Alexandrov Spaces with Integral Current Structure

Jaramillo, Maree; Perales, Raquel; Rajan, Priyanka; Searle, Catherine; Siffert, Anna

doi:10.48550/arxiv.1703.08195

Cited by 2 publications

(3 citation statements)

References 27 publications

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“…Sormani and Wenger defined a larger class of oriented rectifiable weighted metric spaces, called integral current spaces, which include the 0 space, and defined the F convergence of such spaces in [138]. Oriented Alexandrov spaces are integral current spaces as seen in work of Jaramillo, Perales, Rajan, Searle, Siffert, and Mitsuishi [76][109] [108]. Integral Current Spaces need not be connected nor have geodesics, and they even include the 0 space [138].…”

Section: Geometric Notions Of Convergencementioning

confidence: 99%

Conjectures on Convergence and Scalar Curvature

Sormani¹,

Curvature²,

Convergence³

2021

Preprint

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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.

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Section: Geometric Notions Of Convergencementioning

confidence: 99%

Conjectures on Convergence and Scalar Curvature

Sormani¹,

Curvature²,

Convergence³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…The intrisic flat distance has also been studied by Jaramillo et al [13], Lakzian [14], Munn [18], Portegies [23] and the author [21], to mention a few. Gromov proposes to use intrinsic flat distance to solve some of his conjectures [10], [11].…”

Section: Introductionmentioning

confidence: 98%

“…Recently Jaramillo et al [13] applying work of Mitsuichi [17] proved that n dimensional closed oriented Alexandrov spaces (X, d) can be endowed with a n dimensional integral current structure T with weight one and no boundary such that (X, d, T ) is a n dimensional integral current space. This shows the existence of sequences of integral current spaces as in Theorem 0.1.…”

Section: Introductionmentioning

confidence: 99%

The Sormani–Wenger intrinsic flat convergence of Alexandrov spaces

Perales²

2018

J. Topol. Anal.

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We study sequences of integral current spaces (X j , d j , T j ) such that the integral current structure T j has weight 1 and no boundary and, all (X j , d j ) 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 n 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 n dimensional integral current space coming from an n dimensional sphere of radius r > 0 in Euclidean space equals r n times the filling volume of the n dimensional integral current space coming from the n dimensional sphere of radius 1.

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Alexandrov Spaces with Integral Current Structure

Cited by 2 publications

References 27 publications

Conjectures on Convergence and Scalar Curvature

Conjectures on Convergence and Scalar Curvature

The Sormani–Wenger intrinsic flat convergence of Alexandrov spaces

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