Vol. 303, No. 1, 2019

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Contrasting various notions of convergence in geometric analysis

Brian Allen and Christina Sormani

Vol. 303 (2019), No. 1, 1–46

We explore the distinctions between Lp 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 Lp sense. We then prove a theorem which requires Lp bounds from above and C0 bounds from below on the warping functions to obtain enough control for all these limits to agree.

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Gromov–Hausdorff convergence, Sormani–Wenger intrinsic flat convergence, convergence of Riemannian manifolds, warped products
Mathematical Subject Classification 2010
Primary: 53C23
Received: 9 August 2018
Revised: 20 February 2019
Accepted: 14 May 2019
Published: 21 December 2019
Brian Allen
University of Hartford
Hartford, CT
United States
Christina Sormani
Department of Mathematics
CUNY Graduate Center
New York, NY
United States