Relating Notions of Convergence in Geometric Analysis

TL;DR

This paper explores different notions of convergence for Riemannian manifolds, showing their relationships and differences, and establishes conditions under which these notions coincide, particularly involving $L^p$ convergence, volume convergence, and boundedness.

Contribution

It demonstrates that under boundedness and convergence of volume or metric tensors, various geometric convergence notions agree, and provides examples where they do not.

Findings

$L^p$ convergence and volume convergence imply Gromov-Hausdorff and Intrinsic Flat convergence under boundedness.

Examples show that these convergence notions can differ even for conformal metric sequences.

Boundedness conditions are crucial for the equivalence of convergence types.

Abstract

We relate $L^p$ convergence of metric tensors or volume convergence to a given smooth metric to Intrinsic Flat and Gromov-Hausdorff convergence for sequences of Riemannian manifolds. We present many examples of sequences of conformal metrics which demonstrate that these notions of convergence do not agree in general even when the sequence is conformal, $g_j=f_j^2g_0$, to a fixed manifold. We then prove a theorem demonstrating that when sequences of metric tensors on a fixed manifold $M$ are bounded, $(1-1/j)g_0 \le g_j \le K g_0$, and either the volumes converge, $\operatorname{Vol}_j(M)\rightarrow \operatorname{Vol}_0(M)$, or the metric tensors converge in the $L^p$ sense, then the Riemannian manifolds $(M,g_j)$ converge in the measured Gromov-Hausdorff and volume preserving Intrinsic Flat sense to $(M,g_0)$.