#### 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 ${L}^{p}$ 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 ${L}^{p}$ sense. We then prove a theorem which requires ${L}^{p}$ bounds from above and ${C}^{0}$ bounds from below on the warping functions to obtain enough control for all these limits to agree.

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