Contrasting Various Notions of Convergence in Geometric Analysis

TL;DR

This paper compares different types of convergence in geometric analysis, showing they can differ but can be aligned under certain bounds on warping functions.

Contribution

It clarifies the relationships between various convergence notions and establishes conditions for their equivalence in warped product manifolds.

Findings

Different convergence notions do not always agree.

$L^p$ bounds alone are insufficient for convergence agreement.

Combined $L^p$ and $C^0$ bounds ensure convergence types coincide.

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.