Contrasting Various Notions of Convergence in Geometric Analysis

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.

7 figures by Penelope Chang of Hunter College High School. v2: Referee comments addressed. To appear in Pacific Journal of Mathematics v3: Further clarification of Remark 2.2 provided