)

while (theorems 5.7 and corollary 6.4)

|2 − e^{2λ}τ^{2} + N − 2| ≤ C_{E}C_{λ}C_{σ}(ε + ε_{1})^{2}  (9.48)

and

∫_{Σ_{t}} |γ′|μ_{σ} ≤ Vσ^{\frac{1}{2}} ||γ′|| ≤ Vσ^{\frac{1}{2}} e^{−λ_{m}}||γ′||_{g},  (9.49)

Also

||ν|| ≤ C_{E}C_{σ}|τ|(ε + ε_{1})  (9.50)

and

||Dγ|| = ||Dγ||_{g} ≤ ε  (9.51)

Using section 7.1 we find that:

|σ^{ab}∂_{t}σ_{ab}| ≤ C_{σ}C_{E}|τ|(ε + ε_{1}).  (9.52)

The same type of inequalities applies to the scalar function ∂_{t}ω̄, but we must now use also the identities

e^{−2γ̄} ∫_{Σ_{t}} ω′μ_{g} ≡ ∫_{Σ_{t}} e^{−2γ}ω′e^{2(γ−γ̄)}μ_{g} ≡ ∫_{Σ_{t}}{e^{−2γ}ω′ + (e^{2(γ−γ̄)} − 1)e^{−2γ}ω′}μ_{g}

e^{2γ̄} ∫_{Σ_{t}} e^{−4γ}ω′μ_{g} ≡ ∫_{Σ_{t}} e^{−2γ}ω′e^{2(γ̄−γ)}μ_{g} ≡ ∫_{Σ_{t}}{e^{−2γ}ω′+(e^{−2(γ−γ̄)}−1)e^{−2γ}ω′}μ_{g}

to obtain an inequality which bounds −∂_{t}ω̄ ∫_{Σ_{t}} e^{−4γ}ω′μ_{g} with higher order terms in the energies, using the bound

|e^{±2(γ−γ̄)} − 1| ≤ 2|γ − γ̄|e^{2|γ−γ̄|} ≤ C_{E,λ,σ}(ε + ε_{1}).  (9.53)

We have proved that

R ≤ ∫_{Σ_{t}}{−N|Du|^{2} + N|u′|^{2} + τu′.(u − ū)}μ_{g} + A_{2}  (9.54)

with:

|A_{2}| ≤ C_{E,λ,σ}(ε + ε_{1})^{3}.  (9.55)

Theorem 9.2 There exist numbers α > 0 and k > 0 such that

\frac{dE_{α}}{dt} − kτE_{α} ≤ |τA|  (9.56)

with

|τA| ≤ |τ|C_{E,λ,σ}(ε + ε_{1})^{3}.  (9.57)

1. If Λ > \frac{1}{8} the best choice is

α = \frac{1}{4}, k = 1  (9.58)

2. Λ ≤ \frac{1}{8}. Then α and k are such that:

0 < α < \frac{4}{8 + Λ_{σ}^{−1}} ≤ \frac{1}{4}, 0 < k < 1 − \frac{1 − 4α}{(1 − 2Λ_{σ}^{−1}α^{2})^{\frac{1}{2}}}  (9.59)

A number α satisfying the conditions of the above theorem is also such

that 0 < α < \frac{Λ \frac{1}{2} ; σ}{\sqrt{2}} .

Proof. using 10.54 and the expression 4.8 of \frac{dE}{dt} we find that:

\frac{dE_{α}}{dt} ≤ τ ∫_{Σ_{t}}\{|h|^{2} + (1 − 2α)|u′|^{2} + 2α|Du|_{g}^{2}] − ατu′.(u − ū)\}μ_{g} + |τA| (9.60)

where

A ≡ αA_{1} + A_{2}.  (9.61)

with

A_{2} ≡ ∫_{Σ_{t}} \frac{1}{2}(N − 2)[|h|^{2} + (1 − 2α)|u′|^{2} + 2α|Du|_{g}^{2}]μ_{g}  (9.62)

We deduce from the corollary 6.4 (L^{∞} estimate of N − 2) that A_{2} satisfies the same type of estimate than A_{1}, hence:

|τA| ≤ |τ|C_{E}C_{λ}C_{σ}(ε + ε_{1})^{3}.  (9.63)

We look for a positive number k such that the difference \frac{dE_{α}}{dt} − kτE_{α} can be estimated with higher order terms in the energies. We deduce from 9.60 that:

\frac{dE_{α}}{dt} − kτE_{α} ≤ τ\{||h||_{g}^{2} + (1 − 2α − \frac{k}{2})||u′||_{g}^{2} + (2α − \frac{k}{2})||Du||_{g}^{2} (9.64)

+α ∫_{Σ_{t}} |τ|(1 − k)u′.(u − ū)μ_{g}} + |τA|  (9.65)

We have seen that:

|τ ∫_{Σ_{t}} u′.(u − u)μ_{g}| ≤ \sqrt{2}Λ_{σ}^{−1/2}||u′||_{g}||Du||_{g} + A_{1},  (9.66)

Since τ < 0, it will hold that

\frac{dE_{α}}{dt} − kτE_{α} ≤ |τA|, A ≡ A_{1} + A_{2} + A_{3}.  (9.67)

if the quadratic form

Q_{α,k}(x) ≡ (1 − 2α − \frac{k}{2})x_{0}^{2} + (2α − \frac{k}{2})x_{1}^{2} − α(1 − k)\sqrt{2}Λ_{σ}^{−1/2}x_{0}x_{1} (9.68)

is non negative.

The quadratic form Q_{α,k} is non negative if:

k ≤ 4α, and k ≤ 2(1 − 2α)  (9.69)

and k is such that its discriminant is negative, that is:

2α^{2}Λ_{σ}^{−1}(1 − k)^{2} − 4(2α − \frac{k}{2})(1 − 2α − \frac{k}{2}) ≤ 0  (9.70)

The inequalities 9.69 imply

k ≤ 1,  (9.71)

The inequality 9.70 reads

(1 − 2Λ_{σ}^{−1}α^{2})k^{2} − (1 − 2Λ_{σ}^{−1}α^{2})2k − 2Λ_{σ}^{−1}α^{2} + 8α(1 − 2α) > 0 (9.72)

We have already supposed that 1 − 2Λ_{σ}^{−1}α^{2} > 0, the inequality above is therefore equivalent to:

k^{2} − 2k + 1 − \frac{(1 − 4α)^{2}}{(1 − 2Λ_{σ}^{−1}α^{2})} > 0  (9.73)

that is

k < 1 − \frac{1 − 4α}{(1 − 2Λ_{σ}^{−1}α^{2})^{\frac{1}{2}}}  (9.74)

There will exist such a k > 0 if

\frac{1 − 4α}{(1 − 2Λ_{σ}^{−1}α^{2})^{\frac{1}{2}}} < 1  (9.75)

Since α > 0 this inequality reduces to:

−2Λ_{σ}^{−1}α − 16α + 8 > 0.  (9.76)

i.e.

α < \frac{4}{8 + Λ_{σ}^{−1}}  (9.77)

We remark that this inequality imposes the hypothesis first made on α, since elementary calculus shows that, for any Λ, it holds that:

\frac{Λ^{\frac{1}{2}}}{\sqrt{2}} ≤ \frac{4}{8 + Λ^{−1}} ,  (9.78)

the inequality being attained only for Λ = \frac{1}{8}.

We distinguish two cases

1. Λ_{σ} > \frac{1}{8}. In this case it is possible to take α = \frac{1}{4}, k = 1 and obtain immediately

\frac{dE_{\frac{1}{4}}}{dt} − τE_{\frac{1}{4}} ≤ |τA|.  (9.79)

2. Λ_{σ} ≤ \frac{1}{8}. We have then:

\frac{4}{8 + Λ_{σ}^{−1}} ≤ \frac{1}{4}.  (9.80)

We choose α such that it satisfies the inequality 9.77, which implies in this case α < \frac{1}{4}, and then k > 0 such that it satisfies 9.74.

10 Corrected second energy.

We define a corrected second energy E_{α}^{(1)} by the formula

E_{α}^{(1)}(t) = E^{(1)}(t) + ατE_{c}^{(1)}(t), E_{c}^{(1)}(t) ≡ ∫_{Σ_{t}} Δ̂_{g}u.u′μ_{g}  (10.1)

10.1 Lower bound.

We have, according to previous notations,

Δ̂_{g}u.u′ ≡ 2Δ_{g}γγ′ + \frac{1}{4}e^{−4γ}Δ_{g}ωω′ + b_{1}  (10.2)

with

b_{1} ≡ e^{−4γ}g^{ab}(∂_{a}ω∂_{b}ωγ′ − ∂_{a}γ∂_{b}ωω′).  (10.3)

hence, using the lemma 5.2:

B_{1} ≡ | ∫_{Σ_{t}} b_{1}μ_{g}| ≤ |τ|C_{E,λ,σ}(ε + ε_{1})^{3}.  (10.4)

The Cauchy Schwarz inequality and the Poincaré inequality (γ̄′ is a constant on Σ_{t} and on a compact manifold ∫_{Σ_{t}} Δ_{g}γμ_{g} = 0) give that:

| ∫_{Σ_{t}} Δ_{g}γγ′μ_{g}| = | ∫_{Σ_{t}} Δ_{g}γ(γ′ − γ̄′)μ_{g}| ≤ e^{λ_{M}} Λ_{σ}^{−1/2}||Δ_{g}γ||_{g}||Dγ′|| (10.5)

while

| ∫_{Σ_{t}}(Δ_{g}ω)e^{−4γ}ω′μ_{g}| = | ∫_{Σ_{t}} Δ_{g}ω(e^{−4γ}ω′ − \overline{e^{−4γ}ω′})μ_{g}|
≤ e^{λ_{M}} Λ_{σ}^{−1/2}||Δ_{g}ω||_{g}||D(e^{−4γ}ω′)||

It holds that

||D(e^{−4γ}ω′)|| = ||e^{−4γ}(Dω′ − 4Dγω′)||  (10.6)
≤ e^{−2γ_{m}}(||e^{−2γ}Dω′)|| + 4||e^{−2γ}ω′||_{4}||Dγ||_{4}  (10.7)

while

||Δ_{g}ω||_{g} ≤ e^{2γ_{M}} ||e^{−2γ}Δ_{g}ω||_{g}  (10.8)

Using the bound (lemma 9.1) of γ_{M} − γ_{m} and the inequalities on the L^{4} norms ||.||_{4} (lemma 5.1), we find an inequality of the form:

| ∫_{Σ_{t}}(Δ_{g}ω)e^{−4γ}ω′μ_{g}|| ≤ e^{λ_{M}} Λ_{σ}^{−1/2}||e^{−2γ}Δ_{g}ω||_{g}||e^{−2γ}Dω′)|| + B_{2}

where B_{2} satisfies an inequality of the same type as B_{1}. We have shown that

| ∫_{Σ_{t}} Δ_{g}u.u′μ_{g}| ≤ e^{λ_{M}} Λ_{σ}^{−1/2}||Δ_{g}u||_{g}||Du′|| + B_{1} + B_{2}.

The estimates of the lemma 5.1 and the inequalities

||Δ_{g}u||_{g} ≤ ||Δ̂_{g}u||_{g} + || |Du|^{2}||_{g}, ||Du′|| ≤ || ˆ∇u′|| + || |Du|^{2}||g^{\frac{1}{2}} || |u′|^{2}||g^{\frac{1}{2}}

(10.9)

give

e^{λ_{M}} Λ_{σ}^{−1/2}||Δ_{g}u||_{g}||Du′|| ≤ e^{λ_{M}} Λ_{σ}^{−1/2}||Δ̂_{g}u||_{g}|| ˆ∇u′|| + B_{3}

with

B_{3} ≤ |τ|C_{E,λ,σ}(ε + ε_{1})^{3}.  (10.10)

Using the estimate 6.23 of e^{λ_{M}} |τ| − \sqrt{2} we have:

| ∫_{Σ_{t}} Δ_{g}u.u′μ_{g}| ≤ \sqrt{2}|τ|^{−1}Λ_{σ}^{−1/2}||Δ̂_{g}u||_{g}|| ˆ∇u′|| + B_{1} + B_{2} + B_{3} + B_{4}. (10.11)

with

B_{4} = |e^{λ_{M}} − \sqrt{2}|τ|^{−1}|Λ_{σ}^{−1/2}||Δ̂_{g}u||_{g}|| ˆ∇u′|| ≤ |τ|C_{σ}(ε + ε_{1})^{3}

We deduce from these estimates, with Q_{α,Λ} the same quadratic form as in 9.18 but with y_{1} ≡ |τ|^{−1}||Δ̂_{g}u||_{g}, y_{0} ≡ |τ|^{−1}|| ˆ∇u′||, that:

τ^{−2}E_{α}^{(1)}(t) ≥ Q_{α,Λ}(y_{0}, y_{1}) − C_{E,σ,λ}(ε + ε_{1})^{3}  (10.12)

Theorem 10.1 If α is chosen satisfying 9.19 there exists a number η_{4} > 0 and L > 0 such that

E_{α} + τ^{−2}E_{α}^{(1)} ≥ L(ε^{2} + ε_{1}^{2})  (10.13)

as soon as

ε + ε_{1} ≤ η_{4}.  (10.14)

Proof. We have found that

ψ(t) ≡ E_{α}(t) + τ^{−2}E_{α}^{(1)}(t) ≥ Q_{α,Λ}(y, x) − (A + B)  (10.15)

where Q_{α,Λ}(x, y) is the quadratic form

Q_{α,Λ}(x, y) ≡ Q_{α,Λ}(x) + Q_{α,Λ}(y).  (10.16)

and A + B admits a bound of the form

|A + B| ≤ C_{E,σ,λ}(ε^{2} + ε_{1}^{2})^{\frac{3}{2}} .  (10.17)

We have

Q_{α,Λ}(x, y) > K(ε^{2} + ε_{1}^{2}) ≡ \frac{1}{2}K(x_{0}^{2} + x_{1}^{2} + y_{0}^{2} + y_{1}^{2})  (10.18)

if the quadratic form Q_{K} defined in the section 9.1 is positive definite. The conditions on α and the corresponding limitation on K as the same as in the section 9.1, and the proof continues along the same line.

10.2 Decay of the second corrected energy.

We have (recall \frac{dτ}{dt} = τ^{2}) :

\frac{dE_{α}^{(1)}}{dt} ≡ \frac{dE^{(1)}}{dt} + ατR^{(1)} R^{(1)} ≡ \frac{d}{dt}E_{c}^{(1)} + τE_{c}^{(1)}

that is:

R^{(1)} = ∫_{Σ_{t}}{ˆ∂_{0}Δ̂_{g}u.u′ + Δ̂_{g}u.(ˆ∂_{0}u′ − Nτ.u′ + τu′)}μ_{g}.  (10.19)

We have found in lemma 4.1 that

ˆ∂_{0}Δ̂_{g}u^{A} ≡ g^{ab} ˆ∇_{a} ˆ∂_{0}∂_{b}u^{A} + NτΔ̂_{g}u^{A} + F_{1}^{A} + F̂_{1}^{A},  (10.20)

with

F̂_{1}^{A} ≡ g^{ab}R_{CB}^{A}_{D}∂_{0}u^{C}∂_{a}u^{B}∂_{b}u^{D}

and

F_{1}^{A} ≡ 2∂_{c}u^{A}(h_{g}^{ac}∂_{a}N + N∇_{a}k^{ac}) + 2Nh_{g}^{ab} ˆ∇_{a}∂_{b}u^{A}  (10.21)

that is, using the equation

(3)R_{0}^{c} ≡ −N∇_{a}k^{ac} = ∂_{0}u.∂^{c}u
F_{1}^{A} ≡ 2∂_{c}u^{A}(h_{g}^{ac}∂_{a}N − ∂_{0}u.∂^{c}u) + 2Nh_{g}^{ab} ˆ∇_{a}∂_{b}u^{A}.  (10.22)

Partial integration gives, using also the identity ˆ∂_{0}∂_{b}u ≡ ˆ∇_{b}∂_{0}u ≡ ˆ∇_{b}(Nu′),

∫_{Σ_{t}}(ˆ∂_{0}Δ̂_{g}u).u′μ_{g} = ∫_{Σ_{t}}{−N| ˆ∇u′|_{g}^{2}−∂^{a}N ˆ∇_{a}u′.u′+NτΔ̂_{g}u.u′+(F_{1}+F̂_{1}).u′}μ_{g}

(10.23)

On the other hand, if u satisfies the equation

−ˆ∂_{0}u′ + ˆ∇^{a}(N∂_{a}u) + τNu′ = 0  (10.24)

it holds that:

∫_{Σ_{t}} Δ̂_{g}u.(ˆ∂_{o}u′ − τNu′)μ_{g} = ∫_{Σ_{t}}{N|Δ̂_{g}u|^{2} + ∂^{a}N∂_{a}u.Δ̂_{g}u}μ_{g} We have found that:

R^{(1)} = ∫_{Σ_{t}}{−N|D̂u′|^{2} + N|Δ̂_{g}u|^{2} + τ(N + 1)Δ̂_{g}u.u′}μ_{g} + R̃^{(1)} (10.25)

with

R̃^{(1)} ≡ ∫_{Σ_{t}}{−∂^{a}N ˆ∇_{a}u′.u′ + (F_{1} + F̂_{1}).u′ + ∂^{a}N∂_{a}u.Δ̂_{g}u}μ_{g}.  (10.26)

Using the expression of \frac{dE^{(1)}}{dt} we find that:

\frac{dE_{α}^{(1)}}{dt} = τ ∫_{Σ_{t}}\{N(2−2α)J_{0}+N(2α+1)J_{1}+(N +1)ατΔ̂_{g}u.u′]\}μ_{g}+Z_{1}+ατR̃^{(1)}

which implies:

\frac{dE_{α}^{(1)}}{dt} − (2 + k)τE_{α}^{(1)} = τ\{∫_{Σ_{t}}\{(2N − 2 − k − 2αN)J_{0} + (2αN + N − 2 − k)J_{1}

+ατ(N + 1 − 2 − k)Δ̂_{g}u.u′}μ_{g} + Z_{1} + ατR̃^{(1)}  (10.27)

which we write:

\frac{dE_{α}^{(1)}}{dt} − (2 + k)τE_{α}^{(1)} ≤ τ\{∫_{Σ_{t}}\{(2 − k − 4α)J_{0} + (4α − k)J_{1}

+ατ(1 − k)Δ̂_{g}u.u′}μ_{g} + Z_{1} + ατR̃^{(1)} + Z_{2}.  (10.28)

with Z_{2} ≡ τ ∫_{Σ_{t}}{(N − 2)(2 − 2α)J_{0} + (N − 2)(2α + 1)J_{1} + (N − 2)ατΔ̂_{g}u.u′]}μ_{g}.

(10.29)

We have found in section 8.2 that

|Z_{1}| ≤ |τ|^{3}C_{E,λ,σ}(ε + ε_{1})^{3}  (10.30)

It results immediately from the estimate of N −2, section 6.2, that Z_{2} satisfies an inequality of the same type.

Some terms of R̃^{(1)} are bounded using the L^{∞}(g) estimate 6.6 of DN, which gives that

| ∫_{Σ_{t}}{−∂^{a}N ˆ∇_{a}u′.u′ + ∂^{a}N∂_{a}u.Δ̂_{g}u}μ_{g}| ≤ τ^{2}C_{E,σ,λ}ε^{2}ε_{1}(ε + ε_{1}). (10.31)

The estimate of the remaining ones uses similar techniques as those of section 9 and lead to an inequality of the form

|R̃^{(1)}| ≤ τ^{2}C_{E,σ,λ}(ε + ε_{1})^{4}.  (10.32)

Theorem 10.2 Under the conditions on α and k given in the theorem 9.2 the following inequality holds:

\frac{dE_{α}^{(1)}}{dt} − (2 + k)τE_{α}^{(1)} ≤ |τ|^{3}B  (10.33)

with

|B| ≤ C_{E,λ,σ}(ε + ε_{1})^{3}.

Proof. We have seen that (10.11)

| ∫_{Σ_{t}} τΔ̂_{g}u.u′μ_{g}| ≤ \sqrt{2}Λ_{σ}^{− \frac{1}{2}} ||Δ̂_{g}u||_{g} || ˆ∇u′||_{g} + Z_{3}  (10.34)

with

|Z_{3}| ≤ τ^{2}C_{E,σ,λ}(ε + ε_{1})^{3}.  (10.35)

Therefore we deduce from 10.28 and the definition of y_{0}, y_{1} that

\frac{dE_{α}^{(1)}}{dt} − (2 + k)τE_{α}^{(1)} ≤ τ^{3}Q_{α,k}^{(1)}(y) + |τ|^{3}B

with

B ≡ Z_{1} + ατR̃^{(1)} + Z_{2} + |τ|αZ_{3}, |B| ≤ C_{E,σ,λ}(ε^{2} + ε_{1}^{2})^{\frac{3}{2}}  (10.36)

and

Q_{α,k}^{(1)}(y) ≡ (1 − \frac{k}{2} − 2α)y_{0}^{2} + (2α − \frac{k}{2})y_{1}^{2} + \sqrt{2}Λ_{σ}^{− \frac{1}{2}} ατ(1 − k)y_{0}y_{1}. (10.37)

This quadratic form in y is non negative under the same conditions as the form Q_{α,k}(x). The conclusion follows, since τ < 0.

11 Decay of the total energy.

We make the following a priori hypothesis, for all t ≥ t_{0} for which the considered quantities exist

• Hypothesis H_{σ} : 1. The t dependent numbers C_{σ} are uniformly bounded by a constant M.

2. There exist Λ > 0 such that Λ_{σ} ≥ Λ.

• We choose α such that

α = \frac{1}{4} if Λ > \frac{1}{8}, α < \frac{4}{8 + Λ^{−1}} ≤ \frac{1}{4} if Λ ≤ \frac{1}{8}.  (11.1)

• Hypotheses H_{E}^{η} : The t dependent energies ε^{2} and ε_{1}^{2} having been supposed bounded by a number c_{E} we suppose, moreover that they satisfy the inequalities 5.25, 6.3, 9.25, 10.14.

We have seen (theorem 5.7) that the hypothesis H_{λ} is then satisfied.

We denote by M_{i} any given positive number dependent on the bounds of these H’s hypothesis but independent of t.

We have defined ψ(t) to be the total corrected energy namely:

ψ(t) ≡ E_{α}(t) + τ^{−2}E_{α}^{(1)}

We have seen (10.13) that ψ(t) bounds the total energy ϕ(t) ≡ ε^{2} + ε_{1}^{2} by an inequality of the form:

ϕ(t) ≡ ε^{2} + ε_{1}^{2} ≤ M_{0}ψ(t), M_{0} = L^{−1}.  (11.2)

Lemma 11.1 Under the hypotheses H_{σ} and H_{E}^{η} the function ψ satisfies a differential inequality of the form

\frac{dψ}{dt} ≤ −\frac{k}{t} (ψ − M_{1}ψ^{3/2})  (11.3)

Proof. The inequalities 9.56 and 10.33 together with the choice τ = −\frac{1}{t}, and the bound 10.13.

Theorem 11.2 Under the hypotheses H_{σ} and H_{E}^{η} there exists a number k > 0 such that the total energy E_{tot}(t) ≡ ϕ(t) ≡ ε^{2} + ε_{1}^{2} satisfies an estimate of the form

t^{k}ϕ(t) ≤ M_{2}ϕ(t_{0})  (11.4)

if it is small enough initially.

Proof. We suppose that ψ_{0} ≡ ψ(t_{0}) satisfies

ψ_{0}^{1/2} < \frac{1}{M_{1}}  (11.5)

Then ψ starts decreasing, continues to decrease as long as it exists, therefore (ψ − M_{1}ψ^{3/2}) > 0 and the inequality 11.3 is equivalent to

\frac{dz}{z − M_{1}z^{3}} + \frac{k}{2} \frac{dt}{t} ≤ 0, with ψ = z^{2}.

This inequality gives by integration:

log\{\frac{z}{(1 − M_{1}z)z_{0}} \frac{(1 − M_{1}z_{0})}{z_{0}}\} + \frac{1}{2}k log \frac{t}{t_{0}} ≤ 0

equivalently

\{\frac{z}{(1 − M_{1}z)} \frac{(1 − M_{1}z_{0})}{z_{0}}\}\{\frac{t}{t_{0}} \}^{\frac{1}{2} k} ≤ 1  (11.6)

and, a fortiori,

t^{k}ψ ≤ \frac{t_{0}^{k}ψ_{0}}{(1 − M_{1}z_{0})^{2}} .  (11.7)

Hence, using 11.2, the decay estimate

t^{k}ϕ(t) ≤ M_{2}ϕ_{0}.  (11.8)

12 Teichmüller parameters.

Instead of considering as in [CB-M1], [CB-M2] the Dirichlet energy of the metric σ we use directly the estimate 7.7 of dQ/dt which we now write, using 11.8:

|\frac{dQ}{dt}| ≤ C_{σ,E}t^{−(1+ \frac{k}{2} )}ϕ(t_{0})^{1/2}  (12.1)

Therefore:

Theorem 12.1 There exists M_{3} such that

|Q(t) − Q(t_{0})| ≤ M_{3}ϕ(t_{0})^{1/2}.  (12.2)

13 Global existence.

Theorem 13.1 Let (σ_{0}, q_{0}) ∈ C^{∞}(Σ_{0}) and (u_{0}, u^{.}_{0}) ∈ H_{2}(Σ_{0}, σ_{0})×H_{1}(Σ_{0}, σ_{0}) be initial data for the Einstein equations with U(1) isometry group on the initial manifold M_{0} ≡ Σ_{0} × U(1) , with Σ_{0} compact, orientable and of genus greater than one, σ_{0} chosen such that R(σ_{0}) = −1. Suppose the initial integral condition 2.1 (with n = 0) satisfied. Then there exists a number η_{0} > 0 such that if

ϕ(t_{0}) ≡ E_{tot}(t_{0}) < η_{0}  (13.1)

these Einstein equations have a solution on M × [t_{0}, ∞), with initial values determined by σ_{0}, q_{0}, u_{0}, u^{.}_{0}. The parameter t is t = −τ^{−1}, with τ the mean extrinsic curvature of Σ × {t} in the lorentzian metric ^{(3)}g on Σ × [t_{0}, ∞).

This solution is unique^{9} up to the choice of a section of Tecichmuller space and a gauge choice for A.

Proof. We first prove that E_{tot}(t) is uniformly bounded, and decays to zero (without a priori hypothesis). We have obtained in the previous sections, under the hypotheses H_{E}^{η} and H_{σ}, the following result: there are numbers M_{i} depending only on c_{E} and c_{σ} such that

t^{k}E_{tot}(t) ≤ M_{2}E_{tot}(t_{0})  (13.2)

and

|Q(t) − Q(t_{0})| ≤ M_{3}ϕ(t_{0})^{1/2}.  (13.3)

Now consider the pair of t dependent numbers

(ϕ(t), ζ(t)), ζ(t) ≡ |Q(t) − Q(t_{0})|

The inequalities 13.3, 13.4 show that the hypothesis (where c_{E} satisfies H_{E}^{η})

ϕ(t) ≤ c_{E}, ζ(t) ≤ c_{σ}

9The global uniqueness theorem of CB-Geroch says that it is geometrically unique in the class of globally hyperbolic spacetimes.

imply that there exists η_{0} > 0 such that ϕ(t_{0}) ≤ η_{0} implies that the pair belongs to the subset U_{1} ⊂ R^{2} defined by the inequalities:

U_{1} ≡ { ϕ(t) < c_{E}, ζ(t) < c_{σ}}.

Therefore for such an η_{0} the pair belongs either to U_{1} or to the subset U_{2} defined by

U_{2} ≡ { ϕ(t) > c_{E} or ζ(t) > c_{σ}}

These subsets are disjoint. We have supposed that for t = t_{0} it holds that (ϕ(t_{0}), ζ(t_{0})) ∈ U_{1} hence, by continuity in t, ( ϕ(t), ζ(t)) ∈ U_{1} for all t.

We have now proved that the total energy is uniformly bounded, and σ_{t} uniformly equivalent to σ_{0}.

To complete the proof of existence of the spacetime for t ∈ [t_{0}, ∞) we need the following lemma.

Lemma. The H_{2} norm of the pair of scalar functions (γ, ω) is uniformly bounded, as well of the H_{1} norm of (∂_{t}γ, ∂_{t}ω).

Proof of lemma. We have already proven in section 9 the uniform bound of ||Dγ|| and ||D^{2}γ|| in terms of the total energy. On the other hand it holds that

γ − γ_{0} = ∫_{t_{0}}^{t} ∂_{t}γdt  (13.4)

hence

||γ − γ_{0}|| ≤ ∫_{t_{0}}^{t} ||∂_{t}γ||dt  (13.5)

Using previous estimates, the fall off of the energy and the property

||∂_{t}γ|| ≤ e^{−λ_{m}}||∂_{t}γ||_{g}  (13.6)

we find that there exists a number M such that

||γ − γ_{0}|| ≤ M ∫_{t_{0}}^{t} t^{−(1+k)}dt,  (13.7)

which completes the proof of the uniform bound of ||γ||_{H_{2}}, hence also of γ in C^{0}.

When γ_{M} is uniformly bounded one can bound ||Dω|| ≤ e^{2γ_{M}} ||e^{−2γ}Dω|| with the first energy and ||D^{2}ω|| with the total energy, in a manner analogous as the one used for ||D^{2}γ||. We just recalled the estimate of ||∂_{t}γ||, the estimate of ||∂_{t}ω|| is similar, when γ has been bounded. It is also easy to bound ||D∂_{t}ω|| and ||D∂_{t}γ||.

Corollary 13.2 1. This solution is globally hyperbolic, future timelike and null complete.

2. It is asymptotic to a flat solution:

(4)g = −4dt^{2} + 2t^{2}σ_{∞} + θ_{∞}^{2}  (13.8)

with σ_{∞} a metric on Σ independent of t and of scalar curvature −1, and θ_{∞} a 1-form on Σ × S^{1} of the type

θ_{∞} = C(dx^{3} + H),  (13.9)

where C is a constant and H is a harmonic 1-form on (Σ, σ_{0}).

Proof. 1. The orthogonal trajectories to the space sections M ×{t} have an infinite proper length since the lapse N is bounded below by a strictly positive number. It can be checked that the conditions given in C.B and Cotsakis for global hyperbolicity, and for future and null completeness are satisfied by (Σ × R,^{(3)} g).

2. The theorem 5.7 and the decay of ε + ε_{1} show that λ tends to 2t^{2} in C^{0} norm when t tends to infinity.

The decay estimate of \frac{dQ}{dt} show that Q tends to a point Q_{∞} in T_{eich} when t tends to infinity, σ tends to σ(Q_{∞}).

The lapse and shift estimates 6.2 and 7.9, 7.11 show that N tends to 2 and ν tends to zero in C^{0} norm when t tends to infinity.

The integral formula for γ shows that γ(t, .) − γ_{0}(.) tends to a function on Σ, γ̂_{∞}(.), in L^{2} norm when t tends to infinity, hence γ tends to γ_{∞} = γ_{0} + γ̂_{∞} in this norm, therefore a fortiori γ(t, .) tends to γ_{∞}(.) in the sense of distributions on Σ. We know on the other hand that ||Dγ|| tends to zero, hence Dγ tends to zero in the sense of distributions. Since derivation in this sense is a continuous operator it holds that Dγ_{∞} = 0, therefore γ_{∞} is a constant.

An analogous reasonning holds for ω. The value of ω_{∞} does not appear in the expression of F.

The estimates of section 2.2 of Â and A_{t} (in Coulomb gauge) show that they both tend to zero in C^{0} norm on Σ. The differential formula giving the c_{i}(t) shows then that the 1 form Ã tends in C^{0} norm to the harmonic form on Σ, H_{∞} = c_{i,∞}H_{(i)}The spacetime metric is asymptotic to the metric

(4)g = e^{−2γ_{∞}}(−4dt^{2} + 2t^{2}σ_{∞}) + e^{2γ_{∞}}(dx^{3} + H_{∞})^{2}  (13.10)

which takes the indicated form by rescaling of t.

Aknowledgements. We thank the University of the Aegean in Samos, the Schrodinger Institute in Vienna and the Institut des Hautes Etudes Scientifiques in Bures sur Yvette which made fruitful discussions with V. Moncrief possible.

References

[A-M-T] L. Andersson, V. Moncrief and A. Tromba On the global evolution problem in 2+1 gravity J. Geom. Phys. 23 1997 n^{∘}3 − 4,1991-205

[CB1] Y. Choquet-Bruhat Global wave maps on curved spacetimes, in ”Mathematical and Quantum Aspects of Relativity and Cosmology” Cotsakis and Gibbons ed. LNP 535, Springer 1998, 1-30

[CB2] Y. Choquet-Bruhat Wave maps in General Relativity in ”On Einstein path” A. Harvey ed. Springer 1998, 161-185.

[CB-C] Y. Choquet-Bruhat and S. Cotsakis Global hyperbolicity and completeness, J. Geom. Phys. 43 n^{∘}4, 2002, 345-350.

[CB-M 1] Y. Choquet-Bruhat and V. Moncrief, Future global in time einsteinian spacetimes with U(1) isometry group Ann. Henri. Poincaré 2 (2001), 1007-1064.

[CB-M 2] Y. Choquet-Bruhat and V. Moncrief, Non linear stability of einsteinian spacetimes with U(1) isometry group, in Partial differential equations and mathematical physics, in honor of J. Leray, Kajitani and Vaillant ed. to appear 2003, Birkahauser.

[CB-M 3] Y. Choquet-Bruhat and V. Moncrief Existence theorem for solutions of Einstein equations with 1 parameter spacelike isometry group, Proc. Symposia in Pure Math, 59, 1996, H. Brezis and I.E. Segal ed. 67-80.

[F-T] A. Fisher and A. Tromba Teichmuller spaces, Math Ann. 267 1984, 311-345, or Y. Choquet-Bruhat and C DeWitt-Morette Analysis Manifolds and Physics II enlarged edition (2000), p. 478-483.

[M1] V. Moncrief Reduction of Einstein equations for vacuum spacetimes with U(1) spacelike isometry group, Annals of Physics 167 (1986), 118-142

[M2] V. Moncrief Reduction of the Einstein - Maxwell and the Einstein - Maxwell - Higgs equations for cosmological spacetimes with U(1) spacelike isometry group. Class. Quantum Grav. 7 (1990) 329-352.

[M-S] S. Muller and M. Struwe Global existence of wave maps in 2+1 dimensions with finite energy data, Top. Met. in non lin. An. 7, n^{∘}2 1996, 245-261.

YCB Université Paris 6, 4 Place Jussieu 75232 Paris France Email YCB@CCR.jussieu.fr