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Abstract
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We explore the distinctions between
convergence of metric tensors on a fixed Riemannian manifold versus
Gromov–Hausdorff, uniform, and intrinsic flat convergence of the corresponding
sequence of metric spaces. We provide a number of examples which demonstrate
these notions of convergence do not agree even for two dimensional
warped product manifolds with warping functions converging in the
sense. We then prove a
theorem which requires
bounds from above and
bounds from below on the warping functions to obtain enough control for all these
limits to agree.
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Keywords
Gromov–Hausdorff convergence, Sormani–Wenger intrinsic flat
convergence, convergence of Riemannian manifolds, warped
products
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Mathematical Subject Classification 2010
Primary: 53C23
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Milestones
Received: 9 August 2018
Revised: 20 February 2019
Accepted: 14 May 2019
Published: 21 December 2019
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