Embeddings, Immersions and the Bartnik Quasi-Local Mass Conjectures

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.

Recent progress and problems on the Bartnik quasi-local mass

“…This open problem is raised in [10], cf. also the recent results and survey given in [7]. The same question holds with respect to P 0 (γ,H) , but here much less is known and the problem appears much harder.…”

“…It is proved in [7] that, if non-empty, C (γ,H) is a smooth, infinite dimensional Banach manifold in a natural C m,α δ Hölder space topology, and that m ADM in (2.7) is a smooth functional on C (γ,H) . Moreover, it is shown that critical points of H RT or m ADM on the constraint manifold C (γ,H) are exactly the static vacuum Einstein metrics (g, u) satisfying the elliptic boundary conditions (γ, H).…”

“…In addition, minimizers of the Bartnik mass (1.1), or (1.3), are static vacuum solutions with u > 0 and u → 1 at infinity, cf. [7].…”

“…This is discussed further in §2. A full proof along these lines is given in [7]. A completely different proof of part of this conjecture was given by Corvino in [17] and more recently in [18].…”

“…there is an isometric immersion of Ω (or its universal cover) into R 3 . (This cannot be improved to an embedding Ω ⊂ R 3 by the work in [7]). The property QL2 follows immediately from the definition and QL3 also follows from the work in [19].…”