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Transactions of the American Mathematical Society

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Stability of a quasi-local positive mass theorem for graphical hypersurfaces of Euclidean space

Authors: Aghil Alaee, Armando J. Cabrera Pacheco and Stephen McCormick
Journal: Trans. Amer. Math. Soc. 374 (2021), 3535-3555
MSC (2020): Primary 53C20; Secondary 83C99
Published electronically: February 23, 2021
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Abstract: We present a quasi-local version of the stability of the positive mass theorem. We work with the Brown-York quasi-local mass as it possesses positivity and rigidity properties, and therefore the stability of this rigidity statement can be studied. Specifically, we ask if the Brown-York mass of the boundary of some compact manifold is close to zero, must the manifold be close to a Euclidean domain in some sense?

Here we consider a class of compact $ n$-manifolds with boundary that can be realized as graphs in $ \mathbb{R}^{n+1}$, and establish the following. If the Brown-York mass of the boundary of such a compact manifold is small, then the manifold is close to a Euclidean hyperplane with respect to the Federer-Fleming flat distance.

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Additional Information

Aghil Alaee
Affiliation: Department of Mathematics and Computer Science, Clark University, Worcester, Massachusetts 01610; and Center of Mathematical Sciences and Applications, Harvard University, Cambridge, Massachusetts 02138

Armando J. Cabrera Pacheco
Affiliation: Department of Mathematics, Universität Tübingen, 72076 Tübingen, Germany

Stephen McCormick
Affiliation: Matematiska institutionen, Uppsala universitet, 751 06 Uppsala, Sweden

Received by editor(s): January 21, 2020
Received by editor(s) in revised form: September 5, 2020
Published electronically: February 23, 2021
Additional Notes: The first author acknowledges the support of the Gordon and Betty Moore Foundation, the John Templeton Foundation, and the AMS-Simons travel grant. The second author is grateful to the Carl Zeiss Foundation for its generous support.
Article copyright: © Copyright 2021 American Mathematical Society