Embeddings, Immersions and the Bartnik Quasi-Local Mass Conjectures

2016/11/30 by Michael T. Anderson, Jeffrey L. Jauregui · 12 citations
Mathematics · Physics and Astronomy · #Boundary (topology) #Class (philosophy) #Einstein #Extension (predicate logic) #Geometric Analysis and Curvature Flows #Mass formula #Nonlinear Partial Differential Equations #Point processes and geometric inequalities #Scalar (mathematics) #gr-qc #math.DG

paper · pdf · doi:10.1007/s00023-019-00786-3

published in Annales Henri Poincaré 20(5), 1651-1698 (Birkhäuser) · 38 pages, 4 figures

openalex created_date 2016/12/08 · arxiv created 2019/02/27 · openalex publication_date 2019/03/20 · arxiv updated 2019/10/16 · openalex updated_date 2026/08/05

Abstract

Given a Riemannian 3-ball ( B, g) of non-negative scalar curvature, Bartnik conjectured that ( B, g) admits an asymptotically flat (AF) extension (without horizons) of the least possible ADM mass, and that such a mass-minimizer is an AF solution to the static vacuum Einstein equations, uniquely determined by natural geometric conditions on the boundary data of ( B, g). We prove the validity of the second statement, i.e.~such mass-minimizers, if they exist, are indeed AF solutions of the static vacuum equations. On the other hand, we prove that the first statement is not true in general; there is a rather large class of bodies ( B, g) for which a minimal mass extension does not exist.

Citations

• A Course in Minimal Surfaces
• Positive Mass Theorem and the Boundary Behaviors of Compact Manifolds with Nonnegative Scalar Curvature
• On the proof of the positive mass conjecture in general relativity
• The mass of an asymptotically flat manifold
• Isometric Embeddings into the Minkowski Space and New Quasi-Local Mass
• A new proof of the positive energy theorem
• Embedded Minimal Surfaces, Exotic Spheres, and Manifolds with Positive Ricci Curvature