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
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Abstract | References | Similar Articles | Additional Information
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 -manifolds with boundary that can be realized as graphs in
, 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
Email:
aalaeekhangha@clarku.edu, aghil.alaee@cmsa.fas.harvard.edu
Armando J. Cabrera Pacheco
Affiliation:
Department of Mathematics, Universität Tübingen, 72076 Tübingen, Germany
Email:
cabrera@math.uni-tuebingen.de
Stephen McCormick
Affiliation:
Matematiska institutionen, Uppsala universitet, 751 06 Uppsala, Sweden
Email:
stephen.mccormick@math.uu.se
DOI:
https://doi.org/10.1090/tran/8297
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