Noncompact Fill-Ins of Bartnik Data

2022/11/11 by Lee, Dan A., Lesourd, Martin, Unger, Ryan · 2 citations
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2211.06280

Abstract

We generalize Y. Shi and L.-F. Tam's \citeShiTam nonnegativity result for the Brown-York mass, by considering nonnegative scalar curvature (NNSC) fill-ins that need only be complete rather than compact. Moreover, the NNSC fill-ins need not even be complete as long the incompleteness is ``shielded'' by a region with positive scalar curvature and occurs occurs sufficiently far away. We accomplish this by generalizing P.~Miao's~\citeMiao02 positive mass theorem with corners to asymptotically flat manifolds that may have other complete ends, or possibly incomplete ends that are appropriately shielded. We can similarly extend other results on the compact NNSC fill-in problem to allow for complete (or shielded) NNSC fill-ins. In particular, we prove the following generalization of a theorem of Miao~\citeMiao20: Given any metric γ on a closed manifold Σ^n-1, there exists a constant λ such that for any complete (or shielded) NNSC fill-in (Ωⁿ, g) of (Σ^n-1,γ), we have min_ΣH ≤ λ, where H is the mean curvature of Σ with respect to g.

Cited by

• Topological Rigidity of Contractible 3-Manifolds and Handlebody Interiors under Nonnegative Scalar Curvature
• Horizons in noncompact fill-ins of nonnegative scalar curvature

Related

• Nonexistence of NNSC fill-ins with large mean curvature
• On the Fill-in of Nonnegative Scalar Curvature Metrics
• Total mean curvature of the boundary and nonnegative scalar curvature fill-ins
• Incompressible hypersurface, positive scalar curvature and positive mass theorem
• Bartnik mass via vacuum extensions