Intrinsic Flat Stability of Manifolds with Boundary where Volume Converges and Distance is Bounded Below

TL;DR

This paper establishes conditions under which a sequence of Riemannian manifolds with boundary converges in the intrinsic flat sense to a limit manifold, emphasizing volume preservation and boundary metric convergence.

Contribution

The paper introduces new criteria for intrinsic flat convergence of manifolds with boundary, extending stability results for the positive mass theorem.

Findings

Volume preserving intrinsic flat convergence under specified conditions

Examples illustrating the necessity of hypotheses

Limitations of Gromov-Hausdorff convergence in this setting

Abstract

Given a compact, connected, and oriented manifold with boundary $M$ and a sequence of smooth Riemannian metrics defined on it, $g_j$, we prove volume preserving intrinsic flat convergence of the sequence to the smooth Riemannian metric $g_0$ provided $g_j$ always measures vectors strictly larger than or equal to $g_0$, the diameter of $g_j$ is uniformly bounded, the volume of $g_j$ converges to the volume of $g_0$, and $L^{\frac{m-1}{2}}$ convergence of the metrics restricted to the boundary. Many examples are reviewed which justify and explain the intuition behind these hypotheses. These examples also show that uniform, Lipschitz, and Gromov-Hausdorff convergence are not appropriate in this setting. Our results provide a new rigorous method of proving some special cases of the intrinsic flat stability of the positive mass theorem.