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.

arXiv comment7 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

References in corpus (3)

• Warped Tori with Almost Non-Negative Scalar Curvature
• Inverse Mean Curvature Flow and the Stability of the Positive Mass Theorem and Riemannian Penrose Inequality Under $L^2$ Metric Convergence
• Stability of the Positive Mass Theorem and Riemannian Penrose Inequality for Asymptotically Hyperbolic Manifolds Foliated by Inverse Mean Curvature Flow

Cited by in corpus (5)

• Properties of the Null Distance and Spacetime Convergence
• Almost non-negative scalar curvature on Riemannian manifolds conformal to tori
• Warped Tori with Almost Non-Negative Scalar Curvature
• Sobolev stability of the PMT and RPI using IMCF
• Sobolev Inequalities and Convergence For Riemannian Metrics and Distance Functions