Warped tori with almost non-negative scalar curvature

  • Brian Allen
  • Lisandra Hernandez-Vazquez
  • Davide Parise
  • Alec Payne
  • Shengwen Wang
Original Paper


For sequences of warped product metrics on a 3-torus satisfying the scalar curvature bound \(R_j \ge -\frac{1}{j}\), 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.


Scalar torus rigidity theorem Almost rigidity Stability Gromov–Hausdorff convergence Sormani–Wenger intrinsic flat convergence Warped products Uniform convergence Flat torus Convergence of Riemannian manifolds 

Mathematics Subject Classification




This research began at the Summer School for Geometric Analysis, located at and supported by the Fields Institute for Research in Mathematical Sciences. The authors would like to thank the organizers of this summer school, Spyros Alexakis, Walter Craig, Robert Haslhofer, Spiro Karigiannis and McKenzie Wang. The authors would like to thank Christina Sormani for her direction, advice, and constant support. Specifically, Christina brought this team together during the Field’s institute to work on this project, organized workshops at the CUNY graduate center which the authors participated in, and provided travel support to the team members. Christina Sormani is supported by NSF grant DMS-1612049. Brian Allen is supported by the USMA. Davide Parise is partially supported by the Swiss National Foundation grant 200021L_175985.


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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.United States Military AcademyWest PointUSA
  2. 2.Stony Brook UniversityStony BrookUSA
  3. 3.Ecole Polytechnique Fédérale de Lausanne (EPFL)LausanneSwitzerland
  4. 4.Courant InstituteNew York UniversityNew YorkUSA
  5. 5.John Hopkins UniversityBaltimoreUSA

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