The Positive Energy Theorem for Asymptotically Hyperboloidal Initial Data Sets with Toroidal Infinity and Related Rigidity Results

Rigidity and Non-Rigidity of $\mathbb{H}^n/\mathbb{Z}^{n-2}$ with Scalar Curvature Bounded from Below

“…with the restricted metric g H | X ; then no Riemannian manifold M n , g with boundary isometric to ∂X can have scalar curvature R g ≥ −n(n − 1) and mean curvature 1 of ∂M greater than that of ∂X.…”

“…In both cases, the proofs rely on the µ-bubble technique. (f ) It would be interesting to compare Theorem 1.5 with some recent progress in proving positive mass and rigidity results for ALH manifolds (see [1,16,17,26]); in this latter development, manifolds are often assumed to have nonempty inner boundary with the mean curvature bound H ≤ n − 1 (now H is computed with respect to the inner unit normal); such mean curvature bounds serve as barrier conditions in the method of 'marginally outer trapped surfaces' (MOTS), which can be viewed as a generalization of the µ-bubble technique.…”