Abstract
By works of Schoen–Yau and Gromov–Lawson any Riemannian manifold with nonnegative scalar curvature and diffeomorphic to a torus is isometric to a flat torus. Gromov conjectured subconvergence of tori with respect to a weak Sobolev type metric when the scalar curvature goes to 0. We prove flat and intrinsic flat subconvergence to a flat torus for noncollapsing sequences of 3-dimensional tori \(M_j\) that can be realized as graphs of certain functions defined over flat tori satisfying a uniform upper diameter bound and scalar curvature bounds of the form \(R_{g_{M_j}} \ge -1/j\). We also show that the volume of the manifolds of the convergent subsequence converges to the volume of the limit space. We do so adapting results of Huang–Lee, Huang–Lee–Sormani and Allen–Perales–Sormani. Furthermore, our results also hold when the condition on the scalar curvature of a torus \((M, g_M)\) is replaced by a bound on the quantity \(-\int _T \min \{R_{g_M},0\} d{\mathrm {vol}_{g_T}}\), where \(M=\text {graph}(f)\), \(f: T \rightarrow \mathbb {R}\) and \((T,g_T)\) is a flat torus. Using arguments developed by Alaee, McCormick and the first named author after this work was completed, our results hold for dimensions \(n \ge 4\) as well.
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Notes
The mean curvature convention is that spheres have positive mean curvature with respect to the inner pointing normal vector.
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Acknowledgements
The authors would like to thank Prof. Gromov and Prof. Sormani for including the third named author in their Emerging Topics on Scalar Curvature Workshop at IAS where both Gromov and Huang provided key feedback that lead to weakened hypotheses on our main theorems. We would also like to thank Sormani for funding our earlier travels to workshops in Montreal and to the Institute of Mathematics of the National Autonomous University of Mexico where we completed part of this project (DMS-1309360, DMS-1612049). We would like to thank the organizers of the Summer School on Geometric Analysis in July 2017 at the Fields Institute in Toronto where we first began working on this problem with Robin Neumayer. We would like to thank R. Neumayer and R. Haslhofer for their many helpful discussions. The third named author thanks the hospitality of the Scuola Normale Superiore di Pisa where part of this project was written while she was visiting Prof. Luigi Ambrosio. We would like to thank Prof. Sormani for suggesting this problem. We are very grateful for her continuous encouragement and support. Finally we would like to thank the anonymous referee for giving important comments and remarks that improved the final version of this article.
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Communicated by C. De Lellis.
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A. J. Cabrera Pacheco: AJCP is grateful to the Carl Zeiss Foundation for its generous support. C. Ketterer: CK is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—Projektnummer 396662902, “Synthetische Krümmungsschranken durch Methoden des Optimal Transports”. The authors were partially supported by NSF DMS-1309360 and NSF DMS-1612049.
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Cabrera Pacheco, A.J., Ketterer, C. & Perales, R. Stability of graphical tori with almost nonnegative scalar curvature. Calc. Var. 59, 134 (2020). https://doi.org/10.1007/s00526-020-01790-w
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DOI: https://doi.org/10.1007/s00526-020-01790-w