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Existence of static vacuum extensions for Bartnik boundary data near Schwarzschild spheres

Spyros Alexakis, Zhongshan An, Ahmed Ellithy, Lan-Hsuan Huang

Abstract

We obtain existence and local uniqueness of asymptotically flat, static vacuum extensions for Bartnik data on a sphere near the data of a sphere of symmetry in a Schwarzschild manifold.

Existence of static vacuum extensions for Bartnik boundary data near Schwarzschild spheres

Abstract

We obtain existence and local uniqueness of asymptotically flat, static vacuum extensions for Bartnik data on a sphere near the data of a sphere of symmetry in a Schwarzschild manifold.

Paper Structure

This paper contains 5 sections, 12 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Conformal formulation and structure equations
  3. Conformal transformation
  4. Global geodesic gauge and structure equations
  5. Vanishing of the potential function deformation

Key Result

Theorem 1

Let $m_0:=\max\{ 0, m\}$, $\alpha\in (0, 1)$, and $q\in (\frac{1}{2}, 1)$. For each $r_0\in (2 m_0, \infty)$, there exist positive constants $\epsilon_0,C$ such that for each $\epsilon\in (0, \epsilon_0)$, if $(\tau,\phi)$ satisfies $\|(\tau,\phi)-(\gamma_{r_0},H_{r_0})\|_{\mathop{\mathrm{\mathcal{C

Theorems & Definitions (23)

  • Theorem 1
  • Theorem 2
  • Lemma 2.1: Cf. Corollary 3.11 of An-Huang:2024
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.7
  • ...and 13 more