scholarly journals Properties of the Intrinsic Flat Distance

2018 ◽  
Vol 29 (3) ◽  
pp. 475-528 ◽  
Author(s):  
J. Portegies ◽  
C. Sormani
Keyword(s):  
Intrinsic Flat ◽  
Intrinsic Flat Distance

scholarly journals The pointed intrinsic flat distance between locally integral current spaces

2019 ◽  
pp. 1-13
Author(s):  
Shu Takeuchi
Keyword(s):  
Distance Function ◽  
Compactness Theorem ◽  
Integral Current ◽  
Intrinsic Flat ◽  
Intrinsic Flat Distance

In this note, we define a distance between two pointed locally integral current spaces. We prove that a sequence of pointed locally integral current spaces converges with respect to this distance if and only if it converges in the sense of Lang–Wenger. This enables us to state the compactness theorem by Lang–Wenger for pointed locally integral current spaces in terms of a distance function.

scholarly journals The Sormani–Wenger intrinsic flat convergence of Alexandrov spaces

2018 ◽  
Vol 12 (03) ◽  
pp. 819-839 ◽  
Author(s):  
Nan Li ◽  
Raquel Perales
Keyword(s):  
Regular Point ◽  
Dimensional Sphere ◽  
Volume Estimate ◽  
Alexandrov Spaces ◽  
Filling Volume ◽  
Integral Current ◽  
Structure Formula ◽  
Intrinsic Flat ◽  
Dimensional Integral ◽  
Uniformly Bounded

We study sequences of integral current spaces [Formula: see text] such that the integral current structure [Formula: see text] has weight [Formula: see text] and no boundary and, all [Formula: see text] are closed Alexandrov spaces with curvature uniformly bounded from below and diameter uniformly bounded from above. We prove that for such sequences either their limits collapse or the Gromov–Hausdorff and Sormani–Wenger Intrinsic Flat limits agree. The latter is done showing that the lower [Formula: see text]-dimensional density of the mass measure at any regular point of the Gromov–Hausdorff limit space is positive by passing to a filling volume estimate. In an appendix, we show that the filling volume of the standard [Formula: see text]-dimensional integral current space coming from an [Formula: see text]-dimensional sphere of radius [Formula: see text] in Euclidean space equals [Formula: see text] times the filling volume of the [Formula: see text]-dimensional integral current space coming from the [Formula: see text]-dimensional sphere of radius [Formula: see text].

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 Intrinsic flat convergence of covering spaces

2016 ◽  
Vol 184 (1) ◽  
pp. 83-114
Author(s):  
Zahra Sinaei ◽  
Christina Sormani
Keyword(s):  
Covering Spaces ◽  
Intrinsic Flat

scholarly journals Properties of the Null Distance and Spacetime Convergence

2021 ◽  
Author(s):  
Brian Allen ◽  
Annegret Burtscher
Keyword(s):  
Convergence Result ◽  
Warped Products ◽  
Invariant Metric ◽  
Lorentzian Manifolds ◽  
Limiting Behavior ◽  
Conformally Invariant ◽  
Integral Current ◽  
Globally Hyperbolic Spacetimes ◽  
Intrinsic Flat ◽  
Hyperbolic Spacetimes

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

scholarly journals Intrinsic flat Arzela–Ascoli theorems

2018 ◽  
Vol 26 (6) ◽  
pp. 1317-1373 ◽  
Author(s):  
Christina Sormani
Keyword(s):  
Intrinsic Flat

scholarly journals An intrinsic flat limit of Riemannian manifolds with no geodesics

2019 ◽  
Vol 204 (1) ◽  
pp. 265-284
Author(s):  
J. Basilio ◽  
D. Kazaras ◽  
C. Sormani
Keyword(s):  
Riemannian Manifolds ◽  
Intrinsic Flat
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