GLUING CONSTRUCTIONS FOR ASYMPTOTICALLY HYPERBOLIC MANIFOLDS WITH CONSTANT SCALAR
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ar X iv :0 71 1. 15 57 v1 [ gr -q c] 9 No v 2 00 7 GLUING CONSTRUCTIONS FOR ASYMPTOTICALLY HYPERBOLIC MANIFOLDS WITH CONSTANT SCALAR CURVATURE PIOTR T. CHRUSCIEL AND ERWANN DELAY Abstract. We show that asymptotically hyperbolic initial data satis- fying smallness conditions in dimensions n ≥ 3, or fast decay conditions in n ≥ 5, or a genericity condition in n ≥ 9, can be deformed, by a de- formation which is supported arbitrarily far in the asymptotic region, to ones which are exactly Kottler (“Schwarzschild- adS”) in the asymptotic region. Contents 1. Introduction 1 2. Definitions, notations and conventions 4 3. A uniform estimate for P ? 7 4. The gluing construction on a moving annulus 12 5. The gluing construction on a fixed annulus 16 6. b-conformal deformations near infinity 18 Appendix A. The asymptotics of P ? 20 A.1. Conformally compact metrics 20 A.2. The (C, k, ?)-asymptotically hyperbolic case 21 Appendix B. Proof of Lemma 3.6 24 References 27 1. Introduction One of the key problems in mathematical general relativity is the under- standing of the space of solutions of the vacuum constraint equations. In this context an important gluing method has been introduced by Corvino and Schoen [12, 13] for vacuum data with vanishing cosmological constant.
Voir
- related gluing
- compact manifold
- constant scalar
- metric b?
- vacuum einstein equations
- parity
- kottler metric
- gluing constructions
- symmetric vacuum initial
- aspect function
GLUING CONSTRUCTIONS FOR ASYMPTOTICALLY HYPERBOLIC MANIFOLDS WITH CONSTANT SCALAR CURVATURE
´ PIOTR T. CHRUSCIEL AND ERWANN DELAY
Abstract.We show that asymptotically hyperbolic initial data satis-fying smallness conditions in dimensionsn≥3, or fast decay conditions inn≥5, or a genericity condition inn≥9, can be deformed, by a de-formation which is supported arbitrarily far in the asymptotic region, to ones which are exactly Kottler (“Schwarzschild- adS”) in the asymptotic region.
Contents
1. Introduction 2. Definitions, notations and conventions 3. A uniform estimate forP∗ 4. The gluing construction on a moving annulus 5. The gluing construction on a fixed annulus 6.b-conformal deformations near infinity Appendix A. The asymptotics ofP∗ A.1. Conformally compact metrics A.2. The (C k σ)-asymptotically hyperbolic case Appendix B. Proof of Lemma 3.6 References
1.odtrInontiuc
1 4 7 12 16 18 20 20 21 24 27
One of the key problems in mathematical general relativity is the under-standing of the space of solutions of the vacuum constraint equations. In this context an important gluing method has been introduced by Corvino and Schoen [12, 13] for vacuum data with vanishing cosmological constant. The object of this paper is to present related gluing results when the cosmo-logical constant Λ is negative. The question we address is the possibility of deforming an asymptotically hyperbolic Riemannian manifold of constant scalar curvature, and hence a time-symmetric vacuum initial data set, to one with a Kottler metric (sometimes known as Schwarzschild – anti de Sit-ter metric) outside of a compact set. We establish deformation or extension theorems in dimensionsn≥3 under a smallness condition for metrics suffi-ciently close to (generalised) Kottler metrics, or under smallness and parity
Date: February 2, 2008.
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´ P.T. CHRUSCIEL AND E. DELAY
conditions for metrics close to a standard hyperbolic metric, or assuming a rapid decay condition in dimensionsn≥5. More precisely, we considern-dimensional manifolds containing asymp-totic ends (1.1)Mext:= (r0∞)×N whereN Weis a compact manifold. are interested in constant scalar curva-ture metrics which asymptote, asrgoes to infinity, to a background metric bof the form 2 b (1.2)b=r2dr+k+r2b b wherek∈ {0±1}, and wherebis a (r–independent) metric onNgniyfsitas b b Ric(b) =k(n−2)b family of examples is provided by the (generalised). A Kottler metrics, (1.3)bm=dr2b r2+krn−2+r2b − Note thatb0=b, withbas in (1.2). For the purpose of the next theorem define the manifoldMto be M= (r0 r2]×N and suppose thatgis a constant negative scalar curvature metric onMclose tob, or tobm. There are two natural questions: First, chooser1sginfyisatr0< r1< r2, can one deformg, keeping the scalar curvature fixed, so that the resulting metric coincides withg on (r0 r1]×N, and withbm, for somem, near{r2} ×N this case we? In setM′= (r0 r1]×N,M′′= [r2∞)×N, and we refer to this case as the deformation problem. Next, letr3> r2, can one extendgto a new metric of constant scalar curvature on (r0∞)×Nso that the extended metric coincides withbm, for somem, on [r3∞)×N this case we set? InM′=M,M′′= [r3∞)×N, and we refer to this case as theextension problem. We show that those problems can always be solved whengis sufficiently b close tob, except perhaps when (N b) is a round sphere andm= 0, in which case we need to impose a restrictive condition: For (r q)∈Mlet ψ(r q) = (r φ(q)), whereφ metric Ais the antipodal map of the sphere.g onMwill be said to be parity-symmetric ifψ∗g=g the end of Section 5. At we prove: b Theorem1.1.Letn≥3,N∋ℓ >⌊2⌋+ 4,λ∈(01),m∈R. If(N b)is a round sphereandm= 0, we suppose moreover thatgis parity-symmetric. There existsε >0such that ifkg−bmkCℓλ(M)< ε, then there exists onMext aCℓλmetric of constant negative scalar curvature which coincides withg onM′, and which is a Kottler metric onM′′. Ifgis smooth, then so is the solution of the deformation problem.
