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Compactness and Stability Conjectures
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Christina SormaniTitle: Compactness and Stability ConjecturesAbstract: Gromov has conjectured that a sequence of compact three dimensional Riemannian manifolds with nonnegative scalar curvature converges in the intrinsic flat sense to a limit space with generalized nonnegative scalar curvature. He has also conjectured the stability of the scalar torus rigidity theorem: that a sequence of three tori with scalar >-1/j has a subsequence which converges in the intrinsic flat sense to a flat torus. We will survey examples and results in this direction including joint work with Allen and Perales introducing a notion we call VADB convergence that has been applied by Cabrera Pacheco, Perales, and Ketterer to prove the stability of the scalar torus rigidity theorem. We will suggest a number of open problems applying VADB convergence to test both these conjectures and others.
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