scholarly journals Intrinsic flat stability of the positive mass theorem for graphical hypersurfaces of Euclidean space

2017 ◽  
Vol 2017 (727) ◽  
Author(s):  
Lan-Hsuan Huang ◽  
Dan A. Lee ◽  
Christina Sormani
Keyword(s):  
Euclidean Space ◽  
Scalar Curvature ◽  
Positive Mass Theorem ◽  
Flat Manifold ◽  
Positive Mass ◽  
Asymptotically Flat ◽  
Intrinsic Flat

AbstractThe rigidity of the Positive Mass Theorem states that the only complete asymptotically flat manifold of nonnegative scalar curvature and

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

2020 ◽  
pp. 1
Author(s):  
Aghil Alaee ◽  
Armando J. Cabrera Pacheco ◽  
Stephen McCormick
Keyword(s):  
Euclidean Space ◽  
Positive Mass Theorem ◽  
Positive Mass

scholarly journals Bartnik mass via vacuum extensions

2019 ◽  
Vol 30 (13) ◽  
pp. 1940006
Author(s):  
Pengzi Miao ◽  
Naqing Xie
Keyword(s):  
Initial Data ◽  
Scalar Curvature ◽  
Apparent Horizon ◽  
Gauss Curvature ◽  
Positive Mass Theorem ◽  
Data Sets ◽  
Penrose Inequality ◽  
Asymptotically Flat ◽  
Apparent Horizons ◽  
Positive Gauss Curvature

We construct asymptotically flat, scalar flat extensions of Bartnik data [Formula: see text], where [Formula: see text] is a metric of positive Gauss curvature on a two-sphere [Formula: see text], and [Formula: see text] is a function that is either positive or identically zero on [Formula: see text], such that the mass of the extension can be made arbitrarily close to the half area radius of [Formula: see text]. In the case of [Formula: see text], the result gives an analog of a theorem of Mantoulidis and Schoen [On the Bartnik mass of apparent horizons, Class. Quantum Grav. 32(20) (2015) 205002, 16 pp.], but with extensions that have vanishing scalar curvature. In the context of initial data sets in general relativity, the result produces asymptotically flat, time-symmetric, vacuum initial data with an apparent horizon [Formula: see text], for any metric [Formula: see text] with positive Gauss curvature, such that the mass of the initial data is arbitrarily close to the optimal value in the Riemannian Penrose inequality. The method we use is the Shi–Tam type metric construction from [Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature, J. Differential Geom. 62(1) (2002) 79–125] and a refined Shi–Tam monotonicity, found by the first named author in [On a localized Riemannian Penrose inequality, Commun. Math. Phys. 292(1) (2009) 271–284].

scholarly journals Initial Data Rigidity Results

2021 ◽  
Author(s):  
Michael Eichmair ◽  
Gregory J. Galloway ◽  
Abraão Mendes
Keyword(s):  
Lower Bound ◽  
Initial Data ◽  
Scalar Curvature ◽  
Riemannian Manifolds ◽  
Positive Mass Theorem ◽  
Positive Mass ◽  
Rigidity Result ◽  
Trapped Surfaces ◽  
Rigidity Results ◽  
Special Case

AbstractWe prove several rigidity results related to the spacetime positive mass theorem. A key step is to show that certain marginally outer trapped surfaces are weakly outermost. As a special case, our results include a rigidity result for Riemannian manifolds with a lower bound on their scalar curvature.

scholarly journals A positive mass theorem for asymptotically flat manifolds with a non-compact boundary

2016 ◽  
Vol 24 (4) ◽  
pp. 673-715 ◽  
Author(s):  
Sérgio Almaraz ◽  
Ezequiel Barbosa ◽  
Levi Lopes de Lima
Keyword(s):  
Positive Mass Theorem ◽  
Positive Mass ◽  
Asymptotically Flat ◽  
Flat Manifolds ◽  
Asymptotically Flat Manifolds ◽  
Compact Boundary

scholarly journals Stability of the Positive Mass Theorem for Graphical Hypersurfaces of Euclidean Space

2015 ◽  
Vol 337 (1) ◽  
pp. 151-169 ◽  
Author(s):  
Lan-Hsuan Huang ◽  
Dan A. Lee
Keyword(s):  
Euclidean Space ◽  
Positive Mass Theorem ◽  
Positive Mass

scholarly journals Isoperimetry, Scalar Curvature, and Mass in Asymptotically Flat Riemannian 3‐Manifolds

2021 ◽  
Vol 74 (4) ◽  
pp. 865-905
Author(s):  
Otis Chodosh ◽  
Michael Eichmair ◽  
Yuguang Shi ◽  
Haobin Yu
Keyword(s):  
Scalar Curvature ◽  
Asymptotically Flat

scholarly journals On far-outlying constant mean curvature spheres in asymptotically flat Riemannian 3-manifolds

2020 ◽  
Vol 2020 (767) ◽  
pp. 161-191
Author(s):  
Otis Chodosh ◽  
Michael Eichmair
Keyword(s):  
Scalar Curvature ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Existence Results ◽  
Asymptotically Flat ◽  
Weingarten Surfaces ◽  
Differential Geom

AbstractWe extend the Lyapunov–Schmidt analysis of outlying stable constant mean curvature spheres in the work of S. Brendle and the second-named author [S. Brendle and M. Eichmair, Isoperimetric and Weingarten surfaces in the Schwarzschild manifold, J. Differential Geom. 94 2013, 3, 387–407] to the “far-off-center” regime and to include general Schwarzschild asymptotics. We obtain sharp existence and non-existence results for large stable constant mean curvature spheres that depend delicately on the behavior of scalar curvature at infinity.

scholarly journals Complete hypersurfaces in cylinders

1989 ◽  
Vol 46 (2) ◽  
pp. 313-318
Author(s):  
Thomas Hasanis
Keyword(s):  
Euclidean Space ◽  
Scalar Curvature ◽  
Sectional Curvature ◽  
Positive Scalar Curvature ◽  
Positive Scalar ◽  
Coordinate Functions ◽  
Bounded Below ◽  
Complete Hypersurfaces

AbstractWe consider the extent of certain complete hypersurfaces of Euclidean space. We prove that every complete hypersurface in En+1 with sectional curvature bounded below and non-positive scalar curvature has at least (n − 1) unbounded coordinate functions.

Total scalar curvature and rigidity of minimal hypersurfaces in Euclidean space

Mathematika ◽  
2001 ◽  
Vol 48 (1-2) ◽  
pp. 247-251
Author(s):  
Gabjin Yun
Keyword(s):  
Euclidean Space ◽  
Scalar Curvature ◽  
Minimal Hypersurfaces ◽  
Total Scalar Curvature
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