Recent progress and problems on the Bartnik quasi-local mass

Abstract

This paper surveys recent progress on issues related to the Bartnik quasi-local mass mB. In addition we formulate a number of new problems and conjectures regarding foundational properties of the mass mB. This work is dedicated with pleasure to Robert Bartnik in honor of his 60 th birthday.

Existence of static vacuum extensions with prescribed Bartnik boundary data

“…It is clear that our arguments don't work without embeddedness. We note that those inner embedded hypersurfaces are conjectured to be counter-examples to Conjecture 1 by [9, Conjecture 5.2] (see also [6]). On the other hand, Theorem 7 implies that those possible counter-examples can be perturbed inward (with arbitrarily small perturbation) to be static regular hypersurfaces.…”

Embeddings, Immersions and the Bartnik Quasi-Local Mass Conjectures

“…There exists a large family of immersed spheres S 2 in R 3 whose boundary data (γ, H) ∈ B are not realized by a static vacuum extension of minimal mass; cf. also [11] for further discussion and conjectures. Remark 3.2.…”

“…In case the boundary ∂M is (weakly) outer-minimizing in (M, g i ) for i sufficiently large, Theorem 3.5 is proved in [11], based on the discussion in [13]. We sketch the argument for completeness.…”

“…Thus, such static vacuum solutions are maximal, in the sense that they cannot be extended to any larger domain. 11 One may then take a boundary ∂M = Im F as above to obtain a Weyl solution (M, g, u) with induced Bartnik boundary data (γ, H) on ∂M . These solutions correspond to varying domains in a fixed ambient maximal Weyl solution generated by ν in (4.13).…”

“…A different representative (M ′ , g ′ , u ′ ) of (M, g, u) in E m,α S 1 gives the same map F , since the diffeomorphisms in Diff m+1,α 1 (M ) fix the boundary pointwise. 11 One could also take potentials ν as in (4.13) with more general signed measures or even distributions supported in A, generating Weyl metrics in the same way. However, when dµ is a positive measure, boundaries have fill-ins (Ω, gΩ) with non-negative scalar curvature and the ADM mass mg of the solution is always non-negative.…”