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

Brian Allen, Raquel Perales

TL;DR

The paper proves volume-preserving intrinsic flat convergence of a sequence of compact oriented manifolds with boundary under lower metric bounds $g_j\ge g_0$, uniform diameter bounds, and convergence of volumes, along with boundary metric convergence in $L^{\frac{m-1}{2}}$ (and in stronger $L^{\frac{m}{2}}$ cases). It extends the boundary-free VADB framework to manifolds with boundary via a delta-doubling construction and almost-everywhere distance convergence, then embeds the manifolds into a common space to obtain VF convergence to the prescribed limit $(M,g_0)$. The work includes new boundary-extended theorems, revisits the core VADB machinery, and develops a detailed proof strategy that culminates in applications to the intrinsic flat stability of the positive mass theorem for graphs. The results demonstrate that VF convergence, rather than Gromov-Hausdorff convergence, is the natural notion in this boundary-rich setting and provide a robust method for proving PMT stability in graph and related geometries.

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.

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

TL;DR

The paper proves volume-preserving intrinsic flat convergence of a sequence of compact oriented manifolds with boundary under lower metric bounds , uniform diameter bounds, and convergence of volumes, along with boundary metric convergence in (and in stronger cases). It extends the boundary-free VADB framework to manifolds with boundary via a delta-doubling construction and almost-everywhere distance convergence, then embeds the manifolds into a common space to obtain VF convergence to the prescribed limit . The work includes new boundary-extended theorems, revisits the core VADB machinery, and develops a detailed proof strategy that culminates in applications to the intrinsic flat stability of the positive mass theorem for graphs. The results demonstrate that VF convergence, rather than Gromov-Hausdorff convergence, is the natural notion in this boundary-rich setting and provide a robust method for proving PMT stability in graph and related geometries.

Abstract

Given a compact, connected, and oriented manifold with boundary and a sequence of smooth Riemannian metrics defined on it, , we prove volume preserving intrinsic flat convergence of the sequence to the smooth Riemannian metric provided always measures vectors strictly larger than or equal to , the diameter of is uniformly bounded, the volume of converges to the volume of , and 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.

Paper Structure

This paper contains 19 sections, 18 theorems, 250 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Notation
  4. Review of Sormani-Wenger Intrinsic Flat Distance
  5. $L^p$ convergence of Riemannian metrics
  6. A Few Words About VADB
  7. Examples of Sequences with Boundaries
  8. Lower bound on the sequence of metrics removed
  9. Volume Convergence Removed
  10. $L^{\frac{m-1}{2}}$ convergence of the boundary removed
  11. $\mathcal{VF}$ Limit Different to GH Limit
  12. VADB Revisited
  13. New Theorems
  14. Revisiting VADB
  15. Proofs of New Theorems
  16. ...and 4 more sections

Key Result

Theorem 1.1

Suppose we have a fixed compact, oriented, connected, and smooth Riemannian manifold with non empty boundary, $M_0=(M^m,g_0)$, a sequence of continuous Riemannian manifolds $M_j=(M,g_j)$ so that a uniform upper bound on diameter volume convergence and Then $M_j$ converges to $M_0$ in the volume preserving intrinsic flat sense:

Theorems & Definitions (45)

  • Theorem 1.1
  • Theorem 1.2
  • Example 2.1
  • Definition 2.2: Sormani-Wenger SW-JDG
  • Lemma 2.3: Lemma 2.2 in VADB
  • Lemma 2.4: Lemma 2.3 in VADB
  • Theorem 2.5: Theorem 1.1 VADB
  • Theorem 2.6: Theorem 4.4 in Allen-Sormani
  • Example 3.1: Example 3.4 in Allen-Sormani
  • Example 3.2: Example 3.1 in Allen-Sormani-2
  • ...and 35 more