Warped tori with almost non-negative scalar curvature

Allen, Brian; Hernandez-Vazquez, Lisandra; Parise, Davide; Payne, Alec; Wang, Shengwen

doi:10.1007/s10711-018-0365-y

Cited by 17 publications

(21 citation statements)

References 19 publications

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“…This example would contradict the conjectured conclusion that the limit space has Euclidean tangent cones. The conjecture has been proven in the special case where the manifold is a warped product by Allen, Hernandez-Vazquez, Parise, Payne, and Wang in [5]. That is they assume the metric tensor is of the form (114) a 2 j (z) dx 2 + b 2 j (z) dy 2 + dz 2 or dx 2 + dy 2 + f 2 j (x, y) dz 2 on S 1 × S 1 × S 1 .…”

Section: Geometric Stability Of the Scalar Torus Rigidity Theoremmentioning

confidence: 95%

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 Stability Of the Scalar Torus Rigidity Theoremmentioning

confidence: 95%

Conjectures on Convergence and Scalar Curvature

Sormani¹,

Curvature²,

Convergence³

2021

Preprint

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“…Contrasting L 2 , GH and SWIF Convergence. In applications where one expects SWIF convergence for a sequence of Riemannian manifolds it has been noticed that one often obtains L 2 convergence or W 1,2 convergence more immediately (See [2,3,8]). This motivated the author and Christina Sormani to investigate the connections between L 2 convergence and SWIF convergence in [4] where we proved the following theorem for warped products.…”

Section: Key Compactnessmentioning

confidence: 99%

Inverse Mean Curvature Flow and the Stability of the Positive Mass Theorem

Allen¹

2018

Preprint

Self Cite

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We study the stability of the Positive Mass Theorem (PMT) in the case where a sequence of regions of manifolds with positive scalar curvature U i T ⊂ M 3 i are foliated by a smooth solution to Inverse Mean Curvature Flow (IMCF) which may not be uniformly controlled near the boundary. Then if→ 0 and extra technical conditions are satisfied we show that U i T converges to a flat annulus with respect to Sormani-Wenger Intrinsic Flat (SWIF) convergence.

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“…Further progresses toward the conjecture formulated by Sormani [26] has been made in various cases. In [2], Allen, Hernandez-Vazquez, Parise, Payne, and Wang studied the warped product case. In [9], Cabrera Pacheco, Ketterer, and Perales studied the case of graphical tori.…”

Section: Introductionmentioning

confidence: 99%