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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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
DOI: https://doi.org/10.1090/tran/8297
Published electronically: February 23, 2021
MathSciNet review: 4237955
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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
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.
Article copyright: © Copyright 2021 American Mathematical Society