Stability of graphical tori with almost nonnegative scalar curvature
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Calculus of Variations
Stability of graphical tori with almost nonnegative scalar curvature Armando J. Cabrera Pacheco1 · Christian Ketterer2 · Raquel Perales3 Received: 20 March 2019 / Accepted: 31 May 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
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 Rg M j ≥ −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 − T min{Rg M , 0}dvolgT , where M = graph( f ), f : T → R and (T , gT ) 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 ≥ 4 as well. Mathematics Subject Classification 53C20 · 53C21 · 53C23
Communicated by C. De Lellis. 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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Christian Ketterer [email protected] Armando J. Cabrera Pacheco [email protected] Raquel Perales [email protected]
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Department of Mathematics, Universität Tübingen, Tübingen, Germany
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Department of Mathematics, University of Toronto, Toronto, Canada
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Instituto de Matemáticas, Universidad Nacional Autónoma de México, Oaxaca, Mexico 0123456789().: V,-vol
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1 Introduction The celebrated scalar torus rigidity theorem says that any Riemannian manifold that is diffeomorphic to an n dimensional torus and has nonnegative scalar curvature must be isometric to a flat torus. This rigidity statement follows from the fact that an n-torus cannot carry a metric of positive scalar curvature. The results were proven for n ≤ 7 using minimal surfaces theory by Schoen and Yau [24,25], and by Gromov and Lawson [14] using the Lichnerowicz formula for spin manifolds for n ≥ 8. In [17] Gromov addressed the corresponding stability problem. Conjecture 1.1 (Gromov, Sect. 5.4 [17]) “There is a particular ’Sobolev type weak metric’ in the space of n-manifolds X , such that, for example,
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