REFINED GLUING FOR VACUUM EINSTEIN CONSTRAINT EQUATIONS

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REFINED GLUING FOR VACUUM EINSTEIN CONSTRAINT EQUATIONS ERWANN DELAY AND LORENZO MAZZIERI Abstract. We first show that the connected sum along submanifolds introduced by the second author for compact initial data sets of the vacuum Einstein system can be adapted to the asymptotically Euclidean and to the asymptotically hyperbolic context. Then, we prove that in any case, and generically, the gluing procedure can be localized, in order to obtain new solutions which coincide with the original ones outside of a neighborhood of the gluing locus. Contents 1. Introduction 1 1.1. Conformal gluing 2 1.2. Localized gluing 5 2. The geometric construction 5 3. The momentum constraint: existence of solutions 8 3.1. Asymptotically Euclidean manifold 9 3.2. Asymptotically Hyperbolic manifold 10 4. The momentum constraint: uniform a priori bound 11 5. The energy constraint 16 5.1. Analysis of the linearized Lichnerowicz operator 17 5.2. Fixed point argument 20 6. Localized gluing 22 7. Constraint with cosmological constant and constant scalar curvature metrics 23 8. Concluding remarks 23 References 24 1. Introduction It is well known [8] that a vacuum solution (Z, ?) for the Einstein system Ric? = 0, where Z is an (m+1)-dimensional manifold and ? is a Lorentzian metric, may be obtained from solutions to the vacuum Einstein constraint equations on an m-dimensional space-like Riemannian submanifold (M, g˜) of Z (for further details see eg.

• tensor µ


• gluing can


• construction gives initial


• m?2m?2 ?


• vacuum einstein


• tt-tensor µ?


• can actually


• conformal gluing


• momentum constraint

Sujets

• Diffeomorphism constraint

Informations

• Publié par: pefav
• Nombre de lectures: 15
• Langue: English

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REFINED GLUING FOR VACUUM EINSTEIN CONSTRAINT EQUATIONS

ERWANN DELAY AND LORENZO MAZZIERI

Abstract.We first show that the connected sum along submanifolds introduced by the second author for compact initial data sets of the vacuum Einstein system can be adapted to the asymptotically Euclidean and to the asymptotically hyperbolic context. Then, we prove that in any case, and generically, the gluing procedure can be localized, in order to obtain new solutions which coincide with the original ones outside of a neighborhood of the gluing locus.

Contents

1. Introduction 1.1. Conformal gluing 1.2. Localized gluing 2. The geometric construction 3. The momentum constraint: existence of solutions 3.1. Asymptotically Euclidean manifold 3.2. Asymptotically Hyperbolic manifold 4. The momentum constraint: uniform a priori bound 5. The energy constraint 5.1. Analysis of the linearized Lichnerowicz operator 5.2. Fixed point argument 6. Localized gluing 7. Constraint with cosmological constant and constant scalar curvature metrics 8. Concluding remarks References

1.Introduction

1 2 5 5 8 9 10 11 16 17 20 22

23 23 24

It is well known [8] that a vacuum solution (Z γ) for the Einstein system Ric γ = 0, whereZis an (m+ 1)-dimensional manifold andγis a Lorentzian metric, may be obtained from solutions to the vacuum Einstein constraint equations on anm-dimensional space-like Riemannian submanifold (M˜g) ofZ(for further details see eg. [4]). To fix the notations, we say that the ˜ triple (M˜gΠ), whereMis a smooth manifold,g˜ is a Riemannian metric ˜ and Π is a symmetric (20) tensors is a solution to the vacuum Einstein

Date: March 22, 2010.

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E. DELAY AND L. MAZZIERI

constraints equations if the following relationships are satisfied (1)J(g˜=div ˜ Π): ˜g ˜ Π−d  tr ˜g Π˜  = 0 (2)ρ(˜g): ˜ Π=R ˜g − | ˜ Π| ˜g 2 +  tr g˜ Π˜  2 = 0. Here div g˜ and tr ˜g are respectively the divergence operator and the trace op- erator computed with respect to the metricg˜ andR g˜ is the scalar curvature ˜ of the metricg˜. In the following we will also refer to the triple (M˜gΠ) as (vacuum) initial data set or Cauchy data set. We also point out that when the evolution (Z γdata set is considered according to the) of the initial ˜ hyperbolic formulation of the vacuum Einstein system, then ˜gand Π turn out to be respectively the Riemannian metric and the second fundamental form induced by the Lorentzian metricγon the space-like sliceM. Of interest in this paper are constant mean curvature (briefly CMC) initial ˜ data sets, this means thatτ:= tr ˜g  the case the system InΠ is a constant. above becomes equivalent to a semi-decoupled system. In fact, following [8], ˜ [20] and [21], one can split the second fundamental form Π into trace free and pure trace parts (3) Π˜  = ˜ τ ˜g µ+ m whereµ˜ is a symmetric 2-tensor such that tr g˜ ˜µ Applying= 0. the so called conformal method, we set (4) ˜ =:u 4/m−2 gandµ˜ =: −2 µ  g u where the conformal factoruis a positive smooth function onM. It is ˜ now straightforward to check thatg˜ and Π verify the Einstein constraint equations (1) and (2) if and only ifg µandusatisfy (5) ( i r d t g v g µ µ 0= 0= − m − 22 m−1 τ 2 u m (6) Δ g u−c m R g u+c m |µ| g 2 u − 3 m −c m m m−2 +2 = 0 withc m = 4 (m m − − )1 2  that our Laplacian is negative definite.. Notice Therefore, starting with a metricgand a real numberτ, one can obtain a τ-CMC solution to the Einstein constraints by producing a symmetric (20)- tensorµwhich verifies (5) (for short a TT-tensor) together with a smooth positive solutionuto (6), which will play the role of the conformal factor. ˜ Using then (4) it is easy to recover the triple (M g˜Π). In this context and due to the physical meaning, the equation (1) (or equivalently the second equation in (5)) is known as the momentum con- straint, whereas the equation (2) is the so called Hamiltonian constraint and modulo the conformal transformations it corresponds to the Lichnerowicz equation (6).

1.1.Conformal gluing.In the spirit of [31], we suppose now to start with two Cauchy data sets in the semi-decoupled formulation of the con- straints, namely two solutions (M i  g i  µ i  u i  τ),i= 12 to equations (5) and (6) with the same constant mean curvatureτand we construct the gen- eralized connected sum of them-dimensional manifoldsM 1 andM 2 along