The no-additional-components conjecture for outermost regions

From papers

Let MM' be an outermost region of a geometrostatic manifold, written as

M=R3i=nnΩi,M'={\mathbb R}^3\setminus\bigcup_{i=-n'}^n\Omega_i,

where each Ωi\Omega_i is diffeomorphic to a three-ball, has stable minimal boundary Σi=Ωi\Sigma_i=\partial\Omega_i, and every component Ωα\Omega_\alpha in the outermost-region description is one of these regions. The no-additional-components conjecture. For every Ωα\Omega_\alpha there exists an index i{1,,n}i\in\{1,\ldots,n\} such that piΩαp_i\in\Omega_\alpha and

M=R3i=1nΩi.M'={\mathbb R}^3\setminus\bigcup_{i=1}^n\Omega_i.

Thus every component removed from the outermost region should contain at least one distinguished point pip_i, so no additional components with indices i0i\leq 0 occur. The claim concerns the topology of outermost regions in this setting and is stated without a resolution in the source.

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Sources & referencesView supporting material

Primary source

Christina Sormani and Iva Stavrov Allen, “Geometrostatic Manifolds of Small ADM Mass”, arXiv:1707.03008 (2018).

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