Intrinsic flat stability of the positive mass theorem for graphical hypersurfaces of Euclidean space

Abstract

The rigidity of the Positive Mass Theorem states that the only complete asymptotically flat manifold of nonnegative scalar curvature and zero mass is Euclidean space. We study the stability of this statement for spaces that can be realized as graphical hypersurfaces in E n+1 . We prove (under certain technical hypotheses) that if a sequence of complete asymptotically flat graphs of nonnegative scalar curvature has mass approaching zero, then the sequence must converge to Euclidean space in the pointed intrinsic flat sense. The appendix includes a new Gromov-Hausdorff and intrinsic flat compactness theorem for sequences of metric spaces with uniform Lipschitz bounds on their metrics.

Contrasting various notions of convergence in geometric analysis

“…In Section 2 we review the definitions of Gromov-Hausdorff (GH) and metric measure (mGH) convergence, Intrinsic Flat Convergence (F) and Volume Preserving Intrinsic Flat (VF ) convergence and key theorems relating them in the simplified setting where all the spaces are Riemannian manifolds. Theorem 2.3 specializes a result of Gromov [Gro81b] and Huang-Lee-Sormani [HLS17] stating that Riemannian manifolds with biLipschitz bounds on their distances have subsequences which converge in the uniform, GH, and F sense to the same limit space. Theorem 2.4 states that if a sequence of Riemannian manifolds converges in the GH and F sense to the same Riemannian limit space, and if the volumes of the manifolds converge to the volume of the limit, then the sequences converge in the VF and mGH sense as well.…”

“…A key new result is given in Theorem 4.4 which shows that a metric lower bound combined with volume convergence implies pointwise convergence of d j (p, q) → d 0 (p, q) for almost every (p, q) ∈ M × M . Due to the uniform bounds on the metric assumed in Theorem 1.2 we are then able to show uniform, GH, and SWIF convergence to a length space by applying a Theorem of Huang, Lee, and the second named author in the appendix of [HLS17]. By combining with the pointwise almost every convergence of distances we are able to conclude that the length space guaranteed by compactness must be the metric respect to the desired background Riemannian metric.…”

Sobolev bounds and convergence of Riemannian manifolds

“…At the end of this review we state a theorem concerning integral current spaces, Theorem 6.2, which we can then immediately see implies Theorem 1.4. We conclude with the proof of that theorem using serious analysis from [HLS17] and [AK00]. 6.1.…”

“…Here we assume only uniform Hölder bounds on the distance functions and so cusps may form. As in [HLS17] and [Gro81], we obtain subsequences which converge in the uniform and GH sense to some (X, d ∞ ). However now our SWIF limit, X ∞ , may be a proper subset of the GH limit, X.…”

Geometrostatic Manifolds of Small ADM Mass

“…0, then .M 0 k ; g k / converge in the pointed intrinsic flat sense to euclidean space, .R 3 ; ı; 0/, assuming the manifolds are centered on well-chosen points x k that do not disappear down increasingly deep wells. It is unknown whether the setting considered in [5] can include multiple black holes. Lan-Hsuan Huang, Dan Lee, and the first author have proven this conjecture in the graph setting assuming additional hypotheses including one that requires all level sets to be outward minimizing [5].…”