Warped Tori Almost Rigidity

Some members of this team met at CUNYGC GR Workshop July 31-August 2, 2017

Brian Allen, Alec Payne, Lisandra Hernandez, and Shengwen Wang met at NYU on Sept 14-15.

Lisandra Hernandez met with Prof Sormani on Sept 26 at Stony Brook.

Brian Allen met with Prof Sormani on Oct 12 at CUNYGC.

Lisandra Hernandez met with Prof Sormani on Oct 13 at Stony Brook.

Davide Parise is continuing to contribute via email.

They met again at the SCGP Spring School at Stony Brook in March 2018.

They have completed the following paper and it has been accepted for publication:

• Warped Tori of Almost Nonnegative Scalar Curvature" originally named "Almost Rigidity of Warped Tori"

  • by Brian Allen, Lisandra Hernandez-Vazquez, Davide Parise, Alec Payne, Shengwen Wang; 21 pages, (arxiv preprint)

  • accepted by Geometreia Dedicata

  • We show that for warped products on a 3-torus, there is almost rigidity of the Scalar Torus Rigidity Theorem: for sequences of warped product metrics on a 3-torus satisfying the scalar curvature bound Rj≥−1j, uniform upper volume and diameter bounds, and a uniform lower area bound on the smallest minimal surface, we find a subsequence which converges in both the Gromov-Hausdorff and the Sormani-Wenger Intrinsic Flat sense to a flat 3-torus. (NSF DMS 1612049) (Team website). Jeff Cheeger was not happy with the use of the term “almost rigidity” both in title and within the paper and requested the change. The term was first used in his 1996 paper with Colding and had a more restricted meaning there. It was too late to change previous papers but moving forward we should avoid using this term.

Davide Parise presented this at the Montreal meeting.

* Warning: The term "almost rigidity theorem" was introduced by Cheeger-Colding in their 1996 paper. In these lectures, I used the term more generally including theorems that they would call "stability theorems". Indeed Gromov also states his conjectures as "stability conjectures". I avoided the use of the word "stability" in my lectures due to confusions arising with other usages of the word stability when dealing with manifolds that have scalar curvature bounds. In April 2018, Jeff Cheeger requested that we not use the term "almost rigidity" in this new expanded way but only use it to refer to theorems in which the limit space is not predetermined (as explained in their original paper). Although the conjectures stated here and various teams listed above use the term "almost rigidity" it is best to publish your results using the term "stability" or just simply state the theorems without naming them.

Sormani was partially supported by NSF DMS \#1006059