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Static vacuum extensions with prescribed Bartnik boundary data near a general static vacuum metric

Zhongshan An, Lan-Hsuan Huang

Abstract

We introduce the notions of static regular of type (I) and type (II) and show that they are sufficient conditions for local well-posedness of solving asymptotically flat, static vacuum metrics with prescribed Bartnik boundary data. We then show that hypersurfaces in a very general open and dense family of hypersurfaces are static regular of type (II). As applications, we confirm Bartnik's static vacuum extension conjecture for a large class of Bartnik boundary data, including those that can be far from Euclidean and have large ADM masses, and give many new examples of static vacuum metrics with intriguing geometry.

Static vacuum extensions with prescribed Bartnik boundary data near a general static vacuum metric

Abstract

We introduce the notions of static regular of type (I) and type (II) and show that they are sufficient conditions for local well-posedness of solving asymptotically flat, static vacuum metrics with prescribed Bartnik boundary data. We then show that hypersurfaces in a very general open and dense family of hypersurfaces are static regular of type (II). As applications, we confirm Bartnik's static vacuum extension conjecture for a large class of Bartnik boundary data, including those that can be far from Euclidean and have large ADM masses, and give many new examples of static vacuum metrics with intriguing geometry.
Paper Structure (19 sections, 41 theorems, 196 equations)

This paper contains 19 sections, 41 theorems, 196 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The structure at infinity
  4. Regge-Teitelboim functional and integral identities
  5. $h$-Killing vectors and the geodesic gauge
  6. Static-harmonic gauge and orthogonal gauge
  7. The gauges
  8. The gauged operator
  9. Static regular and "trivial" kernel
  10. Cauchy boundary condition
  11. Relaxed Cauchy boundary condition
  12. Infinite-order boundary condition
  13. Existence and local uniqueness
  14. Perturbed hypersurfaces
  15. Extension of local $h$-Killing vector fields
  16. ...and 4 more sections

Key Result

Theorem 3

Let $(M, \bar{g}, \bar{u})$ be an asymptotically flat, static vacuum triple with $\bar{u} > 0$, and let $\Omega$ be as defined in the notation above. Suppose that Then there exist positive constants $\epsilon_0,C$ such that for each $\epsilon\in (0, \epsilon_0)$, if $(\tau,\phi)$ satisfies $\|(\tau,\phi)-(\bar{g}^\intercal,H_{\bar{g}})\|_{\mathop{\mathrm{\mathcal{C}}}\nolimits^{2,\alpha}(\Sigma)\

Theorems & Definitions (87)

  • Conjecture 1: Bartnik's static extension conjecture
  • Conjecture 2: Local well-posedness
  • Theorem 3
  • Definition 4: Static regular
  • Theorem 5
  • Remark 1.1
  • Definition 6
  • Theorem 7
  • Corollary 8
  • Theorem 9: Cf. Anderson:2008
  • ...and 77 more