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 The Jang Equation and the Positive Mass Theorem in the Asymptotically Hyperbolic Setting

2021 ◽  
Author(s):  
Anna Sakovich
Keyword(s):  
Initial Data ◽  
Spatial Dimension ◽  
Positive Mass Theorem ◽  
Positive Mass ◽  
Asymptotically Hyperbolic ◽  
Jang Equation

AbstractWe solve the Jang equation with respect to asymptotically hyperbolic “hyperboloidal” initial data. The results are applied to give a non-spinor proof of the positive mass theorem in the asymptotically hyperbolic setting. This work focuses on the case when the spatial dimension is equal to three.

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 A positive mass theorem for asymptotically hyperbolic manifolds with inner boundary

2015 ◽  
Vol 26 (12) ◽  
pp. 1550101 ◽  
Author(s):  
Oussama Hijazi ◽  
Sebastián Montiel ◽  
Simon Raulot
Keyword(s):  
Riemannian Manifolds ◽  
Hyperbolic Manifolds ◽  
Positive Mass Theorem ◽  
Positive Mass ◽  
Asymptotically Hyperbolic Manifolds ◽  
Spin Manifolds ◽  
Asymptotically Hyperbolic

In this paper, we prove an optimal Positive Mass theorem for Asymptotically Hyperbolic spin manifolds with compact inner boundary. This improves a previous result of Chruściel and Herzlich [The mass of asymptotically hyperbolic Riemannian manifolds, Pacific J. Math. 212(2) (2003) 231–264].

scholarly journals The Conformal Flow of Metrics and the General Penrose Inequality

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Qing Han ◽  
Marcus Khuri
Keyword(s):  
Lower Bound ◽  
Initial Data ◽  
Total Mass ◽  
Einstein Equations ◽  
Data Sets ◽  
Penrose Inequality ◽  
General Initial Data ◽  
Time Symmetry ◽  
Conformal Flow ◽  
Special Case

The conformal flow of metrics has been used to successfully establish a special case of the Penrose inequality, which yields a lower bound for the total mass of a spacetime in terms of horizon area. Here we show how to adapt the conformal flow of metrics, so that it may be applied to the Penrose inequality for general initial data sets of the Einstein equations. The Penrose conjecture without the assumption of time symmetry is then reduced to solving a system of PDE with desirable properties.

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].

A Note on a Lower Bound for the Multiplicative Odds Theorem of Optimal Stopping

2014 ◽  
Vol 51 (03) ◽  
pp. 885-889 ◽  
Author(s):  
Tomomi Matsui ◽  
Katsunori Ano
Keyword(s):  
Lower Bound ◽  
Optimal Stopping ◽  
Secretary Problem ◽  
Bernoulli Trials ◽  
Optimal Stopping Problem ◽  
Maximum Probability ◽  
Optimal Stopping Rule ◽  
Optimal Probability ◽  
Optimal Stopping Theory ◽  
Special Case

In this note we present a bound of the optimal maximum probability for the multiplicative odds theorem of optimal stopping theory. We deal with an optimal stopping problem that maximizes the probability of stopping on any of the last m successes of a sequence of independent Bernoulli trials of length N, where m and N are predetermined integers satisfying 1 ≤ m < N. This problem is an extension of Bruss' (2000) odds problem. In a previous work, Tamaki (2010) derived an optimal stopping rule. We present a lower bound of the optimal probability. Interestingly, our lower bound is attained using a variation of the well-known secretary problem, which is a special case of the odds problem.

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