An introduction to Einstein constraints andthe seed-to-solution method
Philippe G. LeFloch
Sorbonne Université
Laboratoire Jacques-Louis LionsCentre National de la Recherche Scientifique
Blog: philippelefloch.org

Outline of this talk1. Topic for this lecture
2. Exact vs. asymptotic localization problems
3. A parametrization of localized initial data sets
4. Localization in cone-like domains
5. Main statement with super-harmonic control
6. Concluding remarks
Main collaborators
The-Cang Nguyen (Montpellier)Bruno Le Floch (Paris)

1. TOPIC for THIS LECTURE
Einstein’s constraint equations“prescribed curvature problem”§manifold pM, g , kq with finitely many asymptotic ends§

unknowns: Riemannian metric g and symmetric p0, 2q-tensor field kextrinsic curvature in the dynamical picture§

matter content: scalar field H‹ : M Ñ R`§

Einstein’s Hamiltonian and momentum constraints
2Rg ` pTrg kq
Notation´ |k|2g “ H‹vector field M‹`˘Divg k ´ pTrg kqg “ M‹`˘77:“It is convenient to introduce the p2, 0q-tensor h by hk ´ Trg pkqg
˘21`Hpg , hq :“ Rg ` Trg h ´ |h|2g
Mpg , hq :“ Divg h2`˘:“Gpg , hq
H, M pg , hq “ pH‹ , M‹ q

In the dynamical picture, Gpg , hq is a spacetime vector.

Vast and rich literature§

Conformal method

Lichnerowicz (1960s), Choquet-Bruhat, Chrusciel, Corvino, Delay,Dilts, Galloway, Gicquaud, Holst, Isenberg, Maxwell, Mazzeo, Miao, Pollack, Schoen, etc.
§Variational method

Corvino, Corvino-Schoen, Chrusciel-Delay, Carlotto-Schoen, etc.

Major achievements§

existence of initial data, explicit constructions, physically relevant solutions§

general relativity, Riemannian geometry§

numerous classes of solutions: compact, various types of asymptotic ends§

including gluing techniques, combine two di↵erent solutions together§

A. Carlotto, The general relativistic constraint equations, Living Reviews
in Relativity (2021).

§Very active subject: Lee, Lesourd, Corvino, Pasqualotto, etc.

Shielding gravity: localization at infinity§

asymptotically Euclidean initial data sets
§ phenomena of anti-gravity (or shielding)§§

Carlotto and SchoenChruściel and Delaysolutions that are localized at infinity§

The Positive Mass Theorem implies restrictions on gluing at infinity.

§identically Euclidian near infinity except in a cone with possibly
arbitrary small angle

Other recent developments§§

S. Aretakis, S. Czimek, I. Rodnianski: characteristic gluing problem
Y.-C. Mao and Z.-K. Tao: localization “a la Carlotto-Schoen” in narrow domains

Localization with (super-)harmonic control§

Carlotto and Schoen
psolutions with sub-harmonic control
r with p P§ conjecture: gluing should be possible at harmonic level§§` n´22,n ´ 2Localization results with harmonic and super-harmonic control§

PLF & The-Cang Nguyen, 2020: The seed-to-solution method forthe Einstein constraint equations§Bruno Le Floch & PLF, 2023

˘Figure: Gluing of the Euclidean metric and the Schwarzschild metric.
Left: exact localization with sub-harmonic control
Middle: asymptotic localization with harmonic control
Right: exact localization with harmonic control

2. EXACT vs. ASYMPTOTIC LOCALIZATION PROBLEMSTheorem. The seed-to-solution method(PLF–Nguyen)

Given any seed data set pM, g1 , h1 q on a 3-manifold (with a single end, say):a Riemannian metric g1 and a symmetric two-tensor h1
satisfying (suitable smallness conditions and)1{2 † pG § minp1, pM q
1{2 † pM † `8
g1 “ gEucl ` Opr ´pG q

h1 “ Opr ´pG ´1 q

Hpg1 , h1 q “ Opr ´pM ´2 q

Mpg1 , h1 q “ Opr ´pM ´2 q

there exists a solution pg , hq to the vacuum Einstein equations Gpg , hq “ 0.
§Sub-harmonic decay: pM † 1
g “ g1 ` Opr ´pM q§§h “ h1 ` Opr ´pM ´1 q
Hpg1 , h1 q and Mpg1 , h1 q in L1 pMq
h “ h1 ` Opr ´2 q

Harmonic decay: pM “ 1g “ g1 ` mrr ` opr ´1 q

Super-harmonic decay: pM ° 1
g “ g1 ` mrr ` Opr ´p q

p “ minppG ` 1, pM , 2q≥

h “ h1 ` Opr ´2 q

r “ mpgr 1 , h1 q “ const. M Hpg1 , h1 q dVg1 ` OpGpg1 , h1 q2 q
Mass modulator m

Exact localization problem

Carlotto and Schoen– Vacuum constraint Einstein equations
Decompose asymptotic infinity into three angular regions§§§

Ca : cone with (possibly arbitrarily small) angle a P p0, 2⇡qc
Ca`"
: complement of the same cone with (slightly) larger angle a ` "Ta" : remaining transition regionc
Ca and Ca`": the metric coincides with Euclidean/Schwarzschild ones

solve the vacuum Einstein equations in the transition region Ta,"– Sub-harmonic control in Ta," , that is, r ´p with p P p1{2, 1q– Question raised by Carlotto and Schoen
construct solutions (with prescribed asymptotic)
enjoying the 1{r harmonic decay in all angular directions
1{r p region
Schwarzschild
Euclidean1{r p region

Asymptotic localization problem
slightly relax the localization condition– asymptotic at a super-harmonic rate to prescribed metrics– physically as natural as the exact localization problem

1{r region

asympt. Schwarzschildasympt. Eucl.
1{r region

Theorem. The asymptotic localization problem(PLF–Nguyen)– Vacuum Einstein equations on a manifold M with a single asymptotic end– Decompose asymptotic infinity into three asymptotic angular regions
cCa Y Ca`"Y Ta," Ä R3
By considering (for instance) the Euclidean metric gEucl and the Schwarzschild
metric gSch “ p1 ` 2mSch {r q gEucl (with mass mSch ° 0),
there exists a solution to the vacuum Einstein equations Gpg , hq “ 0:g “ gEucl ` Opr ´q q

cin Ca`"g “ gEucl ` Opr ´1 qin Ta,"g “ gSch ` Opr ´q qin Caq P p1, 2q

3. A PARAMETRIZATION of INITIAL DATA SETSDefinition
A localized reference set pM, g0 , h0 , !q consists of– Riemannian manifold pM, g0 q
– symmetric p2, 0q-tensor h0 on M
– localization weight ! P r0, `8s§§

a smooth open set ⌦ Ä M, referred to as the gluing domain$°0in the gluing domain ⌦’’’&Ñ 0
when approaching B⌦ from ⌦!
’Ñ0
at asymptotic ends’’%
“ `8 in A ⌦! :“ br P r0, `8s§§

asymptotic radius r : M Ñ r1, `8q`
˘1{2r » 1 ` pdg0 p¨, y0 qq2

for some (fixed) point y0 P M

boundary distance b : M Ñ p0, 1slarge integer N • 1#` `˘N ¯» min 1, dB⌦ {rin the gluing domain ⌦b“ `8in the fixed domain A ⌦§

Weighted Hölder spaces Cpl,↵ pM, g0 , !q with}f }l,↵g0 ,!,p “§∞|L|§l supM´¯b ! p`|L| |rL f |g0 `∞|L|“l supM´

b ! p`|L|`↵ JrL f Kg0 ,↵Weighted Lebesgue spaces (and Sobolev spaces, etc.):≥}f }2L2 pM,g0 ,!q :“ M |f |2g0 b ! 2p´n dVg0p§

Weighted Lebesgue-Hölder spaces L2 C!l,↵ pM, g0 q with`˘2
`
˘2 `
˘2l,↵
~f ~g0 ,!,p :“ }f }L2p pM,g0 ,!q ` }f }C l,↵ pM,g ,!qp0¯

Definition
Consider a localized reference set pM, g0 , h0 , !q and some parameters "G , "M ° 0.
Consider decay and accuracy exponents ppG , qG q and ppM , qM qppM , qM q • ppG , qG q

A ppG , qG , pM , qM q-localized seed data set pg1 , h1 , H‹ , M‹ q consists of:§Near-reference data
g1 is a Riemannian metrich1 is a symmetric p2, 0q-tensor}g1 ´ g0 }N,↵
g0 ,!,pG § "G§}h1 ´ h0 }N´1,↵
g0 ,!,qG § "GNear-Einsteinian dataH‹ is a scalar field and M‹ is a vector field and⌫⌫⌫Hpg1 , h1 q ´ H‹ ⌫N´4,↵§ "M
g0 ,!,pM `2⌫⌫⌫Mpg1 , h1 q ´ M‹ ⌫N´3,↵§ "Mg ,!,q `10§M

metrics with possibly slow decay§ treat, together, the Einstein operator on pg1 , h1 q and the matter content
We seek to make a “projection” of each pg1 , h1 q on the “solution manifold”.

Variational framework

Corvino and Schoen

Given pf , V q consider the functionalª ´¯ˇ2 2p´n 1 ˇˇ2 2q´n1 ˇˇ˚
˚
dHpg1 ,h1 q ru, Z sˇ b !
` ˇdMpg1 ,h1 q ru, Z sˇ b !
´ f u ´ g1 pV , Z q dVg1
22
M
Euler-Lagrange equation

pu, Z q minimizer: fourth-order PDEs

˚g ´ g1 “ b ! 2p´n dHpgru, Z s
1 ,h1 qh ´ h1 “ b ! 2q´n dM˚
pg1 ,h1 q ru, Z s

pf , V q “ dGpg1 ,h1 qq rg ´ g1 , h ´ h1 s

Fixed-point’s Picard scheme§

quadratic part of the Einstein operator

decay properties of nonlinearities

Qpg1 ,h1 q rg2 , h2 s :“ Gpg , hq ´ Gpg1 , h1 q ´ dGpg1 ,h1 q rg2 , h2 s§

study sequences pfi , Vi q and pgi , hi q
pfi , Vi q :“ ´Qpg1 ,h1 q rgi´1 , hi´1 s ` pH‹ , M‹ q ´ Gpg1 , h1 q

pgi , hi q :“ pdGq´1pg1 ,h1 q pfi , Vi q˚Operator dGpg0 ,h0 q ˝ (weights) ˝ dGpg0 ,h0 qwith wp “ b ! 2p´n and wq “ b ! 2q´n“‰˚dH wp dH rus “ pn ´ 1q pwpuq ` pri rj wp ´ wp R ij qpri rj u ´ Rij uq´ p wp ´ 2wp Rq u´¯¯
31´i` 2R ri wp ` wp ri R r u `pwp Rq ` p wp qR u
22“‰i11˚dM wq dM rZ s “ ´ wq prj rj Z i ` rj ri Z j q ´ rj wq prj Z i ` ri Z j q
22
In the vicinity of an asymptotically Euclidian end: eij “ijAe,wp rus :“ pn ´ 1q e pwp e uq ` pBi Bj wp qBi Bj u ´ p e wp q e u
11Be,wq rZ si :“ ´ wq p e Z i ` Bj Bi Z j q ´ pBj wq qpBj Z i ` Bi Z j q
22Solution mappingorder of regularity N • 4 and Hölder exponent ↵ P p0, 1sDefinition§§§a localized reference set pM, g0 , h0 , !q and parameters "G , "M ° 0pairs ppG , qG q and ppM , qM q with ppM , qM q • ppG , qG qpair of exponents pp, qq

To any ppG , qG , pM , qM q-localized seed data set pg1 , h1 , H‹ , M‹ q,
the pp, qq-localized seed-to-solution map
associates a scalar-valued field u and a vector-valued field Z :˚g “ g1 ` b ! 2p´n dHpg
pu, Z q
1 ,h1 qh “ h1 ` b ! 2q´n dM˚
pg1 ,h1 q pu, Z qobey the Einstein constraints Gpg , hq “ pH‹ , M‹ q
Lebesgue-Hölder norm⌫⌫⌫g ´ g1 ⌫N,↵ § "Gg0 ,!,p⌫⌫⌫h ´ h1 ⌫N´1,↵ § "Gg0 ,!,q⌫⌫N´4,↵⌫⌫N´3,↵⌫⌫⌫⌫}u}N,↵À
Hpg,hq´H`"
Mpg,hq´M1
1‹ g ,!,p`2
1
1‹ g ,g ,!,q`1
Gg0 ,!,p00 0⌫⌫⌫⌫⌫⌫N´4,↵⌫⌫N´3,↵}Z }N´1,↵
g0 ,!,q À "G Hpg1 , h1 q ´ H‹ g ,!,p`2 ` Mpg1 , h1 q ´ M‹ g ,!,q`100

Our parametrization
Definition
Equivalence relation is defined between two pairs of tensors pg1 , h1 q and pg , hq
˚g “ g1 ` b ! 2p´n dHpg
pu, Z q,0 ,h0 q

h “ h1 ` b ! 2q´n dM˚
pg0 ,h0 q pu, Z q

for some scalar field u and vector field Z

pg , hq „p!,p,qq pg1 , h1 q.

Definition
For any given matter data H‹ , M‹ (possibly vacuum data)
the corresponding solution map S!,p,q
sending a (seed data) element pg1 , h1 q to one of its representative (exact solution)in the same class rpg1 , h1 qs, namely`
˘S!,p,q : pg1 , h1 q P E"G ,"M pG , qG ; pM , qM ﬁÑ pg , hq P rpg1 , h1 qs
is referred to as the localized seed-to-solution parametrization for Einstein’s
constraint equations in the vicinity of the localized reference set pM, g0 , h0 , !q.
§make a specific choice of reference g0 , h0
§ establish weighted Poincaré and weighted Korn inequalities§ conditions required: geometry of the gluing domain, decay/accuracy
exponents

Coercivity and elliptic regularity§§

linearization dGpg1 ,h1 q of the Einstein operator G around pg1 , h1 q
not elliptic unless a gauge choice is made§restrict the deformation pu, Z q to lie in the image of the dual operator
˚dGpg, up to weights that suitably localize the deformation of interest
1 ,h1 q§˚
invertibility of the fourth-order operator dGpg1 ,h1 q ˝ (weights) ˝ dGpg
1 ,h1 q§weighted Hölder-Sobolev spaces§

weighted Poincaré inequality, weighted Korn inequality§elliptic system in the sense of Douglis-Nirenberg, Hölder interior regularity
estimates