On the Bartnik extension problem for the static vacuum Einstein equations

2009/09/30 by Michael T. Anderson, Marcus Khuri, Marcus A. Khuri · 31 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Boundary (topology) #Classical mechanics #Conjecture #Einstein #Einstein equations #Extension (predicate logic) #General relativity #Geometric Analysis and Curvature Flows #Mathematical analysis #Mathematical physics #Mathematics #Nonlinear Partial Differential Equations #Physics #Pure mathematics #Uniqueness #gr-qc #math-ph #math.DG #math.MP

paper · pdf · doi:10.1088/0264-9381/30/12/125005

published in Classical and Quantum Gravity 30(12), 125005 (IOP Publishing) · 33 pages, 1 figure. Minor revision of v2. Final version, to appear in Class. Quantum Gravity

arxiv created 2013/05/04 · openalex publication_date 2013/05/14 · arxiv updated 2015/05/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/08

Abstract

We develop a framework for understanding the existence of asymptotically flat solutions to the static vacuum Einstein equations on with geometric boundary conditions on ∂ M ≃ S 2 . A partial existence result is obtained, giving a partial resolution of a conjecture of Bartnik on such static vacuum extensions. The existence and uniqueness of such extensions is closely related to Bartnik's definition of quasi-local mass.

Citations

• Positive Mass Theorem and the Boundary Behaviors of Compact Manifolds with Nonnegative Scalar Curvature
• Elliptic Partial Differential Equations of Second Order
• The Yamabe problem
• Weakly decaying asymptotically flat static and stationary solutions to the Einstein equations
• The mass of an asymptotically flat manifold
• Unique continuation results for Ricci curvature and applications
• On the Structure of Solutions to the Static Vacuum Einstein Equations
• On boundary value problems for Einstein metrics
• Einstein Manifolds