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

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- by Aghil Alaee, Armando J. Cabrera Pacheco and Stephen McCormick PDF
- Trans. Amer. Math. Soc.
**374**(2021), 3535-3555 Request permission

## 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
- MR Author ID: 1055444
- Email: aalaeekhangha@clarku.edu, aghil.alaee@cmsa.fas.harvard.edu
**Armando J. Cabrera Pacheco**- Affiliation: Department of Mathematics, Universität Tübingen, 72076 Tübingen, Germany
- MR Author ID: 1148286
- ORCID: 0000-0002-1148-5125
- Email: cabrera@math.uni-tuebingen.de
**Stephen McCormick**- Affiliation: Matematiska institutionen, Uppsala universitet, 751 06 Uppsala, Sweden
- MR Author ID: 1085293
- ORCID: 0000-0001-9536-9908
- Email: stephen.mccormick@math.uu.se
- 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.
- © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**374**(2021), 3535-3555 - MSC (2020): Primary 53C20; Secondary 83C99
- DOI: https://doi.org/10.1090/tran/8297
- MathSciNet review: 4237955