On the stability of the positive mass theorem for asymptotically hyperbolic graphs

Pacheco, Armando J. Cabrera

doi:10.1007/s10455-019-09674-9

Cited by 10 publications

(7 citation statements)

References 32 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Remark 10.7. There are also asymptotically hyperbolic versions of this conjecture with Scal ≥ −6 proven by Sakovich and Sormani in the spherically symmetric setting [123] and announced work by Cabrera Pacheco and Perales in the graph setting [36]. It would be intriguing to see if there might be a asymptotically hyperbolic manifolds with Scal ≥ −6 similar to the geometrostatic manifolds studied by Sormani-Stavrov in [137].…”

Section: Geometric Stability Of Zero Mass Rigidity [Lee-sormani]mentioning

confidence: 82%

Conjectures on Convergence and Scalar Curvature

Sormani¹,

Curvature²,

Convergence³

2021

Preprint

View full text Add to dashboard Cite

Here we survey the compactness and geometric stability conjectures formulated by the participants at the 2018 IAS Emerging Topics Workshop on Scalar Curvature and Convergence. We have tried to survey all the progress towards these conjectures as well as related examples, although it is impossible to cover everything. We focus primarily on sequences of compact Riemannian manifolds with nonnegative scalar curvature and their limit spaces. Christina Sormani is grateful to have had the opportunity to write up our ideas and has done her best to credit everyone involved within the paper even though she is the only author listed above. In truth we are a team of over thirty people working together and apart on these deep questions and we welcome everyone who is interested in these conjectures to join us.

show abstract

Section: Geometric Stability Of Zero Mass Rigidity [Lee-sormani]mentioning

confidence: 82%

Conjectures on Convergence and Scalar Curvature

Sormani¹,

Curvature²,

Convergence³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Since this estimate holds for all ε ≤ r this implies that dh (x, y) ≤ dg (x, y), x, y ∈ B g r (p). In particular, since the points p n and q n converge to p we have that for sufficiently large n dh (p n , q n ) ≤ dg (p n , q n ), which contradicts (25).…”

Section: 3mentioning

confidence: 92%

“…The question of geometric stability has already been studied extensively for spacelike slices of spacetimes by Allen [2][3][4], Bray and Finster [17], Bryden, Khuri, and Sormani [18], Cabrera Pacheco [25], Finster [35], Finster and Kath [36], Huang and Lee [48], Huang, Lee, and Sormani [49], Jauregui and Lee [50], Lee [54], Lee and Sormani [55], LeFloch and Sormani [58], Sakovich and Sormani [70], Sormani and Stavrov Allen [76]. In the setting of spacelike slices Lee and Sormani [55] conjecture that Sormani-Wenger intrinsic flat (SWIF) convergence is the correct notion of convergence for stability.…”

Section: Introductionmentioning

confidence: 99%

Properties of the Null Distance and Spacetime Convergence

Allen,

Burtscher

2019

Preprint

View full text Add to dashboard Cite

The notion of null distance for Lorentzian manifolds recently introduced by Sormani and Vega gives rise to an intrinsic metric and induces the manifold topology under mild assumptions on the time function of the spacetime. We prove that warped products of low regularity and globally hyperbolic spacetimes with complete Cauchy surfaces endowed with the null distance are integral current spaces. This metric and integral current structure sets the stage for investigating spacetime convergence analogous to Riemannian geometry. Our main theorem is a general Lorentzian convergence result for warped product spacetimes relating uniform convergence of warping functions to uniform, Gromov-Hausdorff and Sormani-Wenger intrinsic flat convergence of the corresponding null distances. In addition, we show that non-uniform convergence of warping functions in general leads to distinct limiting behavior, such as the convergence of null distances to Gromov-Hausdorff and Sormani-Wenger intrinsic flat limits that disagree.

show abstract

“…He considers manifolds that are foliated by a smooth inverse mean curvature flow, with two boundary components that are leaves of the flow. Stability of the positive mass theorem has also been studied for asymptotically hyperbolic manifolds, by Allen [4]; Dahl, Gicquad and Sakovich [17]; Sakovich and Sormani [39]; and by the second-named author [13]. Further related stability results can be found in [5,10,12,16,21,22].…”

Section: Introductionmentioning

confidence: 98%

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

Alaee¹,

Pacheco²,

McCormick³

2019

Preprint

View full text Add to dashboard Cite

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 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.

show abstract

On the stability of the positive mass theorem for asymptotically hyperbolic graphs

Cited by 10 publications

References 32 publications

Conjectures on Convergence and Scalar Curvature

Conjectures on Convergence and Scalar Curvature

Properties of the Null Distance and Spacetime Convergence

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

Contact Info

Product

Resources

About