CONTRASTING VARIOUS NOTIONS OF CONVERGENCE IN GEOMETRIC ANALYSIS

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.