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
Abstract

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.

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