Contrasting various notions of convergence in geometric analysis

Brian Allen and Christina Sormani

Vol. 303 (2019), No. 1, 1–46

DOI: 10.2140/pjm.2019.303.1

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.

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