We construct a solution operator for the linearized constant scalar curvature equation at hyperbolic space in space dimension larger than or equal to two. The solution operator has good support propagation properties and gains two derivatives relative to standard norms. It can be used for CorvinoSchoentype hyperbolic gluing partly extending the recently introduced MaoOhTao gluing method to the hyperbolic setting.
['General Relativity and Quantum Cosmology', 'Mathematics - Differential Geometry']
A Bogovskitype operator for CorvinoSchoen hyperbolic gluing
{'A Bogovski i-type operator for Corvino-Schoen hyperbolic gluing': 'Piotr T. Chru´sciel, Albachiara Cogo, Andrea N¨utzi', 'Abstract': 'We construct a solution operator for the linearized constant scalar curvature equation at hyperbolic space in space dimension larger than or equal to two. The solution operator has good support propagation properties and gains two derivatives relative to standard norms. It can be used for Corvino-Schoen-type hyperbolic gluing, partly extending the recently introduced Mao-Oh-Tao gluing method to the hyperbolic setting.', '1 Introduction': "Mao, Oh and Tao [11] have shown how to simplify the gluing constructions of Corvino and Schoen [9,10] for three-dimensional asymptotically flat general relativistic vacuum initial data sets. The simplification comes from using Bogovski ˇ i-type solution operators for the linearized equations, which preserve the support of the sources and have good differentiability properties. Similar methods [12] led to simplifications and improvements of the gluing constructions of Carlotto and Schoen [4] in dimensions larger than or equal to three, in particular they provide better asymptotics of the glued solutions.  \nThe aim of this note is to point out the existence of a Bogovski ˇ i-type solution operator for the linearization of the constant scalar curvature equation at hyperbolic space, with the same good support and differentiability properties. It thus can be used in nonlinear problems, e.g. in Corvino-type gluing near hyperbolic space, and in particular to provide simpler proofs of some of the small-data gluing results of [7]. See Section 4 for details.  \nThe constant scalar curvature equation for an n -dimensional asymptotically hyperbolic Riemannian manifold ( M,g ) is  \nR( g ) = -n ( n -1) , (1.1)  \nwhere R( g ) is the scalar curvature of the metric g . Recall that (1.1) is equivalent to the time-symmetric general relativistic constraint equations with cosmological constant 2Λ = -n ( n -1).  \nHere we consider the case where the manifold is the upper half-space  \nM = /C0 n = { x = ( x 1 , . . . , x n ) | x 1 > 0 } .  \nThen a trivial solution of (1.1) is given by the hyperbolic metric  \nb = 1 ( x 1 ) 2 ( dx 1 ⊗ dx 1 + · · · + dx n ⊗ dx n ) .  \nWe consider the linearization of (1.1) at b . This is given by the underdetermined elliptic linear operator P : C ∞ ( /C0 n , S 2 ( T ∗ /C0 n )) → C ∞ ( /C0 n , /CA ),  \nP( h ) = D i D j h ij -D i D i h j j -R ij h ij , (1.2)  \nwhere S 2 ( T ∗ /C0 n ) denotes the bundle of two-covariant symmetric tensors, D i and R ij are the covariant derivative, respectively the Ricci curvature, of b , and where we use the index conventions detailed in Remark 2.1.  \nOur main proposition provides a solution operator for the linear equation  \nP( h ) = f (1.3)  \nwith good support and differentiability properties. We discuss necessary integrability conditions of this equation when the source f has compact support in /C0 n , and one seeks a solution h with compact support. Recall that the space of static KIDs for the hyperbolic metric is given by the kernel of the formal adjoint of P. Explicitly, this is spanned over /CA by the n +1 functions  \n1 + | x | 2 2 x 1 , 1 -| x | 2 2 x 1 , x 2 x 1 , . . . , x n x 1 , (1.4)  \nwhich we will denote by κ 1 , . . . , κ n +1 . For all h and all a = 1 . . . n + 1 the product κ a P( h ) is a divergence, in fact the identity  \nκ a P( h ) = D i V ( a ) i ( h ) (1.5)  \nholds where by definition  \nV ( a ) i ( h ) = κ a ( D j h ij ) -( D j κ a ) h ij -κ a ( D i h j j ) + ( D i κ a ) h j j .  \nTherefore for all h with compact support,  \n∫ /C0 n κ a P( h ) dµ b = 0 (1.6)  \nwith dµ b = 1 ( x 1 ) n dx 1 · · · dx n . These are n +1 necessary integrability conditions for (1.3), in the compact support case. We now state the main proposition, which shows in particular that these integrability conditions are sufficient for existence of a solution with compact support.  \nFor an open subset Ω ⊆ /C0 n we denote by C ∞ c (Ω , /CA ) the space of smooth functions with compact support in Ω, analogously for sections of S 2 ( T ∗ Ω).  \nProposition 1.1. Let Ω ⊆ /C0 n be a bounded connected open set with smooth boundary such that ¯ Ω ⊆ /C0 n . Then there exists an /CA -linear operator  \nG : C ∞ c (Ω , /CA ) → C ∞ c (Ω , S 2 ( T ∗ Ω))  \nwith the following properties:  \n(a1) The map Π : C ∞ (Ω , /CA ) → C ∞ (Ω , /CA ) defined by  \nc c PG = /BD -Π  \nhas rank n + 1 , is a projection ( Π 2 = Π ), and is explicitly given as follows: Let κ 1 , . . . , κ n +1 be the static KIDs (1.4) . There exist functions ϕ 1 , . . . , ϕ n +1 ∈ C ∞ c (Ω , /CA ) such that for all f ∈ C ∞ c (Ω , /CA ) ,  \nΠ( f ) = n +1 ∑ a =1 ϕ a ∫ Ω κ a f dµ b . (1.7)  \n(a2) For all s ∈ /CA and 1 < p < ∞ there exists a constant C > 0 such that for all i = 1 . . . n and f ∈ C ∞ c (Ω , /CA ) ,  \n‖ G( f ) ‖ W s,p ≤ C ‖ f ‖ W s -2 ,p , ‖ [G , ∂ x i ]( f ) ‖ W s,p ≤ C ‖ f ‖ W s -2 ,p ,  \nwhere ‖ · ‖ W s,p are the standard L p -based Sobolev norms. Analogous estimates hold for standard Holder norms with exponent 0 < α < 1 .  \nThe proof appears at the end of Section 3.2.  \nAcknowledgement. Many useful discussions with Bobby Beig and Erwann Delay are acknowledged. We are thankful to the Erwinn Schrodinger Institute in Vienna for hospitality during an initial stage of this work. AC was supported by the DAAD 'Forschungsstipendien fur Doktorandinnen und Doktoranden' during her research stay in Vienna. AN is supported by the Swiss National Science Foundation, project number P500PT-214470. PTC's research was supported in part by the NSF under Grant No. DMS-1928930 while the author was in residence at the Simons Laufer Mathematical Sciences Institute (formerly MSRI) in Berkeley during the Fall 2024 semester.", '2 Linearized scalar curvature equation': 'In this section we derive a simple formula for the linear operator P in (1.2). We will use the conventions detailed in the following remark.  \nRemark 2.1. For a symmetric two-tensor h ∈ C ∞ ( /C0 n , S 2 ( T ∗ /C0 n )) we denote by h ij = h ( ∂ x i , ∂ x j ) its components relative to the coordinates x . Indices are lowered with b ij = b ( ∂ x i , ∂ x j ) and raised with its inverse b ij . All repeated indices are summed over 1 . . . n , regardless of their position. We denote by δ ij = δ ij = δ i j the Kronecker delta. ✷  \nLet h ∈ C ∞ ( /C0 n , S 2 ( T ∗ /C0 n )). Using the fact that R ij = (1 -n ) b ij we have  \nP( h ) = D i D j ( h ij -h /lscript /lscript b ij ) -(1 -n ) h /lscript /lscript  \n.  \nThus if we define ˜ h = h -h /lscript /lscript b then we obtain  \nP( h ) = D i D j ˜ h ij -˜ h i i .  \nThe non-vanishing Christoffel symbols of the hyperbolic metric are  \nΓ 1 11 = -1 x 1 , Γ 1 pq = 1 x 1 δ pq , Γ p 1 q = -1 x 1 δ p q ,  \nwhere p, q = 2 . . . n . Then by direct calculation,  \nD i D j ˜ h ij = ( x 1 ) n +1 ∂ i ∂ j ( 1 ( x 1 ) n +1 ˜ h ij ) +( x 1 ) n ∂ 1 ( 1 ( x 1 ) n +1 δ ij ˜ h ij ) ,  \nwhere ∂ i = ∂ x i . Thus, using ˜ h i i = 1 ( x 1 ) 2 δ ij ˜ h ij ,  \nP( h ) = ( x 1 ) n +1 ∂ i ∂ j ( 1 ( x 1 ) n +1 ˜ h ij ) +( x 1 ) n +1 ∂ 1 ( 1 ( x 1 ) n +2 ( δ ij ˜ h ij ) ) .  \nWe have therefore proved the following lemma:  \nLemma 2.2. Let h ∈ C ∞ ( /C0 n , S 2 ( T ∗ /C0 n )) . Define  \nA = 1 ( x 1 ) n +1 ( h -h i i b ) . (2.1)  \nThen h = ( x 1 ) n +1 ( A + 1 1 -n A i i b ) and  \nP( h ) = ( x 1 ) n +1 ( ∂ i ∂ j A ij + ∂ 1 ( 1 x 1 A ii ) ) .', '3 Construction of the solution operator': 'In this section we construct the operator G of Proposition 1.1. The main tool is a right inverse for the divergence operator on Euclidean space originally introduced by Bogovski ˇ i [2, 3]. We recall it in the next lemma.  \nLemma 3.1. Let Ω ⊆ /C0 n be as in Proposition 1.1. There exists an /CA -linear map B : C ∞ c (Ω , /CA ) → C ∞ c (Ω , /CA n ) with the following properties:  \n- (b1) There exists a function ϕ ∈ C ∞ c (Ω , /CA ) such that ∫ Ω ϕdx = 1 where dx is the Lebesgue measure, and such that for all f ∈ C ∞ c (Ω , /CA ) ,  \n∂ i B i ( f ) = f -ϕ ∫ Ω f dx . (3.1)  \n- (b2) For all s ∈ /CA and 1 < p < ∞ there exists a constant C > 0 such that for all i = 1 . . . n and f ∈ C ∞ c (Ω , /CA ) ,  \n‖ B( f ) ‖ W s,p ≤ C ‖ f ‖ W s -1 ,p , ‖ [B , ∂ i ]( f ) ‖ W s,p ≤ C ‖ f ‖ W s -1 ,p .  \nAnalogous estimates hold for Holder norms with exponent 0 < α < 1 .  \n- (b3) For all i = 1 . . . n and f ∈ C ∞ c (Ω , /CA ) ,  \n∫ Ω B i ( f ) dx = -∫ Ω x i f dx + m i ∫ Ω f dx , (3.2)  \nwhere by definition m i = ∫ Ω x i ϕdx and m ij = ∫ Ω x i x j ϕdx , and where we use the summation convention in Remark 2.1.  \n∫ Ω x j B j ( f ) dx = -1 2 ∫ Ω | x | 2 f dx + 1 2 m jj ∫ Ω f dx , (3.3)  \nProof. Such an operator exists by [14, Section 4.1] (note that Ω can be covered by finitely many open sets that are star-shaped with respect to a ball), or alternatively by [13, Theorem 1]. More precisely, the references  \nimply that there exists an operator ˜ B : C ∞ c ( /CA n , /CA ) → C ∞ c ( /CA n , /CA n ) that satisfies (b1), ˜ B( C ∞ c (Ω , /CA )) ⊆ C ∞ c (Ω , /CA n ), and such that if χ ∈ C ∞ c ( /CA n , /CA ) then ˜ B χ is a pseudodifferential operator of order -1 on /CA n . Define B = ˜ B | C ∞ c (Ω , /CA ) . Then B satisfies (b1), and (b2) using standard mapping properties for pseudodifferential operators, see [15, Chapter 13.6, Proposition 6.5] for Sobolev norms, and [15, Chapter 13.8, Proposition 8.5] for Holder norms.  \nWe show (b3). For (3.2) multiply (3.1) with x i and integrate, which yields  \n∫ Ω x i ∂ j B j ( f ) dx = ∫ Ω x i f dx -m i ∫ Ω f dx .  \nIntegrating by parts yields (3.2). For (3.3) multiply (3.1) with | x | 2 and proceed analogously. ✷  \nGiven such a right inverse B for the divergence operator, we obtain the solution operator G for P( h ) = f in two steps. We first use B to obtain a right inverse for the double divergence on symmetric trace-free two-tensors (Section 3.1). Using this operator one can then solve separately for the trace of h and the trace-free part of h (Section 3.2). In particular, given B the construction of G requires no further analysis.', '3.1 Trace-free symmetric double divergence': 'Let C ∞ c (Ω , S 2 /CA n ) be the space of symmetric n × n matrix valued functions with compact support in Ω. In this section we use the right inverse B of the divergence from Lemma 3.1 to obtain a solution operator for the equation  \n∂ i ∂ j A ij = f , (3.4)  \nwhere the source f has compact support in Ω and where we seek A ∈ C ∞ c (Ω , S 2 /CA n ) with vanishing matrix trace, tr( A ) = A ii = 0. Define  \n· Q ( f ) = ∫ Ω ( 1 , x 1 , x 2 , . . . , x n , | x | 2 ) f dx ∈ /CA n +2 .  \nIntegrating by parts one obtains that for all A ∈ C ∞ c (Ω , S 2 /CA n ) with A ii = 0,  \n· Q ( ∂ i ∂ j A ij ) = 0 . (3.5)  \nThus · Q ( f ) = 0 are n +2 necessary integrability conditions for (3.4).  \nAs a first step we need the next lemma, this is similar to [11, Lemma 2.2].  \nLemma 3.2. Let Ω , B , ϕ , m i , m ij be as in Lemma 3.1. Define the operator S : C ∞ c (Ω , /CA ) → C ∞ c (Ω , S 2 /CA n ) by  \nS ij ( f ) = 1 2 B i (B j ( f )) + 1 2 B j (B i ( f )) .  \nThen:  \n(c1) For all f ∈ C ∞ c (Ω , /CA ) ,  \n∂ i ∂ j S ij ( f ) = f -( ( ϕ + m j ∂ j ϕ ) ∫ Ω f dx -∂ j ϕ ∫ Ω x j f dx ) .  \n- (c2) The estimates in point (a2) of Proposition 1.1 hold for S .\n- (c3) For all f ∈ C ∞ c (Ω , /CA ) ,  \n∫ Ω S ii ( f ) dx = ( m i m i -1 2 m ii ) ∫ Ω f dx -m i ∫ Ω x i f dx + 1 2 ∫ Ω | x | 2 f dx .  \nProof. (c1): Using point (b1) of Lemma 3.1 we have  \n∂ i ∂ j S ij ( f ) = ∂ j ∂ i B i (B j ( f )) = ∂ j B j ( f ) -∂ j ϕ ∫ Ω B j ( f ) dx = f -ϕ ∫ Ω f dx -∂ j ϕ ∫ Ω B j ( f ) dx.  \nTogether with (b3) one obtains (c1).  \n(c2): This follows directly from (b2), for the commutator estimate also use [B k B /lscript , ∂ i ] = B k [B /lscript , ∂ i ] + [B k , ∂ i ] B /lscript for all k, /lscript, i = 1 . . . n .  \n(c3): This follows directly from (b3). ✷  \nSuppose that f ∈ C ∞ c (Ω , /CA ) satisfies the integrability conditions · Q ( f ) = 0 from (3.5). Then by (c1), respectively (c3),  \n∂ i ∂ j S ij ( f ) = f , ∫ Ω S ii ( f ) dx = 0 .  \nThe second identity and point (b1) of Lemma 3.1 imply S ii ( f ) = ∂ j B j (S ii ( f )). This motivates making the following ansatz for a trace-free solution of (3.4):  \nS ij ( f ) + ( X /lscript k ) ij ∂ /lscript B k ( T ) with T = S ii ( f ) , (3.6)  \nwhere ( X /lscript k ) ij ∈ /CA is symmetric in ij , ( X /lscript k ) ij = ( X /lscript k ) ji . We require that:  \n- · The total symmetrization of ( X /lscript k ) ij in /lscript, i, j vanishes, for all k . Then ∂ i ∂ j ( ( X /lscript k ) ij ∂ /lscript B k ( T ) ) = 0, hence (3.6) still solves (3.4).\n- · ( X /lscript k ) ii = -δ /lscript k for all k, /lscript . Then ( X /lscript k ) ii ∂ /lscript B k ( T ) = -∂ /lscript B /lscript ( T ) = -T , hence (3.6) is trace-free.  \nThe following choice for ( X /lscript k ) ij satisfies these properties:  \n( X /lscript k ) ij = 1 n -1 ( 1 2 ( δ /lscripti δ j k + δ /lscriptj δ i k ) -δ ij δ /lscript k ) . (3.7)  \nLemma 3.3. Let Ω , ϕ , B , S be as in Lemma 3.1 and Lemma 3.2, and let ( X /lscript k ) ij be as in (3.7) . Define · S : C ∞ c (Ω , /CA ) → C ∞ c (Ω , S 2 /CA n ) by  \n· S ij ( f ) = S ij ( f ) + ( X /lscript k ) ij ∂ /lscript B k ( S pp ( f ) ) . (3.8)  \nThen:  \n- (d1) For all f ∈ C ∞ c (Ω , /CA ) one has ∂ i ∂ j · S ij ( f ) = ∂ i ∂ j S ij ( f ) .\n- (d2) The estimates of point (a2) of Proposition 1.1 hold for · S .\n- (d3) For all f ∈ C ∞ c (Ω , /CA ) one has · S ii ( f ) = ϕ ∫ Ω S ii ( f ) dx .  \nIn particular, if · Q ( f ) = 0 then A = · S( f ) is trace-free and solves (3.4).  \nProof. Denote the second term on the right of (3.8) by X ij ( f ).  \n- (d1): By ∂ i ∂ j X ij ( f ) = 0, using our choice of ( X /lscript k ) ij .\n- (d2): This follows from points (b2) of Lemma 3.1 and (c2) of Lemma 3.2.\n- (d3): We have  \nX ii ( f ) = -∂ /lscript B /lscript (S ii ( f )) = -S ii ( f ) + ϕ ∫ Ω S ii ( f ) dx,  \nwhere for the last equality we use (b1). This implies (d3). ✷', '3.2 Linearized constant scalar curvature equation': "Define the linear operator ˜ P : C ∞ (Ω , S 2 /CA n ) → C ∞ (Ω , /CA ),  \n˜ P( A ) = ∂ i ∂ j A ij + ∂ 1 ( 1 x 1 A ii ) .  \nBy Lemma 2.2 this agrees with the linearized scalar curvature operator P up to the change of variables (2.1) and the overall factor ( x 1 ) n +1 . Here we use the operator · S from Lemma 3.3 to obtain a solution operator for  \n˜ P( A ) = f , (3.9)  \nwhere f has compact support in Ω and we seek A ∈ C ∞ c (Ω , S 2 /CA n ). Define  \nQ KID ( f ) = ∫ Ω ( 1 , x 2 , . . . , x n , | x | 2 ) f dx ∈ /CA n +1 .  \nBy (1.6), for all A with compact support we have  \nQ KID ( ˜ P( A )) = 0 (3.10)  \nand thus Q KID ( f ) = 0 are n +1 necessary integrability conditions for (3.9). We indicate how one can solve (3.9) when Q KID ( f ) = 0. Decompose  \nA = · A + 1 n τ /BD  \nwhere · A is trace-free symmetric and τ is a function, and where /BD is the n × n identity matrix. Then (3.9) reads  \n∂ i ∂ j · A ij = f -˜ P ( 1 n τ /BD ) , (3.11)  \nwith ˜ P( 1 n τ /BD ) = 1 n ∂ i ∂ i τ + ∂ 1 ( 1 x 1 τ ). Note that for all τ with compact support,  \n∫ Ω x 1 ˜ P ( 1 n τ /BD ) dx = -∫ Ω 1 x 1 τ dx . (3.12)  \nThus by choosing τ appropriately one can achieve that f ' = f -˜ P( 1 n τ /BD ) satisfies both Q KID ( f ' ) = 0 and ∫ Ω x 1 f ' dx = 0, which means that f ' satisfies the n + 2 integrability conditions · Q ( f ' ) = 0 for the trace-free symmetric double divergence in (3.5). Thus · A = · S( f ' ) is trace-free and solves (3.11).  \nThis is implemented in the following proposition.  \nLemma 3.4. Let Ω , ϕ , · S be as in Lemma 3.3. Set ψ = -x 1 ϕ . Define ˜ G : C ∞ c (Ω , /CA ) → C ∞ c (Ω , S 2 /CA n ) by  \n˜ G( f ) = · S ( f -˜ P ( 1 n ψ ( ∫ Ω x 1 f dx ) /BD ) ) + 1 n ψ ( ∫ Ω x 1 f dx ) /BD .  \nThen:  \n(e1) There exist ϕ 1 , . . . , ϕ n +1 ∈ C ∞ c (Ω , /CA ) such that for all f ∈ C ∞ c (Ω , /CA ) ,  \n˜ P( ˜ G( f )) = f -n +1 ∑ a =1 ϕ a Q a KID ( f ) .  \n(e2) The estimates of point (a2) of Proposition 1.1 hold for ˜ G .  \nProof. (e1): Abbreviate τ = ψ ∫ Ω x 1 f dx and f ' = f -˜ P( 1 n τ /BD ). Then  \n∫ Ω x 1 f ' dx = ∫ Ω x 1 f dx -∫ Ω x 1 ˜ P ( 1 n τ /BD ) dx (3.14)  \nQ KID ( f ' ) = Q KID ( f ) -Q KID ( ˜ P ( 1 n τ /BD )) = Q KID ( f ) , (3.13)  \n= ( 1 + ∫ Ω 1 x 1 ψdx ) ∫ Ω x 1 f dx = 0 .  \nFor (3.13) we use (3.10), and for (3.14) we use (3.12) and ∫ Ω 1 x 1 ψdx = -∫ Ω ϕdx = -1. By definition of ˜ P and ˜ G we have  \nwhere for the second equality we use (d1) and (d3). We now rewrite the first term using (c1), and the second term using (c3), then we simplify using (3.13) and (3.14). This yields (e1) with  \n˜ P( ˜ G( f )) = ∂ i ∂ j · S ij ( f ' ) + ∂ 1 ( 1 x 1 · S ii ( f ' ) ) + ˜ P ( 1 n τ /BD ) = ∂ i ∂ j S ij ( f ' ) + ∂ 1 ( 1 x 1 ϕ ) ∫ Ω S ii ( f ' ) dx + ˜ P ( 1 n τ /BD )  \nϕ a =      ϕ + m i ∂ i ϕ -( m i m i -1 2 m ii ) ∂ 1 ( 1 x 1 ϕ ) if a = 1 , -∂ a ϕ + m a ∂ 1 ( 1 x 1 ϕ ) if a = 2 . . . n , -1 2 ∂ 1 ( 1 x 1 ϕ ) if a = n +1 .  \n(e2): By linearity of · S we can write  \n˜ G( f ) = · S( f ) + ( 1 n ψ /BD -· S ( ˜ P ( 1 n ψ /BD )) ) ∫ Ω x 1 f dx .  \nThe first term satisfies the estimates by (d2), for the second they are clear. ✷  \nWe are now ready to pass to the  \nProof (of Proposition 1.1). Define G by  \nG ij ( f ) = ( x 1 ) n +1 ( ˜ G ij + 1 1 -n b ij b k/lscript ˜ G k/lscript ) ( 1 ( x 1 ) n +1 f ) .  \nWe check (a1): We have (1.7) by Lemma 2.2 and (e1); the map Π has rank n + 1 since by (1.7) the rank is ≤ n + 1 and by (1.6) the rank is ≥ n + 1; we have Π 2 = Π since applying Π to (1.7) and using (1.6) yields 0 = Π -Π 2 . We check (a2): This holds by (e2), note that multiplying and dividing by x 1 does not affect the estimates since ¯ Ω ⊆ /C0 n . ✷", '4 Application to Corvino-Schoen-type gluing': "Proposition 1.1 can be used to prove the following nonlinear result. We give an informal statement, since the details are routine.  \nProposition 4.1 (Informal). Let n ≥ 2 and s > n/ 2 . Let Ω ⊆ /C0 n be an open annulus-type region with ¯ Ω ⊆ /C0 n . Let g in and g out be two smooth (or H s = W s, 2 ) metrics on ¯ Ω that have constant scalar curvature -n ( n -1) and that are sufficiently close in H s to the hyperbolic metric b . Then there exists a smooth (or H s ) metric g on ¯ Ω such that:  \n- · R( g ) = -n ( n -1) mod ran(Π) , where Π is defined in Proposition 1.1.\n- · g = g in in a neighborhood of the interior boundary of ¯ Ω .\n- · g = g out in a neighborhood of the exterior boundary of ¯ Ω .  \nProof. This is similar to [11, Proof of Theorem 1.3, Step 1 and 2], hence we only give a sketch. Write R( b + h ) = -n ( n -1) + P( h ) + N( h ) where N( h ) is the nonlinearity, given by terms of quadratic and higher order. Fix an interpolation h between g in -b and g out -b , then make the ansatz g = b + h + h ' where the correction h ' has compact support in Ω. It is constructed as the solution of the fixed point  \nh ' = -G ( P( h ) + N( h + h ' ) ) . (4.1)  \nThen g satisfies all properties stated in the proposition.  \n✷  \nIn Proposition 4.1, the constant scalar curvature equation is only solved modulo the range of Π, which has dimension n +1. The full equation R( g ) = -n ( n -1) holds if and only if the argument of G in (4.1) is in the kernel of Π, equivalently, if and only if for all a = 1 , . . . , n +1,  \n∫ Ω κ a ( P( h ) + N( h + h ' ) ) dµ b = 0  \nwhere κ a are the static KIDs defined in (1.4). By (1.5) and the divergence theorem, this is equivalent to  \n∫ ∂ out Ω V ( a ) i ( g out -b ) dS i b -∫ ∂ in Ω V ( a ) i ( g in -b ) dS i b = -∫ Ω κ a N( h + h ' ) dµ b (4.2)  \nwhere the integrals on the left are flux integrals over the exterior boundary ∂ out Ω and interior boundary ∂ in Ω respectively, both in the outward direction (in the low regularity regime one can use averaged flux integrals).  \nOf particular interest is the case when g in is given and one looks for g out in the family of boosted Schwarzschild-AdS metrics, which depends on n + 1 mass and boost parameters θ = ( θ 1 , . . . , θ n +1 ), see e.g. [7]. The idea is to apply Proposition 4.1 with θ as parameters, and to then adjust the parameters such that the n +1 conditions (4.2) hold.  \nSince the right hand side of (4.2) is small, it is necessary that the flux integrals of g in are approximately equal to the flux integrals of some member of the family of boosted Schwarzschild-AdS metrics. If this is the case, and under appropriate smallness conditions, one can then construct the exact  \nparameters θ for which (4.2) holds by reformulating (4.2) as a fixed point equation for θ . Arguments of this kind are worked out in detail in [7] in the hyperbolic setting, and in [5,6,9,11] in the Euclidean setting. A particularly simple argument (which also shows that the smallness conditions needed can be realised in some cases) can be given for parity-invariant metrics, see Example 4.2 below.  \nIn this manner, Proposition 1.1 can be used to glue a constant-scalarcurvature metric g in to a boosted Schwarzschild-AdS metric, in such a way that the glued metric has constant scalar curvature. This gives a simpler proof of a similar perturbative result in [7], under the much weaker differentiability conditions of Proposition 4.1, and extends the analysis of [7] to include dimension n = 2.  \nExample 4.2 (Gluing parity-invariant metrics). Let Z : /C0 n → /C0 n be the parity isometry of the hyperbolic metric about the point (1 , 0 , . . . , 0). In Proposition 4.1, if the annulus Ω and the metrics g in and g out are Z -invariant, then the glued metric g will be Z -invariant, by using instead of G the Z -invariant 1 2 (G+ Z ∗ G Z ∗ ). Then n of the n +1 equations in (4.2) hold trivially, and (4.2) becomes a scalar equation. Then:  \nIf g in ,/epsilon1 is a one-parameter family of Z -invariant constant-scalarcurvature metrics depending smoothly on a parameter /epsilon1 ∈ ( -1 , 1) , and with g in , 0 = b the hyperbolic metric, then for all /epsilon1 sufficiently close to zero, the metric g in ,/epsilon1 can be glued, with constant scalar curvature, to a Z -invariant Schwarzschild AdS-metric g out .  \n/negationslash  \nTo see this, let g out ,m be the unboosted and Z -invariant Schwarzschild AdSmetric with mass parameter m , in particular g out , 0 = b . By Proposition 4.1 one obtains a Z -invariant glued metric g m,/epsilon1 that depends parametrically on m and /epsilon1 . The scalar equation (4.2) is of the form u ( m,/epsilon1 ) = 0, where u is a smooth function near the origin that satisfies u (0 , 0) = 0. Furthermore ∂u ∂m (0 , 0) = 0 since only the flux integral over ∂ out Ω contributes to this dervative. Hence by the implicit function theorem, u ( m,/epsilon1 ) = 0 can be locally solved for m = m ( /epsilon1 ), implying the claim. ✷  \nRemark 4.3. Non-trivial families g in ,/epsilon1 needed in Example 4.2 can be constructed by starting from any smooth Z -invariant family of metrics (not conformal to b ) passing through b , to obtain a constant-scalar-curvature family by conformal rescaling (Yamabe problem [1]). ✷  \nWe note that it is also possible to use other families of metrics for g out , such as the family of metrics with arbitrary energy-momentum vector constructed in [8].  \nLet us finally mention that given a solution operator for the trace-free symmetric divergence equation with good support and differentiability properties, then the above discussion may be extended to the CMC general relativistic constraint equations (instead of only Schwarzschild-AdS one then has to use Kerr-AdS). In dimension n = 3 such an operator is provided in [13], for n ≥ 3 it will be provided in a forthcoming paper of Isett, Mao, Oh and Tao.", 'References': "- [1] L. Andersson, P.T. Chru'sciel, and H. 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Oh, and Z. Tao, Initial data gluing in the asymptotically flat regime via solution operators with prescribed support properties , (2023), arXiv:2308.13031 [math.AP].\n- [12] Y. Mao and Z. Tao, Localized initial data for Einstein equations , (2022), arXiv:2210.09437 [math.AP].\n- [13] A. Nutzi, A support preserving homotopy for the de Rham complex with boundary decay estimates , (2024), arXiv:2404.18005 [math.DG].\n- [14] S.-J. Oh and D. Tataru, Local well-posedness of the (4 + 1) -dimensional Maxwell-Klein-Gordon equation at energy regularity , Ann. PDE 2 (2016), no. 1, Art. 2, 70. MR 3462105\n- [15] M.E. Taylor, Partial differential equations III , Applied Mathematical Sciences, vol. 117, Springer, 1996.  \nUniversity of Vienna  \nEmail:  \npiotr.chrusciel@univie.ac.at  \nHomepage:  \nhomepage.univie.ac.at/piotr.chrusciel  \nUniversity of T¨ubingen  \nEmail: albachiara.cogo@uni-tuebingen.de  \nStanford University  \nEmail:  \nanuetzi@stanford.edu"}