Intrinsic Flat Convergence of Points and Applications to Stability of the Positive Mass Theorem

We prove results on intrinsic fiat convergence of points-a concept first explored by Sormani (Commun Anal Geom 26(6):1317-1373, 2018). In particular, we discuss compatibility with Gromov-Hausdorff convergence of points-a concept first described by Gromov (Inst Hautes Etudes Sci Publ Math 53:53-73, 1981). We apply these results to the problem of stability of the positive mass theorem in mathematical relativity. Specifically, we revisit the article (Huang et al. in J Reine Angew Math 727:269-299, 2017) on intrinsic flat stability for the case of graphical hypersurfaces of Euclidean space: We are able to fill in some details in the proofs of Theorems 1.4 and Lemma 5.1 of Huang et al. (2017) and strengthen some statements. Moreover, in light of an acknowledged error in the proof of Theorem 1.3 of Huang et al. (2017), we provide an alternative proof that extends recent work of Allen and Perales (Intrinsic flat stability of manifolds with boundary where volume converges and distance is bounded below, 2020. arXiv:2006.13030).

Automated summary

We prove results on the intrinsic flat convergence of points and discuss its compatibility with Gromov-Hausdorff convergence of points. We apply these results to stability problems in mathematical relativity and provide a revised and strengthened analysis of the intrinsic flat stability of graphical hypersurfaces in Euclidean space based on the work of Huang et al. (2017). Additionally, we offer an alternative proof due to an acknowledged error in the original proof.