Ellipticity of Bartnik boundary data for stationary vacuum spacetimes
aa r X i v : . [ m a t h . DG ] J u l ELLIPTICITY OF BARTNIK BOUNDARY DATA FORSTATIONARY VACUUM SPACETIMES
ZHONGSHAN AN
Abstract.
We establish a moduli space E of stationary vacuum metrics in aspacetime, and set up a well-defined boundary map Π in E , assigning a metricclass with its Bartnik boundary data. Furthermore, we prove the boundarymap Π is Fredholm by showing that the stationary vacuum equations (com-bined with proper gauge terms) and the Bartnik boundary conditions form anelliptic boundary value problem. As an application, we show that the Bartnikboundary data near the standard flat boundary data admits a unique (up todiffeomorphism) stationary vacuum extension locally. Introduction
In general relativity, one of the most interesting and well-known notions of quasi-local mass is the Bartnik quasi-local mass. Let Ω be a bounded smooth 3-manifoldwith nonempty boundary Σ. Equip Ω with a Riemannian metric g and a symmetric2-tensor K , which is essentially the second fundamental form of Ω when it is em-bedded in some spacetime. The Bartnik quasi-local mass of the data set (Ω , g, K )is defined as (cf.[B1],[B2]),(1.1) m B [(Ω , g, K )] = inf { m ADM [( M, g, K )] } , where the infimum is taken over all asymptotically flat admissible initial data sets( M, g, K ) such that after gluing M and Ω along the boundary ∂M ∼ = ∂ Ω, themanifold M ∪ Ω forms a complete asymptotically flat spacetime initial data set.By analyzing the constraint equations across the boundary Σ = ∂ Ω ∼ = ∂M ,Bartnik proposed a set of geometric boundary data for (Ω , g, K ) given by,(1.2) ( g Σ , H Σ , tr Σ K, ω n Σ ) . Here g Σ is the induced metric on the boundary Σ obtained from (Ω , g ); H Σ is themean curvature of Σ ⊂ (Ω , g ); tr Σ K is the trace of the restriction K | Σ of the secondfundamental form; and ω n Σ is the connection 1-form of the spacetime normal bundleof Σ, which is defined as, ω n Σ ( v ) = K ( n Σ , v ) , ∀ v ∈ T Σ , where n Σ is the outward unit normal vector field on Σ ⊂ (Ω , g ).Then definition (1.1) of the Bartnik quasi-local mass can be reduced to theinfimum ADM mass taken over all asymptotically flat admissible initial data sets Zhongshan An ( M, g, K ) which satisfy the following boundary conditions:(1.3) g ∂M = g Σ H ∂M = H Σ tr ∂M K = tr Σ Kω n Σ = ω n ∂M . The tuple of geometric boundary data (1.2) is called the Bartnik boundarydata. It also arises naturally from a Hamiltonian analysis of the vacuum Ein-stein equations. In fact, a regularization H of the Regge-Teitelboim Hamiltonianis constructed in [B3]. When the spacetime has empty boundary, by analyzing thefunctional H and following an approach initiated by Brill-Deser-Fadeev (cf.[BDF]),Bartnik proved that stationary metrics are critical points of the ADM energy func-tional on the constraint manifold. However, if the spacetime has non-empty bound-ary, boundary terms arise from the variation of H ; they were explicitly identified byBartnik in [B1], and the boundary terms vanish if and only if the Bartnik boundarydata (1.2) is preserved in the variation.It was conjectured in [B1] that the Bartnik quasi-local mass of a given dataset (Ω , g, K ) must be realized by an admissible extension ( M, g, K ) which can beembedded as an initial data set into a stationary vacuum spacetime. To solve thisconjecture, one of the well-known and fundamental open problems raised by Bartnikin [B1] is the following:(1.4)
Is the Bartnik boundary data elliptic for stationary vacuum metrics?
In this paper, we give a positive answer to this question.A stationary spacetime ( V (4) , g (4) ) is a 4-manifold V (4) with a smooth Lorentzianmetric g (4) of signature ( − , + , + , +), which admits a time-like Killing vector field.In addition, a stationary spacetime is called vacuum if it solves the vacuum Einsteinequation(1.5) Ric g (4) = 0 . Throughout this paper, we assume that the stationary spacetime ( V (4) , g (4) ) is globally hyperbolic , i.e. it admits a Cauchy surface M and V (4) ∼ = R × M . In thiscase, there exists a global time function τ on V (4) so that M = { τ = 0 } and everysurface of constant τ is a Cauchy surface. Since the spacetime is stationary, onecan choose local coordinates { τ, x i } ( i = 1 , ,
3) so that ∂ τ is the time-like Killingvector field. Then the metric g (4) can be written globally in the form(1.6) g (4) = − N dτ + g ij ( dx i + X i dτ )( dx j + X j dτ ) . Notice that since ∂ τ is a Killing vector field, the stationary spacetime ( V (4) , g (4) )is vacuum if and only if the equation (1.5) holds on M . Remark
In the expression of g (4) above, the scalar field N and the vector field X in V (4) are usually called the lapse function and the shift vector of g (4) in this3+1 formalism of the spacetime. The tensor field g is the induced (Riemannian)metric on the Cauchy surfaces { τ = constant } ⊂ ( V (4) , g (4) ). Since the spacetimeis stationary, the fields g, X and N are all independent of the time variable τ , sothey can be regarded as tensor fields on the hypersurface M . Consequently, thevacuum equation (1.5) is an elliptic system (modulo gauge) of the fields ( g, X, N )on M . llipticity of Bartnik boundary data for stationary vacuum spacetimes 3 Let K be the second fundamental form of M ⊂ ( V (4) , g (4) ). The triple ( M, g, K )is called an initial data set of the spacetime. In the case where the boundary ∂M is nonempty, we can impose the Bartnik boundary condition (1.3) on this data set,coupling with the vacuum equation (1.5). So we obtain a boundary value problem(BVP) as, Ric g (4) = 0 on M, g ∂M = γH ∂M = Htr ∂M K = kω n ∂M = τ on ∂M, (1.7)where γ, H, k , and τ are prescribed tensor fields on ∂M . Now, the ellipticity ques-tion (1.4) is essentially asking whether this BVP is elliptic.Another way to formulate question (1.4) is to establish a boundary map. Let B ( ∂M ) denote the space of Bartnik boundary data, i.e. space of tuples ( γ, H, k, τ )on ∂M . Let E be the space of stationary vacuum metrics on V (4) . Then a naturalboundary map Π arises as,Π : E → B ( ∂M ) , Π ( g (4) ) = ( g ∂M , H ∂M , tr ∂M K, ω n ) . (1.8)The map Π being Fredholm is essentially equivalent to that BVP (1.7) is elliptic.However, it is easy to observe that equation (1.5) is not elliptic, since it is in-variant under diffeomorphisms, i.e., if g (4) is a stationary metric that solves (1.5),then the pull back metric Φ ∗ g (4) of g (4) under an arbitrary time-independent dif-feomorphism Φ of V (4) gives another stationary vacuum solution. This means thatwe need to add gauge terms to the BVP (1.7), and at the same time modify thedomain space E in (1.8) to a moduli space.In this paper, we first analyze how to choose the right domain space for theboundary map to be well-defined. We conclude in § E → B ( ∂M ) , Π([ g (4) ]) = ( g ∂M , H ∂M , tr ∂M K, ω n ) . Here the moduli space E is the quotient of E by a particular diffeomorphism group D . We refer to §
2, cf.(2.14), for the exact definition of D ; roughly it is a naturalintermediate group D ⊂ D ⊂ D between the groups of 3-dimensional diffeo-morphisms on M fixing the boundary ∂M and 4-dimensional time-independentdiffeomorphisms on V (4) fixing ∂V (4) . In order to prove ellipticity of the map Π,we establish in § §
4, and from this derive the maintheorem of this paper:
Theorem 1.1.
The moduli space E is a C ∞ smooth Banach manifold of infinitedimension and the boundary map Π is Fredholm. We show in § §
3, completing the proof of Theorem 1.1.To conclude, we apply this ellipticity result in § g (4)0 on R × ( R \ B ) can Zhongshan An be locally uniquely realized by a stationary vacuum metric up to diffeomorphismsin D . Theorem 1.2.
There is a neighborhood
U ⊂ B ( S ) of the standard flat boundarydata ( g , , , such that for any ( γ, H, k, τ ) ∈ U , there is a unique stationaryvacuum metric g (4) ∈ E near ˜ g (4)0 up to diffeomorphisms in D , for which Π ( g (4) ) = ( γ, H, k.τ ) . Throughout, we assume the hypersurface M ∼ = R \ B (exterior problem), to-gether with certain asymptotically flat conditions on the metric g (4) . Meanwhile, allthe methods and results here can be applied equally well in the case where M ∼ = B (interior problem).This paper is a continuation of our previous paper [Az], in which we developeda general method to prove the ellipticity of boundary value problems for the sta-tionary vacuum spacetime. Expanding this method, we study the ellipticity of theBartnik boundary data here. Theorem 1.1 is a generalization of the results provedin [AK], where spacetimes are static. Theorem 1.2 generalizes the result in [A2]for static metrics. We refer to [Mi],[J],[R] for other existence results on stationaryvacuum extensions of boundary data.The results we prove in this paper provide a firm foundation for future workon Bartnik’s conjecture about the quasi-local mass in spacetimes and the existenceproblem of stationary vacuum metrics that satisfy the Bartnik boundary condi-tions. To the author’s knowledge, this is the first ellipticity result of the Bartnikboundary data for general stationary vacuum metrics. Acknowledgements
I would like to express great thanks to my advisor MichaelAnderson for suggesting this problem and for valuable discussions and comments.2. background
Fix a 3-dimensional manifold M ∼ = R \ B . Let V (4) ∼ = R × M . By fixinga diffeomorphism between V (4) and R × M , we can equip V (4) with the naturalcoordinates { ( t, p ) } where t ∈ R and p ∈ M . Fix the hypersurface { t = 0 } ⊂ V (4) and identify it with M . Let S denote the space of Lorenztian metrics g (4) on V (4) which satisfy the following conditions:
1. (globally hyperbolic) The metric g (4) can be expressed globally as (2.1) g (4) = − N dt + g ij ( dx i + X i dt )( dx j + X j dt ) , where { x i } ( i = 1 , , are local coordinates on M . There exists some time indepen-dent function f ∈ C m,αδ ( V (4) ) (cf. § τ = t + f makes τ a time function of the spacetime ( V (4) , g (4) ) in the sense of general relativity andevery surface of constant τ is a Cauchy surface (cf. equation (1.6)). In particular,if f ≡ then expression (2.1) and (1.6) agree.2. (stationary) The vector field ∂ t is a time-like Killing vector field in ( V (4) , g (4) ) .So the triple ( g, X, N ) is independent of t and can be regarded as tensor fieldson M (cf. the remark below equation (1.6)). In addition, since h ∂ t , ∂ t i g (4) = − N + || X || g < , one has (2.2) N > || X || g . llipticity of Bartnik boundary data for stationary vacuum spacetimes 5
3. (asymptotically flat) The metric g (4) decays to the flat (Minkowski) metric atinfinity. Explicitly, N , X and g belong to the weighted H¨older spaces on M , givenby, g ∈ M et m,αδ ( M ) ,N − ∈ C m,αδ ( M ) ,X ∈ T m,αδ ( M ) , (2.3) for some fixed number m ≥ , < α < , and < δ < . We refer to the Appendix § Throughout this paper, we will use h X, Y i g to denote the inner product of twovector fields with respect to the metric g . The (square) norm of a vector field X with respect to the metric g is h X, X i g = || X || g . We will omit the metric in thesubcript when it is clear in the context what metric is being used. Remark . Based on the definition of the space S , it is easy to observe that S is invariance under the action of the diffeomorphism group D (cf.(2.6) below).Consequently, the tensor field g in (2.1), which can be taken as the induced metricon the hypersurface M ⊂ ( V (4) , g (4) ), is not necessarily Riemannian.It is obvious that an element in S is uniquely determined by a triple of fields( g, X, N ) on M . Thus S is an open domain in a Banach space and so admits smoothBanach manifold structure.As is mentioned in the introduction, one can establish BVP (1.7) for g (4) ∈ S ,but in order to make it elliptic, we need to add gauge terms. A standard choice isto use the Bianchi gauge, leading to a modified system with unknown g (4) ∈ S asfollows: Ric g (4) + δ ∗ g (4) β ˜ g (4) g (4) = 0 on M, g ∂M = γH ∂M = Htr ∂M K = kω n = τβ ˜ g (4) g (4) = 0 . on ∂M (2.4)In the system above, we add the term δ ∗ g (4) β ˜ g (4) g (4) in the vacuum equation, and addthe Dirichlet condition of the gauge term β ˜ g (4) g (4) on the boundary. Here the gaugeterm β ˜ g (4) g (4) is the Bianchi operator acting on the metric g (4) with respect to afixed stationary vacuum metric ˜ g (4) , i.e. β ˜ g (4) g (4) = δ ˜ g (4) g (4) + dtr ˜ g (4) g (4) , wherethe reference metric ˜ g (4) ∈ E (cf.(2.5) below). We use δ to denote the divergenceoperator δ = − tr ∇ , and δ ∗ denotes the formal adjoint of the divergence operator,i.e. δ ∗ g (4) Y = L Y g (4) for any Y ∈ T V (4) . Among the Bartnik boundary conditions,here and throughout the following, we use ω n as the abbreviation of ω n ∂M .The effect of adding the gauge term to BVP (1.7) as above is to give a sliceto the action on the solution space of (1.7) by the group D (cf.(2.6) below) ofdiffeomorphisms of the spacetime fixing the boundary ∂V (4) . However, such amodification has two issues. First, it is easy to observe that (2 .
4) is not well posed,because there are 10 interior equations on M but 11 boundary conditions on ∂M — Zhongshan An notice that, the gauge term β ˜ g (4) g (4) defines a vector field in V (4) , so it contributes4 extra boundary equations in (2 . E be the space of stationary vacuum metrics, i.e.(2.5) E = { g (4) ∈ S : Ric g (4) = 0 } . As is explained above, after adding the gauge term β ˜ g (4) g (4) , the boundary map Π defined in (1.8) should be modified to Π as follows,Π : E / D → B ( ∂M ) , Π ([ g (4) ]) = ( g ∂M , H ∂M , tr ∂M K, ω n ) , where the target space B ( ∂M ) is given by B ( ∂M ) = M et m,α ( ∂M ) × [ C m − ,α ( ∂M )] ×∧ m − ,α ( ∂M ) (cf. § D do not always preserve theBartnik boundary data (cf. Proposition 2.2 below), which means that the Bartnikboundary data is not well defined for an element [ g (4) ] — an equivalence class ofmetrics — in the moduli space E / D .Since we are working with stationary metrics, it is natural to require elementsin D to be time-independent and preserve the Killing vector field ∂ t . Thus ageneral element in D can be decomposed into two parts — a diffeomorphism onthe hypersurface M and a translation of time, i.e. D can be defined as, D = { Φ ( ψ,f ) | ψ ∈ D m +1 ,αδ ( M ) and ψ | ∂M = Id ∂M ; f ∈ C m +1 ,αδ ( M ) and f | ∂M = 0;Φ ( ψ,f ) : V (4) → V (4) , Φ ( ψ,f ) [ t, p ] = [ t + f, ψ ( p )] , ∀ t ∈ R , p ∈ M. } , (2.6)Here D m +1 ,αδ ( M ) denotes the group of C m +1 ,α diffeomorphisms of M which areasymptotically Id M at the rate of δ (cf. § Proposition 2.2.
If an element Φ ( ψ,f ) ∈ D has a nontrivial time translationfunction f , then it does not preserve the Bartnik boundary data on ∂M .Proof. Equip M = { t = 0 } with local coordinates { x i } , i = 1 , , . Choose afunction f ∈ C m +1 ,αδ ( M ), and take the diffeomorphism Φ ( Id M ,f ) ∈ D :Φ ( Id M ,f ) : V (4) → V (4) Φ ( Id M ,f ) ( t, x , x , x ) = ( t + f, x , x , x ) . In the following, we will use Φ f as the abbreviation of Φ ( Id M ,f ) . Take a spacetimemetric g (4) ∈ S expressed as, g (4) = − N dt + g ij ( dx i + X i dt )( dx j + X j dt ) . Let ˆ g (4) denotes the pull back metric, i.e. ˆ g (4) = Φ ∗ f g (4) . Then we have,ˆ g (4) = − N [ d ( t + f )] + g ij [ dx i + X i d ( t + f )][ dx j + X j d ( t + f )]= − u dt − u df ⊙ dt + X i dx i ⊙ dt − u ( df ) + X i dx i ⊙ df + g ij dx i dx j , where u = N − | X | g . Here we use ⊙ to denote the symmetrized product betweentwo 1-forms A, B , i.e. A ⊙ B = A ⊗ B + B ⊗ A . From the expression above, one llipticity of Bartnik boundary data for stationary vacuum spacetimes 7 easily observes that the induced metric on M ⊂ ( V (4) , ˆ g (4) ) is given by,(2.7) ˆ g = − u ( df ) + X i dx i ⊙ df + g ij dx i dx j . Plugging f | ∂M = 0 in the equation above, it is obvious that the first Bartnik bound-ary term g ∂M in (2.4) remains the same under such a time translation. However,this is not the case for the other data H ∂M , tr ∂M K and ω n ∂M .Let N denote the future-pointing time-like unit normal vector to the slice M ⊂ ( V (4) , g (4) ) and n denote the outward unit normal of ∂M ⊂ ( M, g ). Define ( ˆN , ˆn )in the same way for M ⊂ ( V (4) , ˆ g (4) ). Then on the boundary ∂M , the pairs ( N , n )and ( ˆN , ˆn ) are related in the following way, (cid:20) d Φ f ( ˆN ) d Φ f ( ˆn ) (cid:21) = (cid:20) a bb a (cid:21) (cid:20) Nn (cid:21) , where a, b are scalar fields on ∂M and a − b = 1. We refer to § ∇ be the Levi-Civita connection of the spacetime ( V (4) , g (4) ), and ˆ ∇ denotesthat of the spacetime ( V (4) , ˆ g (4) ), then ˆ ∇ = Φ ∗ f ( ∇ ). We use H ∂M , tr ∂M K and ω n to denote the Bartnik boundary data of ( V (4) , g (4) ) on ∂M ; and use ˆ H ∂M , tr ∂M ˆ K and ˆ ω ˆ n as that of ( V (4) , ˆ g (4) ). Then we have the following formula for the meancurvature: ˆ H ∂M = tr ∂M ( ˆ ∇ ˆn )= tr ∂M [ ∇ d Φ f ( ˆn )]= tr ∂M [ ∇ ( b N + a n )]= btr ∂M ( ∇ N ) + atr ∂M ( ∇ n )= btr ∂M K + aH ∂M . (2.8)It is easy to show that tr ∂M K is transformed in a similar way as above, i.e.(2.9) tr ∂M ˆ K = atr ∂M K + bH ∂M . As for the last boundary term ω n , one has ∀ v ∈ T ( ∂M ),ˆ ω ˆ n ( v ) = ˆ K (ˆ n , v ) = h ˆ ∇ v ˆ N , ˆ n i ˆ g (4) = h Φ ∗ f ( ∇ ) v ˆN , ˆn i Φ ∗ f g (4) = h∇ d Φ f ( v ) ( a N + b n ) , b N + a n i g (4) = − b · ∇ d Φ f ( v ) a + a · ∇ d Φ f ( v ) b + ( a − b ) h∇ d Φ f ( v ) N , n i g (4) = a ∇ d Φ f ( v ) ( b/a ) + K ( n , d Φ f ( v )) , = a v ( b/a ) + ω n ( v ) . Here the last equality is based on the observation that d Φ f ( v ) = v ∀ v ∈ T ( ∂M ),since Φ f | ∂M = Id ∂M . From the formula above, we conclude that,(2.10) ˆ ω ˆn = a d ∂M ( b/a ) + ω n , where d ∂M ( b/a ) denotes the exterior derivative of the scalar field on ∂M . Alongthe boundary ∂M , one has(2.11) a = 1 + h X, n i n ( f ) p [1 + h X, n i n ( f )] − N | n ( f ) | . Zhongshan An
We refer to the Appendix § a, b .Therefore, if the function f is nontrivial, in the sense that n ( f ) | ∂M = 0 and a = 1 in(2.11), then it is easy to observe from equations (2.8-10) that the Bartnik boundaryconditions are not invariant under the diffeomorphism Φ f . (cid:3) In view of the fact above, one may suggest to reduce the diffeomorphism group D in the definition of the boundary map to a smaller one D consisting of only3-dim diffeomorphism on the slice, i.e.(2.12) D = { Φ ( ψ,f ) ∈ D : f ≡ M } . However, this approach does not work either. Let Π be the associated boundarymap as follows, Π : E / D → B Π ([ g (4) ]) = ( g ∂M , H ∂M , tr ∂M K, ω n ) . Given a fixed boundary condition ( γ, H, k, τ ), and an element g (4) in the pre-imageset Π − [( γ, H, k, τ )], we can take an arbitrary function f ∈ C m +1 ,αδ ( M ) such that f | ∂M = n ( f ) | ∂M = 0, and make time translation Φ f to obtain a new metric ¯ g (4) =Φ ∗ f g (4) . Then, by the previous analysis, ¯ g (4) also belongs to Π − [( γ, H, k, τ )].By taking a smooth curve (parametrized by τ ) of such time translations Φ f ( τ ) ( τ ∈ ( − , g (4) ( τ ) = Φ ∗ f ( τ ) g (4) . Let f τ = ∂∂τ | τ =0 f ,then the infinitesimal deformation of the spacetime metric at τ = 0 is of the form,( g (4) ) ′ = L f τ ∂ t g (4) = df τ ⊙ ( ∂t ) ♭ = df τ ⊙ ( − u dt + X i dx i ) . By construction, g (4) ( τ ) ∈ Π − [( γ, H, k, τ )] for all τ , which implies that ( g (4) ) ′ ∈ Ker D Π . Such a kernel element is nontrivial if it is not tangent to any 3-dimdiffeomorphism variation, i.e. the following equation is not solvable for Z ∈ T m,αδ M ,(2.13) df τ ⊙ ( − u dt + X i dx i ) = L Z g (4) . Since (2 .
13) is an overdetermined system for Z , it is not solvable for generic choicesof f τ . This means that the kernel of D Π should be of infinite dimension, whichindicates that Π is not a Fredholm map.From all the previous analysis, we notice that the Neumann data n ( f ) of the timetranslation function plays an important role in choosing the right diffeomorphismgroup. This suggests defining a new group D as,(2.14) D = { Φ ( ψ,f ) ∈ D : n g ( f ) = 0 on ∂M } . It is in fact an intermediate group in the sense that D ⊂ D ⊂ D . Remark . The vector field n g in (2.14) can be taken as the unit normal vectorof ∂M with respect to any Riemannian metric g on M — the group D does notdepend on the choice of the metric g . In fact, it is easy to observe that D can bedefined in an equivalent way: D = { Φ ( ψ,f ) ∈ D : df = 0 at ∂M } . Notice that n ( f ) = 0 in (2.14) yields a = 1 in (2.11). This further implies that, geo-metrically, elements in the group D are diffeomorphisms of the spacetime ( V (4) , g (4) )which fix the boundary ∂M and the time-like unit normal vector field N along ∂M . llipticity of Bartnik boundary data for stationary vacuum spacetimes 9 Now, define E to be the quotient space, E = E / D . Elements in E are equivalence classes [ g (4) ] given by,[ g (4) ] = { Φ ∗ ( ψ,f ) g (4) : g (4) ∈ E , Φ ( ψ,f ) ∈ D} . Now we can consider the natural boundary map:Π : E → B Π([ g (4) ]) = ( g ∂M , H ∂M , tr ∂M K, ω n ) . (2.15)This map is well defined — the Bartnik boundary data is the same for all themetrics inside one equivalence class [ g (4) ] ∈ E , because the transformation formulas(2 . −
10) show that Bartnik boundary data is preserved under diffeomorphisms in D . In the following sections we will prove this boundary map Π is Fredholm. Remark . By the Remark 2.1, a general spacetime metric g (4) in an equivalenceclass [ g (4) ] ∈ E does not necessarily induce Rimannian geometry on the fixed slice M , which might make the Bartnik boundary data not well-defined on the slice M ⊂ ( V (4) , g (4) ). To solve this problem, we can restrict the boundary map Π in(2.15) to the open subset E ′ ⊂ E , where every equivalent class [ g (4) ] ∈ E ′ admitsa representative g (4) which induces Riemannian metric on M . On the other hand,since the ellipticity of the Bartnik boundary data is only a local property, we canfocus the study in a neighborhood of such metric g (4) ∈ E and think of E locally as aslice of E under the action of D . The boundary map Π being Fredholm implies thatthe linearization D Π of the map Π in (1.8) at g (4) has finite dimensional kerneltransverse to the action of D in the sense that Ker D Π / ∼ has finite dimension.Here two deformations h (4)1 , h (4)2 ∈ Ker D Π are equivalent ( h (4)1 ∼ h (4)2 ) if thedifference ( h (4)1 − h (4)2 ) = δ ∗ g (4) Y for some Y ∈ T D . In the following sections wealways assume that the reference spacetime metric ˜ g (4) is chosen so that the slice M is a Cauchy surface in ( V (4) , ˜ g (4) ).3. The well-defined BVP
Throughout this section, we take ˜ g (4) ∈ E as a fixed reference metric and makethe following assumption: Assumption 3.1.
The BVP with unknown Y ∈ T m,αδ ( V (4) ) , given by, (3.1) ( β g (4) δ ∗ g (4) Y = 0 on MY = 0 on ∂M has only the zero solution Y = 0 when g (4) = ˜ g (4) . In the above, T m,αδ ( V (4) ) denotes the space of C m,α vector fields in V (4) , whichare asymptotically zero at the rate of δ and in addition time-independent(cf. § T in V (4) is time-independent if L ∂ t T =0. In the following, we call the operator β ˜ g (4) δ ∗ ˜ g (4) invertible if Assumption 3.1 holds.Note that since the system (3.1) is elliptic and self-adjoint (cf. § β ˜ g (4) δ ∗ ˜ g (4) : T → T m − ,αδ +2 ( V (4) ) , where the domain space T = { Y ∈ T m,αδ ( V (4) ) : Y = 0 on ∂M } . This is an opencondition. Thus if β ˜ g (4) δ ∗ ˜ g (4) is invertible, then so is the operator β g (4) δ ∗ g (4) for g (4) near ˜ g (4) in the space S . We refer to Remark 4.3 for more insights about theAssumption 3.1.Based on the discussions in §
2, we modify the system (2.4) to a new BVP withunknowns ( g (4) , F ) ∈ S × C m,αδ ( M ) as follows, ( Ric g (4) + δ ∗ g (4) β ˜ g (4) g (4) = 0∆ F = 0 on M, g ∂M = γaH ∂M + btr ∂M K = Hatr ∂M K + bH ∂M = kω n + a d ∂M ( a/b ) = τβ ˜ g (4) g (4) = 0 on ∂M, (3.2)where(3.3) a = 1 + h X, n i F p (1 + h X, n i F ) − N F , and b = p a − , with N and X denoting the lapse function and shift vector of g (4) as in (2.1). In thesecond equation above, ∆ = − trHess denotes the Laplace operator with respect tothe metric g (4) . Here we think of F ∈ C m,αδ ( M ) as a function in V (4) by expandingit time-independently. The argument to follow works in the same way if one sets ∆to be the Laplacian of the induced Riemannian metric g on the slice M . But withthe former choice, the principal symbol which we will compute in § . β g (4) δ ∗ g (4) [ β ˜ g (4) g (4) ] = 0 on M. In addition, the last boundary condition in (3.2) gives,(3.5) β ˜ g (4) g (4) = 0 on ∂M. Combining (3.4) and (3.5), together with the Assumption 3.1, it follows that, β ˜ g (4) g (4) = 0 , ∀ solution g (4) of (3.2) near ˜ g (4) . Therefore, if g (4) is a solution to (3.2) near ˜ g (4) , then it must be Ricci flat andBianchi-free with respect to ˜ g (4) . So we define a solution space C as follows: C := { ( g (4) , F ) ∈ S × C m,αδ ( M ) : Ric g (4) = 0 , β ˜ g (4) g (4) = 0 , ∆ F = 0 on M } . Obviously, (˜ g (4) , ∈ C . Let ˜Π be the boundary map:˜Π : C → B ˜Π( g (4) , F ) = ( g ∂M , aH ∂M + btr ∂M K,atr ∂M K + bH ∂M , ω n + a d ∂M ( b/a )) . This map ˜Π is closely related to the boundary map Π defined in (2.15) near thereference pair (˜ g (4) , ∈ C . In fact, we have the following theorem. Theorem 3.2.
There is a map P : C → E which is locally a diffeomorphism near (˜ g (4) , , and the boundary maps Π and ˜Π are related by ˜Π = Π ◦ P . llipticity of Bartnik boundary data for stationary vacuum spacetimes 11 Proof.
Given an element (ˆ g (4) , ˆ F ) ∈ C , one can take a function f on M such that f | ∂M = 0 and n ( f ) | ∂M = ˆ F | ∂M , and apply the diffeomorphism Φ ( ψ,f ) ∈ D to ˆ g (4) where ψ is an arbitrary diffeomorphism in D m,αδ ( M ) with ψ | ∂M = Id ∂M . Thus,any element (ˆ g (4) , ˆ F ) ∈ C gives rise to a class of elements as follows,(3.6) { Φ ∗ ( ψ,f ) (ˆ g (4) ) : Φ ( ψ,f ) ∈ D , n ( f ) | ∂M = ˆ F | ∂M } . It is easy to observe that the equivalence class above actually defines an element in E . Henceforth we can define a map P as, P : C → E , P (ˆ g (4) , ˆ F ) = [ g (4) ] , where [ g (4) ] is defined as the equivalence class (3 . G : S × D → ( ∧ ) m,αδ ( V (4) ) G ( g (4) , Φ) = β ˜ g (4) Φ ∗ g (4) , where ( ∧ ) m,αδ ( V (4) ) denotes the space of C m,α V (4) which are time-independent and asymptotically zero at the rate of δ (cf. § G at (˜ g (4) , Id V (4) ) is given by, D G| (˜ g (4) ,Id V (4) ) : T S × T D → ( ∧ ) m,αδ V (4) D G| (˜ g (4) ,Id V (4) ) [( h (4) , Y )] = β ˜ g (4) δ ∗ ˜ g (4) Y + β ˜ g (4) h (4) . By the definition of D , the vector field Y ∈ T D is time-independent, asymptoti-cally zero and Y = 0 on ∂M . So the operator β ˜ g (4) δ ∗ ˜ g (4) in the linearization aboveis invertible by the Assumption 3.1. Therefore, by the implicit function theorem,there is a neighborhood U ˜ g (4) of ˜ g (4) in S such that for any g (4) ∈ U , there is aunique element Φ ( ψ,f ) ∈ D near Id V such that the pull back metric Φ ∗ ( ψ,f ) g (4) is gauge-free, i.e. β ˜ g (4) (Φ ∗ ( ψ,f ) g (4) ) = 0 in V (4) . Moreover, if g (4) is vacuum, i.e. g (4) ∈ U ˜ g (4) ∩ E , then the gauge-free metric ˆ g (4) = Φ ∗ ( ψ,f ) g (4) is also vacuum.Trivially it follows, g (4) = (Φ ∗ ( ψ,f ) ) − ˆ g (4) = Φ ∗ ( ψ − , − f ) ˆ g (4) . Let ˆ F ∈ C m,αδ ( M ) be the unique harmonic function (with respect to the metricˆ g (4) ) on M satisfying the Dirichlet boundary condition ˆ F = n ( − f ) on ∂M . Sincethe diffeomorphism Φ ( ψ,f ) is near Id V (4) , n ( f ) is close to zero. Thus F is also nearthe zero function on M . Pairing it with ˆ g (4) , we obtain an element (ˆ g (4) , ˆ F ) ∈ C near (˜ g (4) , g (4)1 , g (4)2 ∈ U ˜ g (4) ∩ E are equivalent under somediffeomorphism Φ ( ψ ,f ) ∈ D , then they correspond to the same gauge-free metricˆ g (4) because of the uniqueness shown above. In addition, since the time translation f makes n ( f ) = 0 on ∂M , g (4)1 and g (4)2 also generate the same harmonic functionˆ F as described above. Therefore, in the neighborhood U ˜ g (4) of ˜ g (4) in S , all themetrics that belong to the same equivalence class [ g (4) ] ∈ E give rise to a uniquepair (ˆ g (4) , ˆ F ) ∈ C near (˜ g (4) , U ˜ g (4) to be the corresponding neighborhood of [˜ g (4) ] in E , i.e. U ˜ g (4) = { [ g (4) ] ∈ E : The equivalence class [ g (4) ] admits a representative g (4)0 such that g (4)0 ∈ U ˜ g (4) } . Then for any [ g (4) ] ∈ U ˜ g (4) we can take an arbitrary representative g (4) ∈ U ˜ g (4) andthen obtain a pair (ˆ g (4) , ˆ F ) in the manner described above. Moreover, as is shown,the pair (ˆ g (4) , ˆ F ) does not depend on the representative we choose in U ˜ g (4) . In thisway, one can establish a map ˜ P locally defined in E near [˜ g (4) ] and mapping U ˜ g (4) to a neighborhood of (˜ g (4) ,
0) in C , given by,˜ P : U ˜ g (4) → C , ˜ P ([ g (4) ]) = (ˆ g (4) , ˆ F ) . It is easy to check that P and ˜ P are the inverse map of each other near (˜ g (4) , C and E are locally diffeomorphic via P .Moreover, based on the formulas (2 . − g (4) ] = P (ˆ g (4) , ˆ F ), then their Bartnik boundary data are related in the following way,( g ∂M ,H ∂M , tr ∂M K, ω n )= (ˆ g ∂M , a ˆ H ∂M + btr ∂M ˆ K, atr ∂M ˆ K + b ˆ H ∂M , ˆ ω n + a d ∂M ( b/a )) , where a, b are given by the formulas in (3 .
3) inside which F = ˆ F and X, N are thelapse and shift of ˆ g (4) . Therefore, near (˜ g (4) ,
0) the boundary maps ˜Π and Π arerelated by, ˜Π = Π ◦ P . (cid:3) From the analysis above, we see that locally the solution space C is a coordinatechart of the moduli space E near the reference metric class [˜ g (4) ], and the map ˜Πis the Bartnik boundary map Π expressed in this local chart. In the following, weshow that the space C admits Banach manifold structure. Theorem 3.3.
The space C admits smooth Banach manifold structure near (˜ g (4) , .Proof. For any stationary vacuum metric g (4) , define H g (4) as the space of harmonicfunctions on M : H g (4) = { f ∈ C m,αδ ( M ) : ∆ g (4) f = 0 on M } . Since ∆ g (4) is invertible when subjected to Dirichlet boundary conditions, it is easyto prove that, H g (4) ∼ = C m,α ( ∂M ) . Thus H g (4) is a smooth Banach manifold.Let P : C → E be the projection P (ˆ g (4) , ˆ F ) = ˆ g (4) . We observe that if (ˆ g (4) , ˆ F ) ∈C near (˜ g (4) , g (4) ∈ E and it satisfies the gauge-free condition β ˜ g (4) ˆ g (4) = 0.By the analysis in the proof of Theorem 3.2, any g (4) ∈ E near ˜ g (4) is isometricvia a diffeomorphism in D to a gauge free element ˆ g (4) . So the projection space P ( C ) is a slice for E under the action of D . Therefore, locally the space C is a llipticity of Bartnik boundary data for stationary vacuum spacetimes 13 fiber bundle over E / D , with the fiber at [ g (4) ] being H g (4) . Thus near the element(˜ g (4) , C ∼ = E / D × H ˜ g (4) . It is proved in [Az] that the moduli space E / D is a smooth Banach manifold, andhence it follows that C admits smooth Banach manifold structure. (cid:3) It then follows directly from Theorem 3.2 that the domain space E of the Bartnikboundary map Π is a smooth Banach manifold. In the following sections, we willshow that the boundary map ˜Π is Fredholm, which then implies so is Π.4. Ellipticity of BVP (3.2)
In this section, we will prove the ellipticity of BVP (3.2), implementing thecriterion developed by Agmon-Douglis-Nirenberg (cf.[ADN]). We use the followingstandard notation. Let ξ denote a 1 − form on M , η denote a nonzero 1 − formtangential to the boundary ∂M , and µ a unit 1 − form normal to the boundary ∂M .The index 0 denotes the direction along ∂t in V (4) , and index 1 , , M . When restricted on the boundary, index 1 denotes the(outward) normal direction to ∂M ⊂ M and indices 2 , ∂M . We use greek letters when 0 is included in the indices, and latin letterswhen there are only tangential components involved.Based on the system (3.2), we define a differential operator F = ( L , B ) withinterior operator L , mapping a pair ( g (4) , F ) to the interior equations in (3.2): L : S × C m,αδ ( M ) → S m − ,αδ +2 ( V (4) ) × C m − ,αδ +2 ( M ) L ( g (4) , F ) = ( 2( Ric g (4) + δ ∗ g (4) β ˜ g (4) g (4) ) , ∆ F );and a boundary operator B mapping ( g (4) , F ) to the boundary equations in (3 . B : S × C m,αδ ( M ) → B B ( g (4) , F ) = ( g ∂M ,aH ∂M + btr ∂M K,atr ∂M K + bH ∂M ,ω n + a d ∂M ( b/a ) ,β ˜ g (4) g (4) ) . In the above, S m − ,αδ +2 ( V (4) ) denotes the space of symmetric 2-tensors in V (4) , whichare time independent, C m − ,α smooth and asymptotically zero at the rate of ( δ +2); B is an abbreviation of the target space of B , given by, B = S m,α ( ∂M ) × [ C m − ,α ( ∂M )] × ∧ m − ,α ( ∂M ) × C m − ,α ( ∂M ) × ∧ m − ,α ( ∂M ) . We refer to § Theorem 4.1.
The linearization D F of F at (˜ g (4) , is elliptic.Proof. We use the characterization of ellipticity in [ADN] to prove the thoerem.We first show in § D F is properly elliptic. Then in § D F satisfies the complementing boundary condition. Properly elliptic condition.
The linearization of the interior operator at (˜ g (4) ,
0) is given by (cf.[Be]) D L : T ˜ g (4) S × C m,αδ ( M ) → S m − ,αδ +2 ( V (4) ) × C m − ,αδ +2 ( M ) D L ( h (4) , G ) = ( D ∗ ˜ g (4) D ˜ g (4) h (4) , ∆ G ) . Let { e i } , ( i = 1 , ,
3) be a local orthonormal basis of the tangent bundle on M .Recall that N denotes the future pointing time-like unit vector perpendicular to M in the spacetime. Based on (2.1), N = N − ( ∂ t − X ), with X, N being the shiftvector and lapse function of ˜ g (4) . Then the Laplacian D ∗ ˜ g (4) D ˜ g (4) h (4) αβ in the abovecan be expressed in the 3 + 1 slice formalism (2.1) of the spacetime as: D ∗ ˜ g (4) D ˜ g (4) h (4) αβ = − D N D N h (4) αβ + Σ i =1 D e i D e i h (4) αβ + O ( h (4) )= − D N ( ∂t − X ) D N ( ∂t − X ) h (4) αβ + Σ i =1 D e i D e i h (4) αβ + O ( h (4) )= − N ∂ X ∂ X h (4) αβ + Σ i =1 ∂ e i ∂ e i h (4) αβ + O ( h (4) ) . Here O ( h (4) ) denotes those terms with lower( ≤
1) order derivatives. A similarformula holds for the term ∆ G , i.e.∆ G = − N ∂ X ∂ X G + Σ i =1 ∂ e i ∂ e i G + O ( G ) . Thus, the matrix of principal symbol for D L is given by,(4.1) L ( ξ ) = a ( ξ ) I × with(4.2) a ( ξ ) = ξ + ξ + ξ − N ( X i ξ i ) . The determinant of this matrix is obviouslydet( L ( ξ )) = [ a ( ξ )] . Notice that || X || N < .
2) and hence, a ( ξ ) = | ξ | − h XN , ξ i ≥ | ξ | − || X || N | ξ | > , Therefore, the interior operator L is properly elliptic.4.2. Complementing boundary condition.
The complementing boundary condition is defined as (cf.[ADN]):
Let L ∗ ( ξ ) be the adjoint matrix of L ( ξ ) and set ξ = η + zµ . The rows of thematrix [ B · L ∗ ]( η + zµ ) are linearly independent modulo l + ( z ) = Q ( z − z k ) , where { z k } are the roots of detL ( η + zµ ) = 0 having positive imaginary parts. Since the principal symbol of L is the identity matrix (up to a scalar) as shownin (4 . B ( η + zµ ) is non-degenerate when z is a root of det L ( η + zµ ) = 0 with positiveimaginary part. llipticity of Bartnik boundary data for stationary vacuum spacetimes 15 The linearization of the boundary operator B at (˜ g (4) ,
0) is given by, B : T S × C m,αδ ( M ) → B D B ( h (4) , G ) = ( h ∂M ( H ∂M ) ′ h (4) + O ( G ) tr ∂M K ′ h (4) + O ( G )( ω n ) ′ h (4) + N d ∂M G + O ( G ) β g (4) h (4) ) . (4.3)Here we use the notation T ′ h (4) to denote the variation of the tensor T with respectto the deformation h (4) . Notice that at (˜ g (4) , a = 1 , b = 0. The formula (3 .
3) ofthe scalar field a involves only the 0 − order information of F . Thus the 2nd and 3rdlines in the expression of D B above, which represent the linearization of Bartnikdata ( aH ∂M + btr ∂M K ) and ( atr ∂M K + bH ∂M ) at ( a = 1 , b = 0), do not containhigh order ( ≥
1) derivatives of G . It is easy to check at ( a = 1 , b = 0), the variation( a d ∂M b/a (cid:1) ′ G = N d ∂M G + O ( G ) , which contributes to the fourth line in D B .Based on (4 . B is of the form:(4.4) B ( ξ ) = × ˜ B × ∗ . Notice that B ( ξ ) is a 11 ×
11 matrix, since the boundary terms in (4.3) contain11 equations in total and 11 (ordered) unknowns. Here for a simple expression ofthe boundary matrix, the unknown vector ( h (4) , G ) is particularly arranged in theorder of: ( G, h (4)00 , h (4)01 , h (4)02 , h (4)03 , h (4)11 , h (4)12 , h (4)13 , h (4)22 , h (4)23 , h (4)33 ) . Obviously, the first boundary term h ∂M = h (4) ij , (2 ≤ i ≤ j ≤ B in (4.4) contain only zeros in the first eight columns and a 3 × B represent the symbol of2nd-5th boundary terms in (4.3), with ˜ B × denoting the first eight columns whichare determined by the G and h (4) αβ (0 ≤ α ≤ , α ≤ β ≤
3) components of thecorresponding boundary terms. Detailed calculation given in § B × is given by˜ B × = − N [( ˆ B ) × ( ˆ B ) × ( ˆ B ) × ] . Here the scalar − N is for the purpose that the matrix following it can be writtenin a simpler way. Since G only appears as N d ∂M G in the fourth boundary term in(4.3), it is easy to derive that the first column of ˜ B is given byˆ B = (cid:2) − N ξ − N ξ (cid:3) t In the above, we have the extra factor − N , because − N has been factored outfrom these rows, which contributes to the factor in the front of the expression of ˜ B . Based on the symbol calculation in § B as,ˆ B = N ξ N ξ N ξ N ξ N ξ N ξ ξ S − ξ X − ξ X − ξ X ξ − ξ X S − ξ X − ξ X ξ − ξ X − ξ X S − ξ X S N ξ − SX ) 2( N ξ − SX ) 2( N ξ − SX ) , and the last three columns are given byˆ B = N ξ N ξ − Nξ X − Nξ X − Nξ X Nξ X − Nξ X − Nξ X − Nξ X Nξ X ξ X X + N ξ − SX ξ X X − SX + 2 N ξ ξ X X − SX + 2 N ξ ξ X X − N ξ ξ X X − SX + 2 N ξ ξ X X ξ X X − N ξ ξ X X ξ X X − SX + 2 N ξ , inside which S = ξ X + ξ X + ξ X .Obviously, to prove the complementing boundary condition, it suffices to verifythat ˜ B ( η + zµ ) is nonsingular when z is a root of a ( η + zµ ) in (4.2) with positiveimaginary part. We can simplify ˜ B using elementary row and column operation ofmatrices (cf. § . B so that det ˜ B = − N det ˆ B . The matrix ˆ B ( ξ ) is given by,(4.5) − ξ − ξ ξ ξ − N ξ ξ ξ X + ξ X − ξ X − N ξ ξ − ξ X ξ X + ξ X ξ S N ξ − SX − SX ξ S N ξ ξ S N ξ S N ξ + SX SX SX N S + N ξ X . Computing the determinant of the matrix above givesdet( ˆ B )( ξ ) = 8 N ( ξ − S N ) ( ξ + ξ ) . If ξ = η + zµ , thendet( ˆ B )( η + zµ ) = 8 N ( z − h X, η + zµ i N ) | η | . If z is a complex root of a ( η + zµ ) = 0, then from (4.2) it follows, | η + zµ | − N h X, η + zµ i = 0 , i.e. | η | + z = h X,η + zµ i N , and thusdet( ˜ B )( η + zµ ) = 8 N ( z − h X, η + zµ i N ) | η | = 8 N ( z − z − | η | ) | η | = 8 N | η | , llipticity of Bartnik boundary data for stationary vacuum spacetimes 17 which is obviously nonzero for η = 0. Thus the complementing boundary conditionholds. This finishes the proof of Theorem 4.1. (cid:3) It then follows from Theorem 4.1 that the linearization of BVP (3.2) is elliptic,which further implies that the boundary map ˜Π defined in § D ˜Π is a Fredholm operator at (˜ g (4) , Theorem 4.2. If ˜ g (4) is a stationary vacuum spacetime metric such that the As-sumption 3.1 holds, then the moduli space E admits smooth Banach manifold struc-ture near [˜ g (4) ] , and the boundary map Π is Fredholm at [˜ g (4) ] . To conclude this section, recall that in § C can be interpretedgeometrically as a local coordinate chart of the moduli space E , and the map ˜Π isexactly the map Π expressed in this chart. However, such a local chart is effectiveonly if Assumption 3.1 holds. In the following section, we will develop an alternativelocal chart at a reference metric ˜ g (4) ∈ E where Assumption 3.1 fails. Furthermore,we show that the ellipticity result still holds in this case. Remark . The operator β ˜ g (4) δ ∗ ˜ g (4) with Dirichlet boundary condition is ellipticand self-adjoint. This is shown in § S, g S ) (cf.(5.9)), this op-erator is in the form of the Laplace operator plus lower order terms – especiallynontrivial 0-order terms generated by the twist tensor of the metric. If the space-time metric ˜ g (4) is static, then the twist tensor is zero and the operator β ˜ g (4) δ ∗ ˜ g (4) reduces to the Laplacian, which is invertible. However, if the metric is not static,the 0-order terms in the operator are not necessarily vanishing or positive, so theymay result in a nontrivial kernel of the operator, in which case Assumption 3.1might fail. On the other hand, because of ellipticity and self-adjointness, this op-erator must be invertible at least for generic metrics in the space E . It would beinteresting to understand whether invertibility holds for all g (4) ∈ E .5. Alternative charts
In this section, we assume that ˜ g (4) is a fixed stationary vacuum metric whereAssumption 3.1 fails.5.1. Perturbation of the metric.
We will use the projection formalism of stationary spacetimes (cf.[Kr],[G]) in thissubsection. In a globally hyperbolic stationary spacetime ( V (4) , g (4) ), the Killingvector field ∂ t generates an isometric and proper R − action on the spacetime. Let S be the orbit space of this action, i.e. S = V (4) / R . Then S is a smooth 3-manifoldand inherits a Riemannian metric g S , which is the restriction of the metric g (4) to the horizontal distribution — the orthogonal complement of span { ∂ t } in T V (4) .It is shown in [G] that there is a one-to-one correspondence between tensor fields T ′ b...da...c on S and tensor fields T b...da...c on V (4) which satisfy( ∂ t ) a T b...da...c = 0 , ..., ( ∂ t ) d T b...da...c = 0 and L ∂ t T b...da...c = 0 . In the following we will identify tensor fields being on S as tensor fields in V (4) satisfying the conditions above. For example, a vector field X ′ ∈ T S corresponds to a vector field X in V (4) such that h X, ∂ t i g (4) = 0 and L ∂ t X = 0. So we will dropthe prime – identify X ′ as X .In the projection formalism, any stationary spacetime metric g (4) is globally ofthe form g (4) = − u ( dt + θ ) + g S . Here g S is the metric on S , which can also be interpreted as a time-independenttensor field in V (4) as discussed above, and θ is a 1-form on S (in the same sense)such that − u ( dt + θ ) is the dual of ∂ t with respect to g (4) . Remark . In our case V (4) ∼ = R × M , there is a natural diffeomorphism betweenthe quotient manifold S and the hypersurface M = { t = 0 } . Thus we can pull backthe radius function on M to S and define weighted H¨older spaces of tensor fieldson S similarly as in § Y ∈ T m,αδ ( V (4) ) satisfies h Y, ∂ t i g (4) = 0and L ∂ t Y = 0, then it can be identified with a tensor field Y ∈ T m,αδ ( S )Suppose that in the projection formalism, ˜ g (4) is expressed as,(5.1) ˜ g (4) = − u ( dt + θ ) + g S . Take a smooth curve (parametrized by ǫ ) of perturbations of ˜ g (4) given by,(5.2) g (4) ǫ = ˜ g (4) + ǫ ( dt + θ ) . First we prove the following property of this family of metrics.
Proposition 5.2.
The metric g (4) ǫ is Bianchi-free with respect to ˜ g (4) , i.e. β ˜ g (4) g (4) ǫ = 0 . Proof.
Clearly by (5.2), β ˜ g (4) g (4) ǫ = ǫβ ˜ g (4) ( dt + θ ) . Let(5.3) α = ( dt + θ ) , then α = − u ξ , where ξ = − u ( dt + θ ) is the dual of ∂ t . Obviously α ( ∂ t ) = 1, α ( v ) = 0 , ∀ v ∈ T S , and hence tr ˜ g (4) α = − u − . As a result, β ˜ g (4) ( α ) = δ ˜ g (4) ( α ) + 12 d ( tr ˜ g (4) α ) = δ ˜ g (4) ( α ) + u − du. (5.4)For the divergence term above, we have δ ˜ g (4) ( α ) = − u {−∇ ∂ t [ α ( ∂ t )] + α ( ∇ ∂ t ∂ t ) } = 1 u ∇ ∂ t α = − u ∇ ∂ t ( 1 u ξ )= − u ∇ ∂ t ξ = − u − du. (5.5)In the first equality above, we have used the fact that ∇ ∂ t ∂ t = u ∇ u (cf. § S and so α ( ∇ ∂ t ∂ t ) = 0. In the last equality, trivially we have ∇ ∂ t ξ = udu . Equations (5.4) and (5.5) now imply that α is Bianchi-free. (cid:3) In addition to Bianchi-free, the family of metrics g (4) ǫ possesses another property— for generic ǫ the operator β ˜ g (4) δ ∗ g (4) ǫ is invertible, in the following sense: llipticity of Bartnik boundary data for stationary vacuum spacetimes 19 Proposition 5.3.
In any neighborhood I of , there is an ǫ ∈ I such that the BVPwith unknown Y ∈ T m,αδ ( V (4) ) given by, (5.6) ( β ˜ g (4) δ ∗ g (4) ǫ Y = 0 on SY = 0 on ∂S has only the trivial solution Y = 0 . To prove this proposition, we state the following lemma first.
Lemma 5.4.
The BVP (5.6) is elliptic (for ǫ small) and formally self-adjoint.Proof. Since δ ∗ g (4) ǫ Y = L Y g (4) ǫ = L Y (˜ g (4) + ǫα ) = δ ∗ ˜ g (4) Y + ǫ L Y α , one has,(5.7) β ˜ g (4) δ ∗ g (4) ǫ Y = β ˜ g (4) δ ∗ ˜ g (4) Y + ǫ β ˜ g (4) L Y α , where α is as defined in (5.3).Notice that any time-independent vector field Y in V (4) can be decomposed intoa vector field on S and another part proportional to ∂ t . Let Y ⊥ = u − h Y, ∂ t i and Y T = Y + u − Y ⊥ ∂ t . Obviously, Y ⊥ is independent of t , so it can be taken as afunction on S . It is also easy to verify that h Y T , ∂ t i g (4) = 0 and L ∂ t Y T = 0, i.e. Y T is a vector fields on S . Thus we have the decomposition(5.8) Y = Y T − u − Y ⊥ ∂ t . We decompose the vector β ˜ g (4) δ ∗ ˜ g (4) Y on the right side of equation (5.7) in the sameway as described above. It is shown in § [ β ˜ g (4) δ ∗ ˜ g (4) Y ] T = ( ∇ g S ) ∗ ∇ g S Y T + u − Y T ( u ) ∇ u − u − ( ∇ g S ) ∇ u Y T + u dθ ( dθ ( Y T )) − u dθ ( ∇ Y ⊥ u ) − [ β ˜ g (4) δ ∗ ˜ g (4) Y ] ⊥ = − u ∆ g S ( Y ⊥ u ) + 3 h∇ Y ⊥ u , ∇ u i − u Y ⊥ | dθ | + u h dθ, ∇ g S Y T i − dθ ( ∇ u, Y T ) , where ∇ g S (and ∆ g S ) denotes connection (and Laplace operator) of g S . Noticethe leading terms of the operator in (5.9) are the Laplacian [( ∇ g S ) ∗ ∇ g S Y T ] and[ − u ∆ g S ( Y ⊥ u )]. Thus β ˜ g (4) δ ∗ ˜ g (4) is an elliptic operator with respect to the Dirichletboundary condition, and so is the operator β ˜ g (4) δ ∗ g (4) ǫ (for ǫ small) according to theexpression (5.7).Let Y , Y ∈ T m,αδ ( V (4) ) be two asymptotically zero time-independent vectorfields which are vanishing along ∂V (4) . Then, Z S h β ˜ g (4) δ ∗ ˜ g (4) Y , Y i g (4) · u · dvol g S = Z S {h [ β ˜ g (4) δ ∗ ˜ g (4) Y ] T , Y T i g S + ( − [ β ˜ g (4) δ ∗ ˜ g (4) Y ] ⊥ ) · Y ⊥ } · u · dvol g S Substituting equations in (5.9) into the integral above and then integrating by partsgives, Z S {h [ β ˜ g (4) δ ∗ ˜ g (4) Y ] T , Y T i g S + ( − [ β ˜ g (4) δ ∗ ˜ g (4) Y ] ⊥ ) · Y ⊥ } · u · dvol g S = Z S {h [ β ˜ g (4) δ ∗ ˜ g (4) Y ] T , Y T i g S + ( − [ β ˜ g (4) δ ∗ ˜ g (4) Y ] ⊥ ) · Y ⊥ } · u · dvol g S + ( Z ∂S + Z ∞ )[ B ( Y , Y ) − B ( Y , Y )] , (5.10)where B ( Y , Y ) = u h∇ n Y T , Y T i ] + 2 u dθ ( n , Y T ) Y ⊥ + u n ( Y ⊥ ) Y ⊥ . It is obviousthat the boundary integral on ∂S is zero, since Y , Y vanish on the boundary. Theboundary term at infinity R ∞ = lim r →∞ R S r , with S r denoting the sphere of radius r on S . It is also zero because the decay rate of the bilinear form B ( Y , Y ) is2 δ + 1 >
2. (cf. Remark 5.1) Thus it follows that, the differential operator (5.9) isformally self-adjoint with respect to the measure u · dvol g S on S . Remark.
One has the following integration by parts formula in the spacetime( V (4) , g (4) ): Z V (4) h∇ ∗ ˜ g (4) ∇ ˜ g (4) Y , Y i ˜ g (4) dvol ˜ g (4) = Z V (4) h∇ ∗ ˜ g (4) ∇ ˜ g (4) Y , Y i ˜ g (4) dvol ˜ g (4) + Z ∂V (4) h ( ∇ ˜ g (4) ) n Y , Y i ˜ g (4) − h ( ∇ ˜ g (4) ) n Y , Y i ˜ g (4) . When the spacetime ( V (4) , ˜ g (4) ) is stationary, the equation above reduces to theequation (5.10) on the quotient manifold ( S, g S ).Using the same method as above, it is easy to check the following equality holdsfor any time-independent symmetric 2-tensor h ∈ S m,αδ +1 ( V (4) ) and vector field Y ∈ T m,αδ ( V (4) ) with Y | ∂V (4) = 0: Z S h β ˜ g (4) h, Y i ˜ g (4) · u · dvol g S = Z S h h, δ ∗ ˜ g (4) Y + 12 ( δ ˜ g (4) Y )˜ g (4) i ˜ g (4) u · dvol g S . (5.11)Thus, as for the second term on the right side of equation (5.7), we have thefollowing equality for all vector fields Y , Y ∈ T m,αδ ( V (4) ) which vanish at ∂V (4) : Z S h β ˜ g (4) L Y α , Y i ˜ g (4) u · dvol g S = Z S h L Y α , δ ∗ ˜ g (4) Y + 12 ( δ ˜ g (4) Y )˜ g (4) i ˜ g (4) u · dvol g S = Z S − u − h L Y α ( ∂ t ) T , δ ∗ ˜ g (4) Y ( ∂ t ) T i g S u · dvol g S = Z S − u − h dθ ( Y T ) − d ( Y ⊥ u ) , − u dθ ( Y T ) − d ( Y ⊥ u )] i g S u · dvol g S = Z S − u − h − u dθ ( Y T ) − d ( Y ⊥ u )] , dθ ( Y T ) − d ( Y ⊥ u ) i g S u · dvol g S llipticity of Bartnik boundary data for stationary vacuum spacetimes 21 = Z S − u h δ ∗ ˜ g (4) Y ( ∂ t ) T , L Y α ( ∂ t ) T i g S u · dvol g S = Z S h δ ∗ Y + 12 ( δ ˜ g (4) Y )˜ g (4) , L Y α i ˜ g (4) u · dvol g S = Z S h Y , β ˜ g (4) L Y ( dt + θ ) i ˜ g (4) u · dvol g S . In the calculation above, the first equality comes from formula (5.11). The secondand third equalities are based on the decompositions (equations (5.12-13) below) ofthe time-independent vector fields L Y α and δ ∗ ˜ g (4) Y . Furthermore, the last threeequalities above are carried out in the reversed way as the first three.The time-independent vector fields L Y α and δ ∗ ˜ g (4) Y can be decomposed as, [ L Y α ] T = 0[ L Y α ]( ∂t, ∂t ) = 0 { [ L Y α ]( ∂t ) } T = dθ ( Y T ) − d ( Y ⊥ u ) , (5.12)and ( δ ∗ ˜ g (4) Y ) T = δ ∗ g S Y T δ ∗ ˜ g (4) Y ( ∂ t , ∂ t ) = − uY T ( u )[ δ ∗ ˜ g (4) Y ( ∂ t )] T = − u dθ ( Y T ) + u d ( Y ⊥ u ) . (5.13)We refer to § (cid:3) Now we give the proof for Proposition 5.3.
Proof.
We prove it by contradiction. Assume that Proposition 5.3 is not true, sothere exists an interval I which contains 0 such that for any ǫ ∈ I , the system (5.6)has a nonzero solution.From Lemma 5.4, we see that system (5 .
6) gives rise to a analytic curve ofelliptic self-adjoint operators parametrized by ǫ . By the perturbation theory for self-adjoint operators (cf.[K],[W]), there exists a smooth curve of nontrivial solutions Y ( ǫ ) ( ǫ ∈ I ) solving the system (5 . ( β ˜ g (4) δ ∗ g (4) ǫ Y ( ǫ ) = 0 on SY ( ǫ ) = 0 on ∂S ∀ ǫ ∈ I. The proof of this is discussed in detail in § Y (0) is a nontrivialsolution to (5 .
6) at ǫ = 0. In the following we will denote it as ˜ Y = Y (0). Takingthe linearization of the system above at ǫ = 0, we obtain:(5.14) ( β ˜ g (4) δ ∗ ˜ g (4) Y ′ + β ˜ g (4) δ ∗ g ′ ˜ Y = 0 on SY ′ = 0 on ∂S, where Y ′ = ddǫ | ǫ =0 Y ( ǫ ) , δ ∗ g ′ ˜ Y = ddǫ | ǫ =0 δ ∗ g ǫ ˜ Y = 12 ddǫ | ǫ =0 L ˜ Y g ǫ = 12 L ˜ Y α . The first equation in (5.14) gives, − β ˜ g (4) δ ∗ ˜ g (4) Y ′ = β ˜ g (4) δ ∗ g ′ ˜ Y .
Since β ˜ g (4) δ ∗ ˜ g (4) is self-adjoint, the equation above yields that, Z V (4) h β ˜ g (4) δ ∗ g ′ ˜ Y , ˜ Y i dvol ˜ g (4) = − Z V (4) h β ˜ g (4) δ ∗ ˜ g (4) Y ′ , ˜ Y i dvol ˜ g (4) = − Z V (4) h Y ′ , β ˜ g (4) δ ∗ ˜ g (4) ˜ Y i dvol ˜ g (4) = 0 . (5.15)Apply integration by parts to the left side of (5 .
15) and obtain,(5.16) Z V (4) h δ ∗ g ′ ˜ Y , δ ∗ ˜ g (4) ˜ Y + 12 ( δ ˜ Y )˜ g (4) i dvol ˜ g (4) = 0 . In the above, δ ∗ g ′ ˜ Y = L ˜ Y α . Now apply the formulas (5.12-13) to L ˜ Y α and δ ∗ ˜ g (4) ˜ Y , and substitute them into (5 . Z S || dθ ( ˜ Y T ) − d ( ˜ Y ⊥ u ) || g S u · dvol g S = 0 . Therefore, we have(5.17) dθ ( ˜ Y T ) = d ( ˜ Y ⊥ u ) . Recall that ˜ Y is a nontrivial solution to system (5 .
6) at ǫ = 0. By applyingthe decomposition equations in (5.9) to the vector field ˜ Y , we express the time-independent system (5.6) (at ǫ = 0) as an equivalent system:(5.18) ∇ ∗ g S ∇ g S ˜ Y T − u ( ∇ g S ) ∇ u ˜ Y T + u ˜ Y T ( u ) ∇ u + u dθ ( dθ ( ˜ Y T )) − u dθ ( ∇ ˜ Y ⊥ u ) = 0∆ g S ( ˜ Y ⊥ u ) − u h∇ u, ∇ ˜ Y ⊥ u i + uY ⊥ | dθ | −h dθ, ∇ g S ˜ Y T i + u dθ ( ∇ u, Y T ) = 0 . Observe that the last two terms in the first equation in (5 .
18) can be manipulatedas: 12 u dθ ( dθ ( ˜ Y T )) − u dθ ( ∇ ˜ Y ⊥ u )= 12 u dθ [ dθ ( ˜ Y T ) − d ( ˜ Y ⊥ u )]= − u dθ ( dθ ( Y T )) , where the last equality is based on (5 . . ∇ ∗ g S ∇ g S ˜ Y T − u ( ∇ g S ) ∇ u ˜ Y T + 1 u ˜ Y T ( u ) ∇ u − u dθ ( dθ ( ˜ Y T )) = 0Pairing the equation above with ˜ Y T yields,12 ∆ g S ( || ˜ Y T || )+ ||∇ g S ˜ Y T || − u ( ∇ g S ) ∇ u || ˜ Y T || + 1 u || ˜ Y T ( u ) || + 12 u || dθ ( ˜ Y T ) || = 0Based on this equation and the fact that ˜ Y T is asymptotically zero and equalsto zero on ∂S , it is easy to derive by the maximum principle that ˜ Y T = 0, and llipticity of Bartnik boundary data for stationary vacuum spacetimes 23 consequently ˜ Y ⊥ = 0 according to the second equation in (5.18). This contradictswith the assumption that ˜ Y is nontrivial. (cid:3) Combining Propositions 5.2 and 5.3, it is straightforward to derive that,
Theorem 5.5.
In any neighborhood of ˜ g (4) ∈ S , there always exists a perturbation g (4)0 of ˜ g (4) such that β ˜ g (4) g (4)0 = 0 (Bianchi-free) and β ˜ g (4) δ ∗ g (4)0 is invertible. Alternative local charts.Theorem 5.6.
Theorem 1.1 still holds without Assumption 3.1.Proof.
In the case Assumption 3.1 fails, we take a perturbation g (4)0 near ˜ g (4) as described in Theorem 5.5. and modify (3.2) to a new BVP with unknowns( g (4) , F ) ∈ S × C m,αδ ( M ) as follows: ( Ric g (4) − δ ∗ g (4)0 β g (4) g (4)0 = 0∆ F = 0 , on M g ∂M = γaH ∂M + btr ∂M K = Hatr ∂M K + bH ∂M = kω n + a d ∂M ( a/b ) = τ − β g (4) g (4)0 = 0 . on ∂M (5.19)By applying Bianchi operator to the first equation above, one obtains,(5.20) ( β g (4) δ ∗ g (4)0 (cid:0) β g (4) g (4)0 (cid:1) = 0 on M, − β g (4) g (4)0 = 0 on ∂M. Since the operator β ˜ g (4) δ ∗ g (4)0 is invertible, so is the operator β g (4) δ ∗ g (4)0 when g (4) is near ˜ g (4) . Thus (5 .
20) implies that β g (4) g (4)0 = 0 on M . So a solution g (4) of (5.19) near ˜ g (4) must be Ricci flat and gauge free with respect to g (4)0 . As in §
3, to associate the perturbed BVP (5.19) with a natural boundary map, we firstconstruct a solution space C near ˜ g (4) given by, C = { ( g (4) , F ) ∈ S × C m,αδ ( M ) : Ric g (4) = 0 , β g (4) g (4)0 = 0 , ∆ F = 0 on M } . Obviously, (˜ g (4) , ∈ C by construction. Next, as in the proof of Theorem 3.2, weneed to prove that any stationary vacuum metric g (4) near ˜ g (4) can be transformedby a diffeomorphism in D so that it satisfies the gauge condition β g (4) g (4)0 = 0.Consider the following map: G : S × D → ( ∧ ) m.αδ ( V (4) ) G ( g (4) , Φ) = β Φ ∗ g (4) g (4)0 . Notice that β Φ ∗ g (4) g (4)0 = Φ ∗ { β g (4) [(Φ ∗ ) − g (4)0 ] } . Thus the linearization of G at (˜ g (4) , Id ) is given by, D G| (˜ g (4) ,Id ) : T S × T D → ( ∧ ) m.αδ ( V (4) ) D G| (˜ g (4) ,Id ) [( h (4) , Y )] = − β ˜ g (4) δ ∗ g (4)0 Y + β ′ h (4) g (4)0 . Since in the linearization above, the operator [ − β ˜ g (4) δ ∗ g (4)0 ] is invertible, it follows bythe implicit function theorem that, for any g (4) near ˜ g (4) , there is a unique elementΦ ∈ D near to Id V (4) such that the gauge term β Φ ∗ g (4) g (4)0 vanishes.Therefore the perturbed solution space C has similar structure as the space C in §
3, i.e. near (˜ g (4) , C is locally a fiber bundle over the quotient space E / D with fiber being the space of harmonic functions in C m,αδ ( M ). Furthermore, basedon the Theorems 3.2 and 3.3, we conclude there exists a local diffeomorphism P such that C ∼ = E near (˜ g (4) ,
0) via P and(5.21) Π = Π ◦ P , where Π is the natural boundary map defined on C given by,Π : C → B Π ( g (4) , F ) = ( g ∂M , aH ∂M + btr ∂M K,atr ∂M K + bH ∂M , ω n + a d ∂M ( b/a )) . As for ellipticity of the system (5.19), notice that linearization of the equality β g (4) g (4) = 0 yields, ( β g (4) ) ′ h (4) g (4) = − β g (4) h (4) . Thus the linearization of the gauge term in (5.19) at (˜ g (4) ,
0) is given by:[ − δ ∗ g (4)0 β g (4) g (4)0 ] ′ h (4) = − δ ∗ g (4)0 ( β ˜ g (4) ) ′ h (4) g (4)0 = − δ ∗ g (4)0 ( β ˜ g (4) ) ′ h (4) (˜ g (4) + g (4)0 − ˜ g (4) )= δ ∗ g (4)0 β ˜ g (4) h (4) − δ ∗ g (4)0 ( β ˜ g (4) ) ′ h (4) ( g (4)0 − ˜ g (4) )Comparing the system (5.19) with the previous one (3.2), it is easy to see that, atthe reference metric ˜ g (4) , the only differences between their linearizations are givenby the term(5.22) [ δ ∗ g (4)0 β ˜ g (4) − δ ∗ ˜ g (4) β ˜ g (4) ]( h (4) ) − δ ∗ g (4)0 ( β ˜ g (4) ) ′ h (4) ( g (4)0 − ˜ g (4) )in the interior equations, and(5.23) − ( β ) ′ h (4) ( g (4)0 − ˜ g (4) )in the boundary equations. We can choose g (4)0 close enough to ˜ g (4) so that theterms in (5.22-23) are very small. Then the principal symbols of the perturbedsystem (5 .
19) is close to that of the system (3.2). It has been proved that (3 . is a Fredholm map andhence so is Π because of the equivalence relation (5.21). This completes the proof. (cid:3) Local existence and uniqueness
In this section we choose V (4) = R × ( R \ B ) ⊂ R , equipped with the standardcoordinates { t, x i } ( i = 1 , ,
3) induced from R . Let M be the hypersurface { t = 0 } .Then ∂M = S , the unit sphere. Set the reference metric ˜ g (4) = ˜ g (4)0 , where ˜ g (4)0 isthe standard flat (Minkowski) metric on R × ( R \ B ), i.e. ˜ g (4)0 = − dt + Σ i ( dx i ) .Since it is static, i.e. its twist tensor in the quotient formalism is zero, it is easy toverify that Assumption 3.1 holds in this case (cf. § llipticity of Bartnik boundary data for stationary vacuum spacetimes 25 ( C , ˜Π) in § g (4)0 ] ∈ E . Obviously, the Bartnikdata of this metic is(6.1) ˜Π(˜ g (4)0 ,
0) = ( g S , , , , where g S is the standard round metric on S . In this section we apply the ellipticityresult proved in the previous sections to show that in a neighborhood of the standardflat boundary data ( g S , , , Theorem 6.1.
The kernel of D ˜Π (˜ g (4)0 , is trivial.Proof. Assume that ( h (4) , G ) ∈ Ker( D ˜Π (˜ g (4)0 , ). Since ( h (4) , G ) ∈ T (˜ g (4)0 , C , it mustbe a vacuum deformation, in the sense that the following equations hold on M :(6.2) ( ( Ric ) ′ h (4) = 0∆ G = 0 . In addition, since elements in C satisfy the gauge condition β ˜ g (4)0 g (4) = 0, the sameequation holds for the deformation h (4) :(6.3) β ˜ g (4)0 h (4) = 0 on M. Since ( h (4) , G ) preserves the Bartnik boundary data, linearization of the boundaryequations are zero, i.e.(6.4) h ∂M = 0( H ∂M ) ′ h (4) = 0( tr ∂M K ) ′ h (4) + 2 G = 0( ω n ) ′ h (4) + ∇ ∂M G = 0 . As we know, a stationary spacetime metric is uniquely determined by the data set( g, X, N ) on the hypersurface M , where g is the induced metric on M , X is the shiftvector and N is the lapse function. For the standard metric ˜ g (4)0 , the correspondingdata is ( g , ,
1) with g being the flat (Euclidean) metric on R \ B . Thus thedeformation h (4) can be decomposed as h (4) = ( h, Y, v ), where h is the deformationof the Riemannian metric g , Y is the deformation of the shift vector and v is thatof the lapse function.The vacuum condition Ric g (4) = 0 is equivalent to the following equations interms of ( g, X, N ) on M (cf.[Mo]): K = − N L X gRic g + ( trK ) K − K − N D N − N L X K = 0 N ∆ N + | K | − N tr ( L X K ) = 0 δK + d ( trK ) = 0 . It is easy to linearize the equations above at ( g , ,
1) and obtain a system interms of ( h, Y, v ), which is equivalent to equation (6.2), given by,(6.5)
Ric ′ h − D v = 0∆ g v = 0 δ g δ ∗ g Y − dδ g Y = 0∆ G = 0 . on M. The gauge equation (6.3) is equivalent to (cf. § ( δ g Y = 0 δ g h + d ( tr g h + 2 v ) = 0 , on M. The boundary conditions (6.4) are equivalent to:(6.7) h ∂M = 0 H ′ h = 0 tr ∂M δ ∗ g Y + 2 G = 0[ δ ∗ g Y ( n )] T + ∇ g T G = 0 . on ∂M. In the last equation above, we use the superscript ′′ T ′′ to denote the restrictionof tensors to the tangent bundle of ∂M . It is proved in [A2] that the first twoequations in (6.5) combined with the first two boundary conditions in (6.7) implythat v = 0 and h = δ ∗ g Z for some vector field Z ∈ C m +1 ,αδ ( M ) vanishing on ∂M .Additionally, h must satisfy the gauge equation in (6.6). It follows that β g δ ∗ g Z = 0,which further implies that Z = 0. So we obtain h = 0 on M .It remains to prove Y = 0 and G = 0. The third equation in (6.5) and the firstequation in (6.6) together imply: δ g δ ∗ g Y = 0 on M. Pair the equation above with Y . Then integration by parts gives,0 = Z M h δ g δ ∗ g Y, Y i g dvol g = Z M | δ ∗ g Y | − Z ∂M δ ∗ g Y ( n , Y ) − Z ∞ δ ∗ g Y ( n , Y )= Z M | δ ∗ g Y | − Z ∂M δ ∗ g Y ( n , Y T ) − Z ∂M δ ∗ g Y ( n , n ) Y ⊥ . (6.8)In the boundary integral, we have decomposed Y as Y = Y T + Y ⊥ n , with Y ⊥ = h Y, n i g . In the second line, the boundary term at infinity R ∞ = lim r →∞ R S r is zerobecause the decay rate of [ δ ∗ g Y ( n , Y )] is 2 δ + 1 > δ ∗ g Y ( n , Y T ) = h [ δ ∗ g Y ( n )] T , Y T i = −h∇ g T G, Y T i = − div g T ( G · Y T ) + G · div g T Y T = − div g T ( G · Y T ) −
12 ( tr ∂M δ ∗ g Y ) · div g T Y T . Here the second equality comes from the last boundary equation in (6.7) and thelast equality is based on the third boundary equation in (6.7). As for the last termin (6.8), notice that we have the following equality on the boundary:0 = δ g Y = − δ ∗ g Y ( n , n ) − tr ∂M δ ∗ g Y, llipticity of Bartnik boundary data for stationary vacuum spacetimes 27 so that δ ∗ g Y ( n , n ) = − tr ∂M δ ∗ g Y . In addition, tr ∂M δ ∗ g Y = tr ∂M δ ∗ g Y T + tr ∂M δ ∗ g ( Y ⊥ n ) = div g T Y T + H g Y ⊥ = div g T Y T + 2 Y ⊥ . Substituting these computations into the in-tegral equation (6.8) gives,0 = Z M | δ ∗ g Y | + Z ∂M
12 ( div g T Y T + 2 Y ⊥ ) · div g T Y T + Z ∂M ( div g T Y T + 2 Y ⊥ ) Y ⊥ = Z M | δ ∗ g Y | + 12 Z ∂M ( div g T Y T ) + 4 Y ⊥ · div g T Y T + 4( Y ⊥ ) = Z M | δ ∗ g Y | + 12 Z ∂M ( div g T Y T + 2 Y ⊥ ) = Z M | δ ∗ g Y | + 12 Z ∂M ( tr ∂M δ ∗ g Y ) . It immediately follows, δ ∗ g Y = 0 on M,tr ∂M δ ∗ g Y = 0 on ∂M. (6.9)The first equation above implies that Y is a Killing vector field of the flat metric g on R \ B . In addition Y must be asymptotically zero since it comes from adeformation of the asymptotically flat metrics in C . Thus it follows Y = 0 on M .The boundary equation in (6.9) implies that G = 0 on ∂M according to (6.7).Furthermore, G is asymptotically zero and harmonic according to (6.5). So G = 0on M . This completes the proof. (cid:3) Next, we prove that the Fredholm map D ˜Π (˜ g (4)0 , is of index 0 by showing theoperator D F = ( D L , D B ) defined in § g (4)0 , D B can be continuously deformed to a newcollection of self-adjoint boundary data.Define the new boundary operator D ˜ B as follows: D ˜ B : T (0 , ˜ g (4)0 ) [ S × C m,αδ ( M )] → B D ˜ B ( h (4) , G ) = ( h ∂M , ∇ n ( h (4) ( n , n )) , n ( G ) , − ∇ n [ h (4) ( ∂ t )] T , − ∇ n h (4) ( ∂ t , ∂ t ) , − ∇ n h (4) ( n ) T , − ∇ n h (4) ( ∂ t , n ) ) . (6.10)Let N denote the space of deformations ( h (4) , G ) of (˜ g (4)0 ,
0) in
S × C m,αδ ( M ) thatare in the kernel of the boundary operator D ˜ B , i.e. N = { ( h (4) , G ) ∈ T (0 , ˜ g (4)0 ) [ S × C m,αδ ( M )] : D ˜ B ( h (4) , G ) = 0 } . Lemma 6.2.
The operator D L : N → [( S ) m − ,αδ +2 × C m − ,αδ +2 ]( M ) , given by D L ( h (4) , G ) = ( D ∗ ˜ g (4)0 D ˜ g (4)0 h (4) , ∆ G ) , is formally self-adjoint.Proof. Let ( h (4) , G ) , ( k (4) , J ) denote two deformations in N . Integration by partsyields: Z M h D L ( h (4) , G ) , ( k (4) , J ) i ˜ g (4)0 dvol g = Z M h D L ( k (4) , J ) , ( h (4) , G ) i ˜ g (4)0 + Z ∂M B [( k (4) , J ) , ( h (4) , G )] − B [( h (4) , G ) , ( k (4) , J )] . Here the boundary term at infinity is zero because of the decay behavior of the thedeformations. The bilinear form B is given by, B [( k (4) , J ) , ( h (4) , G )] = h∇ n k (4) , h (4) i ˜ g (4)0 + n ( J ) G. It is easy to verify that the terms above are zero because ( h (4) , G ) and ( k (4) , J )make all the boundary terms listed in (6.10) vanish. Therefore D L is formallyself-adjoint. (cid:3) Thus it follows that the new operator ( D L , D ˜ B ) is of index 0. Next we show thatthe boundary data in D B can be deformed continuously through elliptic boundarydata to D ˜ B . Define a family of boundary operator D B t , t ∈ [0 ,
1] as follows, D B t : T (˜ g (4)0 , [ S × C m,αδ ( M )] → B D B t ( h (4) , G ) = ( h ∂M , (1 − t )( H ∂M ) ′ h (4) + t ∇ n ( h (4) ( n , n )) , (1 − t )( tr ∂M K ) ′ h (4) + t n ( G ) , −
12 [ ∇ n [ h (4) ( ∂ t )]( e i ) + (1 − t ) ∇ e i [ h (4) ( ∂ t )]( n )] + (1 − t ) e i ( G ) , − ∇ n h (4) ( ∂ t , ∂ t ) + (1 − t )[ 12 ∇ n tr M h + δh ( n )] , − ∇ n h (4) ( n ) T + (1 − t )[ −∇ e i h (4) ( e i ) T + 12 ∇ g T ( trh (4) )] , − ∇ n h (4) ( n , ∂ t ) − (1 − t ) ∇ e i h (4) ( e i , ∂ t ) ) . Here { e i } , i = 2 , T ( ∂M ). It is easy to checkthat D B = D ˜ B . When t = 0, it is obvious that the first three lines of the boundarydata above give the first three of the linearized Bartnik boundary data. It is alsoeasy to check that the fourth line above (at t = 0) has the same principal part asthe linearized Bartnik boundary term ( ω n + d ∂M G ) ′ . Moreover, the last three linesabove are respectively the n , tangential ( ∂M ) and ∂ t components of the gauge term β ˜ g (4)0 h (4) when t = 0. Therefore, D B = D B . Lemma 6.3.
The operator ( D L , D B t ) is elliptic for t ∈ [0 , .Proof. One can carry out the same proof as in §
4. Since the shift vector and lapsefunction of ˜ g (4)0 are simply X = 0 and N = 1, the principal symbol of the interioroperator is (cf.equation (4.2)), L ( ξ ) = ( ξ + ξ + ξ ) I × . llipticity of Bartnik boundary data for stationary vacuum spacetimes 29 Set (
X, N ) = (0 ,
1) in equation (4.4) to compute the principal matrix of boundaryoperator D B . Make a linear combination with the matrix of D B . One can derivethe principal matrix of the boundary operator D B t ( t ∈ [0 , B t ( ξ ) = × ( ˜ B t ) × ∗ , where ˜ B t is given by (up to a factor of − − ), tξ − (1 − t ) ξ − (1 − t ) ξ − tξ − t ) ξ (1 − t ) ξ − − t ) ξ − t ) ξ ξ − − t ) ξ − t ) ξ ξ ξ − t ) ξ − t ) ξ − t ) ξ − t ) ξ − (1 − t ) ξ ξ
00 (1 − t ) ξ − (1 − t ) ξ ξ ξ (1 − t ) ξ (1 − t ) ξ . The determinant of B t ( ξ ) isdet B t ( ξ ) = 132 [ tξ − (2 + t )(1 − t ) ξ ( ξ + ξ )] · [2(2 + t )(1 − t ) ( ξ + ξ ) ξ − tξ ] . Let ξ = zµ + η , where z = i | η | , the root of det L ( zµ + η ) = 0 with positive imaginarypart. Thendet( B t ( zµ + η )) = −
132 [ t + (2 + t )(1 − t ) ] · [2(2 + t )(1 − t ) + 4 t ] | η | , which obviously never vanishes for t ∈ [0 , η = 0. Thus the complementingboundary condition holds for all t ∈ [0 , (cid:3) To conclude, we have the following theorem:
Theorem 6.4.
The boundary map ˜Π is locally a diffeomorphism near (˜ g (4)0 , .Proof. From Lemma 6.2, 6.3 and the homotopy invariance of the index, it followsthat the index of the boundary map ˜Π is 0 at (˜ g (4)0 , D ˜Π (˜ g (4)0 , is trivial. Thus, the linearization D ˜Π (˜ g (4)0 , is an isomorphism. Then the inverse function theorem in Banach spacesgives the theorem. (cid:3) Now the equivalence relation between the maps ˜Π and Π (cf. §
3) gives Theorem1.2. 7.
Appendix
Notations.
On a Riemannian manifold M ∼ = R \ B , we can pull back the standard coordi-nates { x i } ( i = 1 , ,
3) and the radius function r from R \ B to M under a chosendiffeomorphism. Then given m ∈ N , and α, δ ∈ R , the weighted H¨older spaces on M are defined as, C mδ ( M ) = { functions v on M : || v || C mδ = Σ mk =0 sup r k + δ |∇ k v | < ∞} ,C m,αδ ( M ) = { functions v on M : || v || C mδ + sup x,y [min( r ( x ) , r ( y )) m + α + δ ∇ m v ( x ) − ∇ m v ( y ) | x − y | α ] < ∞} ,M et m,αδ ( M ) = { metrics g on M : ( g ij − δ ij ) ∈ C m,αδ ( M ) } ,T m,αδ ( M ) = { vector fields X on M : X i ∈ C m,αδ ( M ) } , ( ∧ ) m,αδ ( M ) = { − forms σ on M : σ i ∈ C m,αδ ( M ) } ,D m,αδ ( M ) = { diffeomorphisms Ψ : M → M, Ψ( x , x , x ) = ( y , y , y ) : y i − x i ∈ C m,αδ ( M ) } . A tensor field is called asymptotically trivial (or zero) at the rate of δ , if it belongsto one of the spaces above. It is well-known that Laplace-type operators withDirichlet boundary condition are Fredholm when acting on these weighted H¨olderspaces (cf.[LM],[Mc],[MV]).On the compact manifold ∂M , we use the standard H¨older norm to define variousBanach spaces of tensor fields as following, C m ( ∂M ) = { functions v on M : || v || C m = Σ mk =0 sup |∇ k v | < ∞} ,C m,α ( ∂M ) = { functions v on M : || v || C m + sup x,y [ ∇ m v ( x ) − ∇ m v ( y ) | x − y | α ] < ∞} ,M et m,α ( ∂M ) = { metrics g on ∂M : g ab ∈ C m,α ( ∂M ) } , ( ∧ ) m,α ( ∂M ) = { − forms σ on ∂M : σ a ∈ C m,α ( ∂M ) } . On the boundary ∂M we use the index 1 to denote the normal direction to ∂M and2 , a, b = 2 , V (4) , g (4) ) be a stationary spacetime such that V (4) ∼ = R × M , with coor-dinates { t, x i } such that R is parametrized by t and M = { t = 0 } . Assume ∂ t is the time-like Killing vector field. In such a spacetime, a tensor field τ is called time-independent if L ∂ t τ = 0. In this case, we can think of τ as a tensor field on M . Define the following spaces of time-independent tensor fields in ( V (4) , g (4) ): C m,αδ ( V (4) ) = { functions f in V (4) : ∂ t f = 0 , f ∈ C m,αδ ( M ) } ,T m,αδ ( V (4) ) = { vector fields Y in V (4) : L ∂ t Y = 0 , Y γ ∈ C m,αδ ( M ) } ,S m,αδ ( V (4) ) = { symmetric 2-tensor fields h (4) in V (4) : L ∂ t h (4) = 0 , h (4) γβ ∈ C m,αδ ( M ) } , ( ∧ ) m,αδ ( V (4) ) = { ω in V (4) : L ∂ t ω = 0 , ω γ ∈ C m,αδ ( M ) } . In the definition above, the greek letters γ, β = 0 , , ,
3, where the index 0 denotesthe ∂ t component of the tensor field and 1 , , M ) components.7.2. Scalar fields a, b in the time translation.
As in proposition 2.1, set up the diffeomorphismΦ f : V (4) → V (4) Φ f ( t, x , x , x ) = ( t + f, x , x , x ) . llipticity of Bartnik boundary data for stationary vacuum spacetimes 31 It maps the hypersurface M = { t = 0 } to a new slice Φ f ( M ) = { t = f } in V (4) . Recall that N is the future-pointing time-like unit normal vector to theslice M ⊂ ( V (4) , g (4) ) and n is the outward unit normal to ∂M ⊂ ( M, g ). Theircorrespondence in the pull-back spacetime ( V (4) , Φ ∗ f ( g (4) )) are ˆ N and ˆ n . Thus d Φ f ( ˆ N ) and d Φ f (ˆ n ) are the unit normal vector fields to the new slice Φ f ( M ) andits boundary Φ f ( ∂M ) in ( V (4) , g (4) ) respectively. Since f | ∂M = 0, ∂M = Φ f ( ∂M )and d Φ f | ∂M = Id ∂M . So along the boundary ∂M the unit normal vectors ( ˆN , ˆn )must be mapped to the same subspace as ( N , n ) in T V (4) , i.e. d Φ f ( ˆN ) , d Φ f ( ˆn ) ∈ span { N , n } . Therefore there exist scalar fields a, b, c, d belongs to C m,α ( ∂M ) so that(7.1) ( d Φ f ( ˆN ) = a N + b n ,d Φ f ( ˆn ) = c N + d n . In addition, by the definition of pull-back metric we have h d Φ f ( ˆN ) , d Φ f ( ˆN ) i g (4) = h ˆ N , ˆ N i Φ ∗ f g (4) = − h d Φ f ( ˆn ) , d Φ f ( ˆn ) i g (4) = h ˆ n , ˆ n i Φ ∗ f g (4) = 1; h d Φ f ( ˆN ) , d Φ f ( ˆn ) i g (4) = h ˆ N , ˆ n i Φ ∗ f g (4) = 0 . So we obtain the following equations for ( a, b, c, d ),(7.2) − a + b = − , − c + d = 1 , − ac + bd = 0 . It further implies that a = d and b = c . Without loss of generality (up to thechoice of directions), we can assume, a = d > , b = c > . Moreover, the vector field d Φ f ( ˆN ) must be orthogonal to d Φ f ( ∂ x i ) with respectto g (4) because h d Φ f ( ˆN ) , d Φ f ( ∂ x i ) i g (4) = h ˆN , ∂ x i i Φ ∗ g (4) = 0 , ∀ i = 1 , , . (7.3)By the definition of Φ f , it follows that d Φ f ( ∂ x i ) = ( ∂ i f ) ∂ t + ∂ x i . On the otherhand, it is easy to verify that,(7.4) h ∂ t − X + N ∇ f, ( ∂ i f ) ∂ t + ∂ x i i g (4) = 0 , ∀ i = 1 , , . where ∇ f denotes the gradient of f with respect to the metric g (4) . Thus, equations(7.3) and (7.4) imply that d Φ f ( ˆN ) must be proportional to ∂ t − X + N ∇ f . We make the direction choice so that d Φ f ( ˆN ) is future pointing, then d Φ f ( ˆN ) = ∂ t − X + N ∇ f q −|| ∂ t − X + N ∇ f || g (4) = ∂ t − X + N ∇ fN q X ( f ) − || N ∇ f || g (4) = ∂ t − X + N ∇ g fN q X ( f ) + X ( f ) − N ||∇ g f || g = ∂ t − X + N ∇ g fN p (1 + X ( f )) − N ||∇ g f || g , where ∇ g f denotes the gradient of f with respect to the induced metric g on M .Therefore, according to the first equation in (7.1), we obtain a = −h N , d Φ f ( ˆN ) i g (4) = − g (4) ( ∂ t − X, ∂ t − X + N ∇ g f ) N p (1 + X ( f )) − N ||∇ g f || = 1 + X ( f ) p (1 + X ( f )) − N ||∇ g f || . In the above, the second equality is because N = ( ∂ t − X ) /N according to the 3+1formalism (2.1). Moreover, since f is chosen to be vanishing on ∂M , so ∇ g f = n ( f ) · n on the boundary. Thus ||∇ g f || g = n ( f ) on the boundary ∂M , and X ( f ) | ∂M = h X, n i n ( f ) and consequently, a = 1 + h X, n i n ( f ) p [1 + h X, n i n ( f )] − N | n ( f ) | on ∂M, which is the formula (2.11). Based on (7.2), we easily derive the formula for b asfollows, b = N n ( f ) p [1 + h X, n i n ( f )] − N | n ( f ) | . Linearization of boundary operator B . For simplicity of notation, we will write h instead of h (4) in this section. Subindex0 denotes the ∂ t direction in V (4) , index 1 denotes the outward normal direction to ∂M and 2 , ∂M .1.With respect to the deformation h , linearization of g ∂M is easily seen to be:( g ∂M ) ′ h = ( h , h , h ) . H ∂M :By the formula of the linearization of mean curvature (cf. § H ∂M ) ′ h = − ∂ h − ∂ h + ∂ ( h + h ) + O ( h ) . llipticity of Bartnik boundary data for stationary vacuum spacetimes 33 K :The defining equation for K is K ij = − N L X g ij , where g ij denotes the Riemannian metric induced from g (4) on M and X is the shiftvector on M . Notice that X is the dual of the 1 − form g (4) ( ∂ t ) | M , i.e. X ♭i = g (4)0 i .Variation of K with respect to h is given by,( K ′ h ) ij = − N ( L ( X ) ′ g ij + L X h ij ) + O ( h ) . As for the variation X ′ , it is given by, X i = g ik g (4)0 k , ( X ′ ) i = ˜ h ik g (4)0 k + g ik h k , where ˜ h is the variation of the inverse g ij . It is easy to see that˜ h ij g jk = − g ij h jk . Therefore, L X ′ g ij = g ik ∇ Mj ( X ′ ) k + g jk ∇ Mi ( X ′ ) k = ∇ Mj { g ik ( X ′ ) k } + ∇ Mi { g jk ( X ′ ) k } = ∇ Mj { g il ˜ h lk g (4)0 k + g il g lk h k } + ∇ Mi { g jl ˜ h lk g (4)0 k + g jl g lk h k } = ∇ Mj {− h il g lk g (4)0 k + h i } + ∇ Mi {− h jl g lk g (4)0 k + h j } = ∇ Mj {− h il X l + h i } + ∇ Mi {− h jl X l + h j } , and L X h ij = ∇ MX h ij + h ik ∇ Mj X k + h jk ∇ Mi X k = ∇ MX h ij + O ( h ) . In the above the connection ∇ M denotes the covariant derivative with respect tothe induced metric g on M . Thus,( K ′ h ) ij = − N [ ∂ i h j + ∂ j h i + ∂ X h ij − X l ( ∂ i h jl + ∂ j h il )] + O ( h ) , and consequently,[ tr ∂M K ] ′ h = tr ∂M ( K ′ h ) + O ( h )= − N [2 ∂ h + 2 ∂ h + ∂ X ( h + h ) − X l (2 ∂ h l + 2 ∂ h l )] + O ( h ) , (cid:0) ( ω n ) ′ h (cid:1) i = [( K ( n ) | ∂M ) ′ h ] i = [ K ′ h ( n ) | ∂M ] i + O ( h )= − N [ ∂ h i + ∂ i h + ∂ X h i − X l ( ∂ h il + ∂ i h l )] + O ( h ) , with i = 2 , .
4. Linearization of the gauge term β ˜ g (4) g (4) . Obviously [ β ˜ g (4) g (4) ] ′ h = β ˜ g (4) h . Take an arbitrary vector field Y ∈ T m,αδ ( V (4) ).Then β g (4) h ( Y ) = δ g (4) h ( Y ) + Y ( trh ) , where the two terms on the right side arecomputed as, δ g (4) h ( Y ) = ∇ N h ( N , Y ) − Σ k =1 , , ∇ k h ( Y ) k + O ( h )= 1 N ∇ ∂ t − X h ( ∂ t − X, Y ) − Σ k =1 , , ∇ k h ( Y ) k + O ( h )= − N ∂ X h ( ∂ t − X, Y ) − Σ k =1 , , ∂ k h ( Y ) k + O ( h ) ,trh = − h ( N , N ) + h + h + h = − N h ( ∂ t − X, ∂ t − X ) + h + h + h = − N ( h + X i X j h ij − X l h l ) + h + h + h . Therefore, for i = 1 , ,
3, the i th component of the linearized gauge term is givenby,[ β ˜ g (4) h (4) ] i = − N [ ∂ i h + X k X j ∂ i h kj − X l ∂ i h l ] + 12 ∂ i ( h + h + h ) − N [ ∂ X h i − X k ∂ X h ki ] − Σ k =1 , , ∂ k h ki + O ( h );and the ∂ t component of the gauge term is given by,[ β ˜ g (4) h (4) ] = − N [ ∂ X h − X k ∂ X h k ] − Σ k =1 , , ∂ k h k + O ( h ) . Summing up all the computations above, we obtain the boundary symbol matrix˜ B in equation (4.4), given by,˜ B = − N N ξ N ξ Nξ Nξ − Nξ X − Nξ X − N ξ Nξ Nξ − Nξ X Nξ X − Nξ X − N ξ Nξ Nξ − Nξ X − Nξ X Nξ X ξ S − ξ X − ξ X − ξ X ξ X X + N ξ − SX ξ X X − SX + 2 N ξ ξ X X − SX + 2 N ξ ξ − ξ X S − ξ X − ξ X ξ X X − N ξ ξ X X − SX + 2 N ξ ξ X X ξ − ξ X − ξ X S − ξ X ξ X X − N ξ ξ X X ξ X X − SX + 2 N ξ S N ξ − SX ) 2( N ξ − SX ) 2( N ξ − SX ) 0 0 0 , inside which S = ξ X + ξ X + ξ X .7.4. Calculation of the boundary symbol matrix.
To compute the determinant of the matrix ˜ B , we can first simplify it by factoringout the common factors in every row: factor out ( − N ) from the first row, 2 N from the second row, N from the third and fourth row, and 2 from the last row.Then we obtain that det ˜ B = − N det ˆ B , with ˆ B given by,ˆ B = − ξ − ξ ξ ξ − ξ X − ξ X − N ξ ξ ξ − ξ X ξ X − ξ X − N ξ ξ ξ − ξ X − ξ X ξ X ξ S − ξ X − ξ X − ξ X ξ X X + N ξ − SX ξ X X − SX + 2 N ξ ξ X X − SX + 2 N ξ ξ − ξ X S − ξ X − ξ X ξ X X − N ξ ξ X X − SX + 2 N ξ ξ X X ξ − ξ X − ξ X S − ξ X ξ X X − N ξ ξ X X ξ X X − SX + 2 N ξ S N ξ − SX N ξ − SX N ξ − SX , We now carry out the following row and column operation to simplify ˆ B . First,multiply the first row of ˆ B by − X and then add it to the second row. Multiply llipticity of Bartnik boundary data for stationary vacuum spacetimes 35 the first row by 2 N and then add it to the fifth row. The matrix becomes:ˆ B = − ξ − ξ ξ ξ − N ξ ξ ξ − ξ X ξ X − ξ X − N ξ ξ ξ − ξ X − ξ X ξ X ξ S − ξ X − ξ X − ξ X ξ X X + N ξ − SX ξ X X − SX ξ X X − SX ξ − ξ X S − ξ X − ξ X ξ X X − N ξ ξ X X − SX + 2 N ξ ξ X X ξ − ξ X − ξ X S − ξ X ξ X X − N ξ ξ X X ξ X X − SX + 2 N ξ S N ξ − SX N ξ − SX N ξ − SX . In ˆ B , multiply the second row by ( − N ) and add it to the last row:ˆ B = − ξ − ξ ξ ξ − N ξ ξ ξ − ξ X ξ X − ξ X − N ξ ξ ξ − ξ X − ξ X ξ X ξ S − ξ X − ξ X − ξ X ξ X X + N ξ − SX ξ X X − SX ξ X X − SX ξ − ξ X S − ξ X − ξ X ξ X X − N ξ ξ X X − SX + 2 N ξ ξ X X ξ − ξ X − ξ X S − ξ X ξ X X − N ξ ξ X X ξ X X − SX + 2 N ξ S N ξ − SX − SX − SX . In ˆ B , multiply the second column by N and add it to the sixth column. Thenmultiply the second column by X i and add it to the (2 + i )th column ( i = 1 , , B = − ξ − ξ ξ ξ − N ξ ξ ξ − ξ X ξ X − ξ X − N ξ ξ ξ − ξ X − ξ X ξ X ξ S − ξ X − ξ X − ξ X ξ X X + 2 N ξ − SX ξ X X − SX ξ X X − SX ξ − ξ X S − ξ X − ξ X ξ X X ξ X X − SX + 2 N ξ ξ X X ξ − ξ X − ξ X S − ξ X ξ X X ξ X X ξ X X − SX + 2 N ξ S N ξ N S . In ˆ B , multiply the i th column by X and add it to the ( i +3)th column ( i = 3 , , X i and add it to column ( i + 1) , ( i = 1 , , B = − ξ − ξ ξ ξ ξ X ξ X − N ξ ξ ξ ξ X + ξ X − ξ X − N ξ ξ ξ − ξ X ξ X + ξ X ξ S − ξ X − ξ X − ξ X N ξ − SX − SX ξ − ξ X S − ξ X − ξ X N ξ ξ − ξ X − ξ X S − ξ X N ξ S N ξ N S + N ξ X . In ˆ B , multiply column 2 by X i and add it to column ( i + 2), ( i = 1 , , N ) − and add it to column 3. Then multiply the first row by X and add it to row 2:ˆ B = − ξ − ξ ξ ξ − N ξ ξ ξ X + ξ X − ξ X − N ξ ξ − ξ X ξ X + ξ X ξ S N ξ − SX − SX ξ S N ξ ξ S N ξ S N ξ + SX SX SX N S + N ξ X . This is the matrix given in (4.5).7.5.
Calculation in the projection formalism.
Take a general stationary metric in V (4) expressed in the projection formalismas, g (4) = − u ( dt + θ ) + g S . In this section, we use ∇ to denote the Levi-Civita connection of g (4) , and ∇ g S todenote that of g S . We first state two simple facts.
1. Since ∂ t is a Killing vector field, it follows that for any vector field Y ∈ T V (4) ,we have h∇ ∂t ∂ t , Y i = −h∇ Y ∂ t , ∂ t i = uY ( u ). Thus,(7.5) ∇ ∂ t ∂ t = u ∇ u. Notice that ∇ u is a vector field on S because u is independent of t .2. For any horizontal vector fields v, w ∈ T S , one has h v, ∂ t i = 0, L ∂t v = 0, andhence h∇ v w, ∂ t i = −h w, ∇ v ∂ t i = h v, ∇ w ∂ t i = −h∇ w v, ∂ t i . Let ξ = − u ( dt + θ ) be the dual of ∂ t . Then by the definition of exterior derivative,we have dξ ( v, w ) = h∇ v ∂ t , w i − h∇ w ∂ t , v i . Combining the quality above, we obtain dξ ( v, w ) = 2 h∇ w v, ∂ t i . On the other hand, dξ = d [ − u ( dt + θ )] = − u dθ − udu ∧ ( dt + dθ ) = − u dθ + 2 u − du ∧ ξ . Thus we can derive that,2 h∇ w v, ∂ t i = dξ ( v, w ) = − u dθ ( v, w ) ξ ([ v, w ]) = h∇ v w − ∇ w v, ∂ t i = 2 h∇ v w, ∂ t i = u dθ ( v, w ) . (7.6)Next we give a proof for the formula (5.12):Let α = dt + θ = − u − ξ , so α ( ∂ t ) = 1 , α ( v ) = 0 ∀ v ∈ T S . Then according tothe the following Lie-derivative formula for time-independent vector feilds
A, B, Y in the spacetime: L Y α ( A, B ) = Y [ α ( A, B )] − α ([ Y, A ] , B ) − α ( A, [ Y, B ]) , it is easy to see that ( L Y α ( ∂t, ∂t ) = 0[ L Y α ] T = 0 . As for the mixed component of L Y α , we can carry out the following computationfor v ∈ T S , L Y α ( ∂ t , v ) = − α ([ Y, v ] , ∂ t )= − α ([ Y, v ])= u − ξ ([ Y, v ]) . (7.7)As discussed in §
5, any vector field Y ∈ T m,αδ ( V (4) ) can be decomposed as,(7.8) Y = Y T − Y ⊥ u ∂t, with Y T ∈ T S and Y ⊥ = 1 u h Y, ∂ t i . Thus, for v ∈ T S , one has, ξ [ Y, v ] = ξ ([ Y T , v ]) − ξ ([ Y ⊥ u ∂ t , v ]) = ξ ([ Y T , v ]) + ξ [ v ( Y ⊥ u ) ∂ t ]= u dθ ( Y T , v ) − u v ( Y ⊥ u ) . In the last equality above, we use the formula in (7.6) to compute ξ ([ Y T , v ]). Plug-ging this to equation (7.7) we obtain[ L Y α ( ∂ t )] T = dθ ( Y T ) − d ( Y ⊥ u )This completes the proof of (5.12).Using the same notation as above, we give a proof of the formula (5.13) as follows. llipticity of Bartnik boundary data for stationary vacuum spacetimes 37 Based on the decomposition (7.8), we have2 δ ∗ g (4) Y = L Y T g (4) − L Y ⊥ u ∂ t g (4) . (7.9)In the following, we assume v, w ∈ T S . For the first term in (7.9), we have L Y T g (4) ( ∂ t , ∂ t ) = 2 h∇ ∂ t Y T , ∂ t i = 2 h∇ Y T ∂ t , ∂ t i = − uY T ( u ) ,L Y T g (4) ( v, w ) = h∇ v Y T , w i + h∇ w Y T , v i = L Y T g S ( v, w ) ,L Y T g (4) ( ∂ t , v ) = h∇ ∂ t Y T , v i + h∇ v Y T , ∂ t i = h∇ Y T ∂ t , v i + h∇ v Y T , ∂ t i = −h∇ Y T v, ∂ t i + h∇ v Y T , ∂ t i = 2 h∇ v Y T , ∂ t i = − u dθ ( Y T , v ) . In the last equality, we apply (7.6) to the term 2 h∇ v Y T , ∂ t i . Summing up theequations,(7.10) L Y T g (4) ( ∂ t , ∂ t ) = − uY T ( u )[ L Y T g (4) ( ∂ t )] T = − u dθ ( Y T )[ L Y T g (4) ] T = L Y T g S . As for the second term on the right side of (7.9), basic calculation yields, L Y ⊥ u ∂ t g (4) = Y ⊥ u L ∂ t g (4) + d ( Y ⊥ u ) ⊙ ξ = d ( Y ⊥ u ) ⊙ ξ. Thus,(7.11) L Y ⊥ u ∂ t g (4) ( ∂ t , ∂ t ) = 0[ L Y ⊥ u ∂ t g (4) ( ∂ t )] T = − u d ( Y ⊥ u )[ L Y ⊥ u ∂ t g (4) ] T = 0 . Equations (7.10) and (7.11) together give (5.13).At last we derive the decomposition (5.9) of the Bianchi gauge operator.We assume g (4) is in addition vacuum, which is equivalent to the following systemin the projection formalism, (cf.[G],[H1],[H2]),(7.12) Ric g S = u D g S u + 2 u − ( ω − | ω | g S · g S )∆ g S u = 2 u − | ω | g S δ g S ω + 3 u − h du, ω i g S = 0 dω = 0 , where ω is the twist tensor defined as, ω = − u ⋆ g S dθ. Here we use subscript ′′ g S ′′ to denote geometric operators (connection and Lapla-cian) of the Riemannian metric g S on the quotient manifold S . First observe that,from the last equation in (7.12), it follows that0 = dω = d ( u ⋆ g S dθ ) = d ⋆ g S ( u dθ ) = δ g S ( u dθ ) = u δ g S dθ − u dθ ( ∇ u ) . Thus, we obtain uδ g S dθ = 3 dθ ( ∇ u ) . (7.13) Moreover, based on the second equation in (7.12), one easily obtains,∆ g S u = 12 u | dθ | g S . (7.14)Now we analyze the operator β g (4) δ ∗ g (4) acting on a time-independent vector field Y , which is decomposed as in (7.8). To begin with, because the metric g (4) isvacuum, a standard Bochner-Weitzenbock formula gives,2 β g (4) δ ∗ g (4) Y = ∇ ∗ ∇ Y − Ric g (4) ( Y ) = ∇ ∗ ∇ Y. Based on the formula of the Laplace operator, we have, ∇ ∗ ∇ Y = 1 u [ ∇ ∂ t ∇ ∂ t Y − ∇ ∇ ∂t ∂ t Y ] − Σ i [ ∇ e i ∇ e i Y − ∇ ∇ ei e i Y ] , (7.15)where e i ( i = 1 , ,
3) are taken to be geodesic normal basis on S . In the following,we compute the tensors on the right side of (7.15) term by term.1.We start with the first two terms in (7.15). Since [ Y, ∂ t ] = 0, we have ∇ ∂ t Y = ∇ Y ∂ t . Thus, the first term in (7.15) gives ∇ ∂ t ∇ ∂ t Y = ∇ ∂ t ∇ Y ∂ t = ∇ ∇ Y ∂ t ∂ t = ∇ ∇ Y T ∂ t ∂ t − Y ⊥ u ∇ ∇ ∂t ∂ t ∂ t = ∇ ∇ Y T ∂ t ∂ t − Y ⊥ u ∇ u ∇ u ∂ t . (7.16)In the above, we use the decomposition (7.8) and the fact that ∇ ∂ t ∂ t = u ∇ u . Inthe same way, the second term in (7.15) gives, ∇ ∇ ∂t ∂ t Y = ∇ u ∇ u Y = ∇ u ∇ u Y T − ∇ u ∇ u ( Y ⊥ u ∂ t )= ∇ u ∇ u Y T − h u ∇ u, ∇ Y ⊥ u i · ∂ t − Y ⊥ u ∇ u ∇ u ∂ t (7.17)Subtract (7.17) from (7.16). We get ∇ ∂ t ∇ ∂ t Y − ∇ ∇ ∂t ∂ t Y = ∇ ∇ Y T ∂ t ∂ t − u ∇ ∇ u Y T + h u ∇ u, ∇ Y ⊥ u i · ∂ t (7.18)Based on (7.6) and (7.8),(7.19) ∇ v ∂ t = ( ∇ v ∂ t ) T − u − h∇ v ∂ t , ∂ t i · ∂ t = − u dθ ( v ) + u − v ( u ) · ∂ t ∀ v ∈ T S.
Thus the first term on the right side of (7.18) can be written as, ∇ ∇ Y T ∂ t ∂ t = ∇ − u dθ ( Y T )+ u − Y T ( u ) · ∂ t ∂ t = ∇ − u dθ ( Y T ) ∂ t + u − Y T ( u ) ∇ ∂ t ∂ t = 14 u dθ ( dθ ( Y T )) − udθ ( Y T , ∇ u ) · ∂ t + Y T ( u ) ∇ u. (7.20)For two vector fields v, w ∈ T S , we have(7.21) ∇ v w = [ ∇ v w ] T + h∇ v w, ∂ t i · ∂ t − u = ( ∇ g S ) v w + 12 dθ ( w, v ) · ∂ t . Applying the decomposition above to the second term on the right side of (7.18)yields, u ∇ ∇ u Y T = u ( ∇ g S ) ∇ u Y T − udθ ( ∇ u, Y T ) · ∂ t (7.22) llipticity of Bartnik boundary data for stationary vacuum spacetimes 39 Plug (7.20),(7.22) into (7.18), ∇ ∂ t ∇ ∂ t Y − ∇ ∇ ∂t ∂ t Y = 14 u dθ ( dθ ( Y T )) − udθ ( Y T , ∇ u ) · ∂ t + Y T ( u ) ∇ u − u ( ∇ g S ) ∇ u Y T + 12 udθ ( ∇ u, Y T ) · ∂ t + h u ∇ u, ∇ Y ⊥ u i · ∂ t = 14 u dθ ( dθ ( Y T )) + Y T ( u ) ∇ u − u ( ∇ g S ) ∇ u Y T + h u ∇ u, ∇ Y ⊥ u i · ∂ t (7.23)2.As for the third term in (7.15), we first use the decomposition (7.8) to get ∇ e i ∇ e i Y = ∇ e i ∇ e i Y T − ∇ e i ∇ e i ( Y ⊥ u ∂ t ) . (7.24)Apply formula (7.21) to the first term on the right side above, ∇ e i ∇ e i Y T = ∇ e i [( ∇ g S ) e i Y T + 12 dθ ( Y T , e i ) · ∂ t ]= ( ∇ g S ) e i ( ∇ g S ) e i Y T + 12 dθ (( ∇ g S ) e i Y T , e i ) · ∂ t + [ ∇ e i dθ ( Y T , e i )] · ∂ t + 12 dθ ( Y T , e i ) · ∇ e i ∂ t = ( ∇ g S ) e i ( ∇ g S ) e i Y T + 12 dθ (( ∇ g S ) e i Y T , e i ) · ∂ t + [ ∇ e i dθ ( Y T , e i )] · ∂ t + 12 dθ ( Y T , e i ) · ( − u dθ ( e i ) + u − e i ( u ) · ∂ t )= ( ∇ g S ) e i ( ∇ g S ) e i Y T − u dθ ( Y T , e i ) · dθ ( e i )+ [ 12 dθ (( ∇ g S ) e i Y T , e i ) + 12 u − dθ ( Y T , e i ) e i ( u ) + 12 ∇ e i dθ ( Y T , e i )] · ∂ t . (7.25)Here in the third equality, we use the formula (7.19). As for the second term in(7.24), we first write it as, ∇ e i ∇ e i ( Y ⊥ u ∂ t ) = ∇ e i [ e i ( Y ⊥ u ) ∂ t + Y ⊥ u ∇ e i ∂ t ]= e i ( e i ( Y ⊥ u )) ∂ t + 2 e i ( Y ⊥ u ) ∇ e i ∂ t + Y ⊥ u ∇ e i ∇ e i ∂ t . (7.26)Apply formula (7.19) to the second term above,2 e i ( Y ⊥ u ) ∇ e i ∂ t = 2 e i ( Y ⊥ u )[ − u dθ ( e i ) + u − e i ( u ) · ∂ t ] . (7.27) Apply formula (7.19) twice to the third term in (7.26) gives Y ⊥ u ∇ e i ∇ e i ∂ t = Y ⊥ u ∇ e i [ − u dθ ( e i ) + u − e i ( u ) · ∂ t ]= Y ⊥ u ( ∇ g S ) e i [ − u dθ ( e i )] − uY ⊥ dθ ( dθ ( e i ) , e i ) · ∂ t . + Y ⊥ u e i ( u − e i ( u )) · ∂ t + Y ⊥ u e i ( u )( − u dθ ( e i ) + u − e i ( u ) · ∂ t )= Y ⊥ u ( ∇ g S ) e i [ − u dθ ( e i )] − Y ⊥ e i ( u ) dθ ( e i )+ [ 14 uY ⊥ dθ ( e i , dθ ( e i )) + Y ⊥ u e i ( e i ( u ))] · ∂ t . (7.28)Summarizing equations (7.24-28) gives, ∇ e i ∇ e i Y = ( ∇ g S ) e i ( ∇ g S ) e i Y T − u dθ ( Y T , e i ) · dθ ( e i )+ [ 12 dθ (( ∇ g S ) e i Y T , e i ) + 12 u − dθ ( Y T , e i ) e i ( u ) + 12 ∇ e i dθ ( Y T , e i )] · ∂ t − e i ( e i ( Y ⊥ u )) ∂ t − e i ( Y ⊥ u )[ − u dθ ( e i ) + u − e i ( u ) · ∂ t ] − Y ⊥ u ( ∇ g S ) e i [ − u dθ ( e i )] + 12 Y ⊥ e i ( u ) dθ ( e i ) − [ 14 uY ⊥ dθ ( e i , dθ ( e i )) + Y ⊥ u e i ( e i ( u ))] · ∂ t = ( ∇ g S ) e i ( ∇ g S ) e i Y T − u dθ ( Y T , e i ) · dθ ( e i ) + e i ( Y ⊥ u ) u dθ ( e i ) − Y ⊥ u ( ∇ g S ) e i [ − u dθ ( e i )] + 12 Y ⊥ e i ( u ) dθ ( e i )+ [ 12 dθ (( ∇ g S ) e i Y T , e i ) + 12 u − dθ ( Y T , e i ) e i ( u ) + 12 ∇ e i dθ ( Y T , e i )] · ∂ t − [ e i ( e i ( Y ⊥ u )) + 2 u − e i ( Y ⊥ u ) e i ( u ) + 14 uY ⊥ dθ ( e i , dθ ( e i )) + Y ⊥ u e i ( e i ( u ))] · ∂ t Take negative trace of the expression above, − Σ i ∇ e i ∇ e i Y = ( ∇ g S ) ∗ ∇ g S Y T + 14 u dθ ( dθ ( Y T )) − u dθ ( ∇ Y ⊥ u )+ Y ⊥ u δ g S [ u dθ ] − Y ⊥ dθ ( ∇ u )+ [ h dθ, ∇ g S Y T i + 12 δ g S ( dθ ( Y T )) − ∆ g S ( Y ⊥ u ) + 14 uY ⊥ | dθ | ] · ∂ t + [ − u − dθ ( Y T , ∇ u ) + 2 u − h∇ Y ⊥ u , ∇ u i − Y ⊥ u ∆ g S u ] · ∂ t . (7.29) llipticity of Bartnik boundary data for stationary vacuum spacetimes 41 Notice that in the third line of (7.29), δ g S [ u dθ ] = u δ g S dθ − udθ ( ∇ u ). In thefourth line of (7.29), δ g S ( dθ ( Y T )) = − δ g S dθ ( Y T ) + h dθ, ∇ g S Y T i . Thus (7.29) canbe rewritten as, − Σ i ∇ e i ∇ e i Y = ( ∇ g S ) ∗ ∇ g S Y T + 14 u dθ ( dθ ( Y T )) − u dθ ( ∇ Y ⊥ u )+ 12 Y ⊥ uδ g S [ dθ ] − Y ⊥ dθ ( ∇ u )+ [ h dθ, ∇ g S Y T i − δ g S dθ ( Y T ) − ∆ g S ( Y ⊥ u ) + 14 uY ⊥ | dθ | ] · ∂ t + [ − u − dθ ( Y T , ∇ u ) + 2 u − h∇ Y ⊥ u , ∇ u i − Y ⊥ u ∆ g S u ] · ∂ t . (7.30)4.The last term in (7.15) is zero because ∇ e i e i = 0 based on formula (7.21).Adding up the equations (7.23) and (7.30), we have [ ∇ ∗ ∇ Y ] T = ( ∇ g S ) ∗ ∇ g S Y T + u − Y T ( u ) ∇ u − u − ( ∇ g S ) ∇ u Y T + u dθ ( dθ ( Y T )) − u dθ ( ∇ Y ⊥ u )+ Y ⊥ uδ g S dθ − Y ⊥ dθ ( ∇ u ) h∇ ∗ ∇ Y, u − ∂ t i = ∆ g S ( Y ⊥ u ) − u − h∇ Y ⊥ u , ∇ u i − h dθ, ∇ g S Y T i− uY ⊥ | dθ | + Y ⊥ u ∆ g S u − u − dθ ( ∇ u, Y T ) + δ g S dθ ( Y T ) . According to equations (7.13) and (7.14), the equations above can be simplified as,(7.31) [ ∇ ∗ ∇ Y ] T = ( ∇ g S ) ∗ ∇ g S Y T + u − Y T ( u ) ∇ u − u − ( ∇ g S ) ∇ u Y T + u dθ ( dθ ( Y T )) − u dθ ( ∇ Y ⊥ u ) h∇ ∗ ∇ Y, u − ∂ t i = ∆ g S ( Y ⊥ u ) − u − h∇ Y ⊥ u , ∇ u i + uY ⊥ | dθ | −h dθ, ∇ g S Y T i + u − dθ ( ∇ u, Y T ) , which is the formula (5.9).We note that in the case where ˜ g (4) = ˜ g (4)0 , the standard flat (Minkowski) metricon R × ( R \ B ). Because θ = 0 , u = 1 for ˜ g (4)0 , equations in (7.31) can be simplifiedas ( [ ∇ ∗ ∇ Y ] T = ( ∇ g ) ∗ ∇ g Y T [ ∇ ∗ ∇ Y ] ⊥ = ∆ g Y ⊥ . Here g denotes the flat metric in R \ B . Based on the decomposition above, it iseasy to see that the solution to ∇ ∗ ∇ Y = 0 with trivial Dirichlet boundary conditionmust be Y = 0. Therefore, the operator β ˜ g (4)0 δ ∗ ˜ g (4)0 is invertible, i.e. the Assumption3.1 holds for ˜ g (4)0 . Perturbation of the operator β ˜ g (4) δ ∗ ˜ g (4) . Here we show that in the beginning of the proof of Proposition 5.3, if it is assumedthat the system (5.6) admits a nontrivial solution for all ǫ ∈ I , then there exists asmooth curve Y ( ǫ ) solving it.In the following discussion, we work with the weighted Sobolev spaces. Since avector Y solving BVP (5.6) must be C ∞ smooth by elliptic regularity, the Banachspace we choose does not affect the final conclusion. Let M and V (4) be the sameas in § p, δ , the weighted Sobolev spaces are defined as, L pδ ( M ) = { functions u on M : || u || p,δ = ( Z M | u | p r δp − n dx ) /p < ∞} ,W k,pδ ( M ) = { functions u on M : Σ ki =0 || D i u || p,δ + i < ∞} ,W k,p ( T V (4) ) = { vector fields Y in V (4) : L ∂ t Y = 0 , Y γ ∈ W k,pδ ( M ) , γ = 0 , , , } . Let W be the space of vector fields that vanish on the boundary: W = { Y ∈ W , δ ( T V (4) ) : Y = 0 on ∂M } . The operator β ˜ g (4) δ ∗ g (4) ǫ give rise to a family of map T ǫ defined as, T ǫ : W → L δ ( T V ) T ǫ ( Y ) = r β ˜ g (4) δ ∗ g (4) ǫ ( Y )It is obvious that T ǫ is an analytic curve of linear operators parametrized by ǫ . It hasbeen proved in § β ˜ g (4) δ ∗ g (4) ǫ is formally self-adjoint, thus so is T ǫ . Moreover, bystandard theory of elliptic operators on non-compact manifold (cf.[MV],[Le]), T ǫ hascompact resolvent. According to [K] (Chapter 7, Theorem 3.9), for an analytic curve T ǫ of self-adjoint operators that have compact resolvent, all (repeated) eigenvaluescan be represented by analytic functions u n ( ǫ ) and there is a sequence of analyticvector-valued functions Y n ( ǫ ) representing the eigenvectors to u n ( ǫ ).If, as assumed in the proof of Proposition 5.3, there is an interval I such that forall ǫ ∈ I the system (5.6) admits a nonzero solution, then 0 is an eigenvalue of T ǫ for all ǫ ∈ I . Based on the analysis above, for each ǫ ∈ I there must be a function u n such that u n ( ǫ ) = 0. However, there are only countably many eigenvalues u n .Thus, among the eigenfunctions u n ( ǫ ), there must be some u n such that u n ( ǫ ) = 0for uncountably ǫ . Since u n ( ǫ ) is analytic in ǫ , u n ≡ ǫ ∈ I . Correspondingly, Y n ( ǫ ) is a smooth curve of 0-eigenvectors for T ǫ ( ǫ ∈ I ). This directly implies thatthere is a smooth curve Y ( ǫ ) solving the system (5.6).7.7. Bianchi operator in the Minkowski spacetime.
We give the proof of equation (6.6). Recall that the spacetime is ( V (4) = R × ( R \ B ) , ˜ g (4)0 ), where ˜ g (4)0 is the standard Minkowski metric. The metric is variedalong the infinitesimal deformation h (4) such that(7.32) β ˜ g (4)0 h (4) = 0 . Under the standard coordinate { t, x i } of the flat spacetime ( V (4) , ˜ g (4)0 ), the Killingvector field ∂ t is of unit norm and it is perpendicular to the hypersurface M = { t = 0 } . On the hypersurface, ∂ x i ( i = 1 , ,
3) is a orthonormal basis of thetangent bundle. Let ˜ ∇ denote the Levi-Civita connection of the flat metric. Then llipticity of Bartnik boundary data for stationary vacuum spacetimes 43 ˜ ∇ ∂ t ∂ t = 0 , ˜ ∇ ∂ t ∂ x i = 0 and ˜ ∇ ∂ xi ∂ x j = 0. As in §
6, we use g to denote the induced(flat) metric on M .While the infinitesimal variation of the spacetime metric is h (4) , the shift vectoris deformed by the vector field Y ∈ T M such that h Y, ∂ x i i g = h (4) ( ∂ t , ∂ x i ). Pairing(7.32) with ∂ t , we obtain,0 = β ˜ g (4)0 h (4) ( ∂ t ) = [ δ ˜ g (4)0 h (4) + 12 dtrh (4) ]( ∂ t ) = δ ˜ g (4)0 h (4) ( ∂ t )= ˜ ∇ ∂ t h (4) ( ∂ t , ∂ t ) − Σ i ˜ ∇ ∂ xi h (4) ( ∂ x i , ∂ t )= ∂ t (cid:0) h (4) ( ∂ t , ∂ t ) (cid:1) − Σ i ∂ x i (cid:0) h (4) ( ∂ x i , ∂ t ) (cid:1) = − Σ i ∂ x i (cid:0) h (4) ( ∂ x i , ∂ t ) (cid:1) = δ g Y. This gives the first equation in (6.6). In the calculation above, the second equalityuses the fact that h (4) is time-independent. The third and last equality are basedon that the metric ˜ g (4)0 and g are flat.Under the deformation h (4) , the induced metric on M is deformed by h whichis the restriction of h (4) on M . The lapse function is deformed by v so that h (4) ( ∂ t , ∂ t ) = − v . Pair (7.32) with ∂ x i . Similar calculation as above gives,0 = β ˜ g (4)0 h (4) ( ∂ x i ) = [ δ ˜ g (4)0 + 12 dtrh (4) ]( ∂ x i )= δ ˜ g (4)0 h (4) ( ∂ x i ) + 12 ∂ x i ( trh (4) )= ˜ ∇ ∂ t h (4) ( ∂ t , ∂ x i ) − Σ k ˜ ∇ ∂ xk h (4) ( ∂ x k , ∂ xi ) + 12 ∂ x i ( trh (4) )= ∂ t (cid:0) h (4) ( ∂ t , ∂ x i ) (cid:1) − Σ k ∂ x k (cid:0) h (4) ( ∂ x k , ∂ x i ) (cid:1) + 12 ∂ x i ( trh (4) )= − Σ k ∂ x k (cid:0) h (4) ( ∂ x k , ∂ x i ) (cid:1) + 12 ∂ x i ( tr g h (4) − h (4) ( ∂ t , ∂ t ))= δ g h ( ∂ x i ) + 12 d ( tr g h + 2 v )This gives the second equation in (6.6). References [ ADN ] S. Agmon, A. Douglis, and L.Nirenberg. Estimates near the boundary for solutionsof elliptic partial differential equations satisfying general boundary conditions. I, II, Comm.Pure App. Math., , (1964) 623-727, (1964), pp. 35-92.[ AK ] M. Anderson and M. Khuri, On the Bartnik extension problem for static vacuum Ein-stein metrics, Classical & Quantum Gravity, , (2013), 125005.[ A
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