Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Monotone Sequences of Metric Spaces with Compact Limits

R. Perales, C. Sormani

TL;DR

The paper studies monotone increasing distance functions $d_j$ on a fixed metric space with a uniform diameter bound, proving that if the pointwise limit $d_ fty$ yields a compact limit $(M,d_ fty)$, then $d_j o d_ fty$ uniformly and $(M,d_j)$ converges to $(M,d_ fty)$ in the Gromov-Hausdorff sense. When the spaces carry an integral current structure with uniformly bounded total mass, the authors establish volume-preserving intrinsic flat convergence to a limit current $(M_ abla,d_ fty,T_ abla)$ such that $ar{M}_ abla=M$, i.e., the SWIF limit fills the GH limit in the compact case. The approach hinges on constructing a common compact metric space and a taxi-product metric to compare the sequence terms, enabling both GH and SWIF analyses without curvature assumptions. The paper also highlights cases where the SWIF limit is a proper subset of the GH limit (e.g., cusp degenerations) and outlines open problems related to convergence without subsequences and the structure of limit currents. Overall, the work advances a constructive framework for simultaneous GH and SWIF convergence under monotone distance growth and mass bounds, with potential applications to geometric compactness problems in Riemannian and metric spaces.

Abstract

In this paper, we consider a fixed metric space (possibly an oriented Riemannian manifold with boundary) with an increasing sequence of distance functions and a uniform upper bound on diameter. When the metric space endowed with the pointwise limit of these distances is compact, then there is uniform and Gromov-Hausdorff (GH) convergence to this limit. When the metric space also has an integral current structure of uniformly bounded total mass (as is true for an oriented Riemannian manifold with boundary that has a uniform bound on total volume), we prove volume preserving intrinsic flat convergence to a subset of the GH limit whose closure is the whole GH limit. We provide a review of all notions and have a list of open questions at the end. Dedicated to Xiaochun Rong.

Monotone Sequences of Metric Spaces with Compact Limits

TL;DR

The paper studies monotone increasing distance functions on a fixed metric space with a uniform diameter bound, proving that if the pointwise limit yields a compact limit , then uniformly and converges to in the Gromov-Hausdorff sense. When the spaces carry an integral current structure with uniformly bounded total mass, the authors establish volume-preserving intrinsic flat convergence to a limit current such that , i.e., the SWIF limit fills the GH limit in the compact case. The approach hinges on constructing a common compact metric space and a taxi-product metric to compare the sequence terms, enabling both GH and SWIF analyses without curvature assumptions. The paper also highlights cases where the SWIF limit is a proper subset of the GH limit (e.g., cusp degenerations) and outlines open problems related to convergence without subsequences and the structure of limit currents. Overall, the work advances a constructive framework for simultaneous GH and SWIF convergence under monotone distance growth and mass bounds, with potential applications to geometric compactness problems in Riemannian and metric spaces.

Abstract

In this paper, we consider a fixed metric space (possibly an oriented Riemannian manifold with boundary) with an increasing sequence of distance functions and a uniform upper bound on diameter. When the metric space endowed with the pointwise limit of these distances is compact, then there is uniform and Gromov-Hausdorff (GH) convergence to this limit. When the metric space also has an integral current structure of uniformly bounded total mass (as is true for an oriented Riemannian manifold with boundary that has a uniform bound on total volume), we prove volume preserving intrinsic flat convergence to a subset of the GH limit whose closure is the whole GH limit. We provide a review of all notions and have a list of open questions at the end. Dedicated to Xiaochun Rong.

Paper Structure

This paper contains 23 sections, 21 theorems, 234 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Pointwise Convergence of Monotone Increasing Distance Functions
  3. Riemannian Distances
  4. Pointwise convergence
  5. Example without Uniform Convergence
  6. GH Convergence to Compact Limits
  7. Distance on the Product Space
  8. Uniform Convergence
  9. Constructing a Common Metric Space
  10. Proof of Theorem \ref{['thm:GH-cmpct']}
  11. A Common Compact Metric Space for the Whole Sequence
  12. Intrinsic Flat Convergence
  13. Lipschitz Functions and Tuples on Metric Spaces
  14. Charts into our Metric Space
  15. Ambrosio-Kirchheim's integral currents on complete metric spaces
  16. ...and 8 more sections

Key Result

Theorem 1.1

Given a compact connected Riemannian manifold, $(M^m,g_0)$, possibly with boundary, with a monotone increasing sequence of Riemannian metric tensors $g_j$ such that with uniform bounded diameter, Then the induced length distance functions $d_j: M\times M\to [0,D_0]$ are monotone increasing and converge pointwise to a distance function, $d_\infty: M\times M\to [0,D_0]$ so that $(M,d_\infty)$ is a

Figures (8)

  • Figure 2.1: This figure depicts Example \ref{['ex:not-unif']}.
  • Figure 3.1: On the left we see $(X_a,d_a)$ above $(X_b,d_b)$ mapping into $(Z_{a,b},d_{Z_{a,b}})$ and on the right we see two paths between points in $Z_{a,b}$ where the first achieves $d_{taxi_b}$ and the second takes a short cut through the image of $X_a$. The shorter of the two paths achieves the minimum in the definition of $d_{Z_{a,b}}$.
  • Figure 3.2: On the left, we see the sequence $(X,d_j)$ above with downward maps $f_{j,(j,\infty)}:(X,d_j)\to (Z_{j,\infty}, d_{Z_{j,\infty}})$ and upward maps $f_{\infty,(j,\infty)}:(X,d_\infty)\to (Z_{j,\infty}, d_{Z_{j,\infty}})$ as constructed in Proposition \ref{['prop:Zab']} and depicted in Figure \ref{['fig:Zab']}. On the right, we glue together all the $(Z_{j,\infty}, d_{Z_{j,\infty}})$ to create $(Z,d_Z)$ as in Proposition \ref{['prop:common-Z']} with maps $\chi_j:X_j\to Z$ and $\chi_\infty:X_j\to Z$.
  • Figure 4.1: Lemma \ref{['lem:ex']} applied with $h=h_{cone}$ of Example \ref{['ex:cone']} on the left (which will include the singular poles) and with $h=h_{cusp}$ of Example \ref{['ex:cusp']} on the right (which will not include the singular poles due to the cusp).
  • Figure 4.2: Example \ref{['fig:lim-cusp']} has $d_j\le d_{j+1}$ because $|h_j'|\le |h_{j+1}'|$ and converges to Example \ref{['ex:cusp']} which has the poles with cusp singularities removed.
  • ...and 3 more figures

Theorems & Definitions (52)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Example 2.6
  • Definition 3.1
  • ...and 42 more