Scalar curvature deformation and mass rigidity for ALH manifolds with boundary

Abstract

We study scalar curvature deformation for asymptotically locally hyperbolic (ALH) manifolds with nonempty compact boundary. We show that the scalar curvature map is locally surjective among either (1) the space of metrics that coincide exponentially toward the boundary, or (2) the space of metrics with arbitrarily prescribed nearby Bartnik boundary data. Using those results, we characterize the ALH manifolds that minimize the Wang-Chruściel-Herzlich mass integrals in great generality and establish the rigidity of the positive mass theorems.

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

“…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.…”