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Optimal localization for the Einstein constraints

Bruno Le Floch, Philippe G. LeFloch

Abstract

We consider asymptotically Euclidean, initial data sets for Einstein's field equations and solve the localization problem at infinity, also called gluing problem. We achieve optimal gluing and optimal decay, in the sense that we encompass solutions with possibly arbitrarily low decay at infinity and establish (super-)harmonic estimates within possibly arbitrarily narrow conical domains. In the localized seed-to-solution method (as we call it), we define a variational projection operator which associates the solution to the Einstein constraints that is closest to any given localized seed data set (as we call it). Our main contribution concerns the derivation of harmonic estimates for the linearized Einstein operator and its formal adjoint which, in particular, includes new analysis on the linearized scalar curvature operator. The statement of harmonic estimates requires the notion of energy-momentum modulators (as we call them), which arise as correctors to the localized seed data sets. For the Hamiltonian and momentum operators, we introduce a notion of harmonic-spherical decomposition and we uncover stability conditions on the localization function, which are localized Poincare and Hardy-type inequalities and, for instance, hold for arbitrarily narrow gluing domains. Our localized seed-to-solution method builds upon the gluing techniques pioneered by Carlotto, Chrusciel, Corvino, Delay, Isenberg, Maxwell, and Schoen, while providing a proof of a conjecture by Carlotto and Schoen on the localization problem and generalize P. LeFloch and Nguyen's theorem on the asymptotic localization problem.

Optimal localization for the Einstein constraints

Abstract

We consider asymptotically Euclidean, initial data sets for Einstein's field equations and solve the localization problem at infinity, also called gluing problem. We achieve optimal gluing and optimal decay, in the sense that we encompass solutions with possibly arbitrarily low decay at infinity and establish (super-)harmonic estimates within possibly arbitrarily narrow conical domains. In the localized seed-to-solution method (as we call it), we define a variational projection operator which associates the solution to the Einstein constraints that is closest to any given localized seed data set (as we call it). Our main contribution concerns the derivation of harmonic estimates for the linearized Einstein operator and its formal adjoint which, in particular, includes new analysis on the linearized scalar curvature operator. The statement of harmonic estimates requires the notion of energy-momentum modulators (as we call them), which arise as correctors to the localized seed data sets. For the Hamiltonian and momentum operators, we introduce a notion of harmonic-spherical decomposition and we uncover stability conditions on the localization function, which are localized Poincare and Hardy-type inequalities and, for instance, hold for arbitrarily narrow gluing domains. Our localized seed-to-solution method builds upon the gluing techniques pioneered by Carlotto, Chrusciel, Corvino, Delay, Isenberg, Maxwell, and Schoen, while providing a proof of a conjecture by Carlotto and Schoen on the localization problem and generalize P. LeFloch and Nguyen's theorem on the asymptotic localization problem.
Paper Structure (149 sections, 43 theorems, 461 equations, 2 figures, 3 tables)

This paper contains 149 sections, 43 theorems, 461 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Localization in Einstein gravity
  3. Main objective
  4. Classes of initial data sets
  5. Anti-gravity phenomenon
  6. Recent developments
  7. Overview of our results
  8. Localization with harmonic control
  9. Einstein's constraint equations
  10. Localized seed-to-solution projection
  11. Conical gluing framework
  12. Harmonic stability
  13. Overview of the main result
  14. A decomposition of the Einstein constraints
  15. Outline and notation
  16. ...and 134 more sections

Key Result

Theorem 1.1

Consider a conical localization data set $(\mathbf M, \Omega, g_0,h_0, \mathbf r, {\boldsymbol{\lambda}})$ (cf.def-conical) together with (projection, geometry, accuracy) exponents $(p, p_G,p_A)$ satisfying, in the gluing domain $\Omega$ and in preferred charts at infinity, the pointwise decay condi with Suppose that the localization function ${\boldsymbol{\lambda}}$ satisfies, at each asymptotic

Figures (2)

  • Figure 1.1: Schematic illustration of the gluing of the Euclidean metric (outside a conical domain) and the Schwarzschild metric (inside a conical domain) in three dimensions. Left: exact localization with sub-harmonic control. Middle: asymptotic localization with harmonic control. Right: exact localization with harmonic control.
  • Figure 1.2: Schematic representation of the ends $\Omega_\iota$ of the gluing domain $\Omega$.

Theorems & Definitions (84)

  • Theorem 1.1: Optimal localization with super-harmonic control. Informal statement
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Remark 1
  • Definition 6
  • Definition 7
  • Theorem 2.1: The localized seed-to-solution projection ---sub-harmonic control
  • ...and 74 more