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The Space-Time Connectivity Theorem for Normal Currents

Paolo Bonicatto, Filip Rindler, Harry Turnbull

TL;DR

This work addresses witnessing weak* convergence of boundaryless normal currents by embedding them in space-time normal currents, extending the classical Federer–Fleming Connectivity to the normal-current setting. The authors develop a comprehensive framework of space-time currents, slices, variation, and a Dynamical Deformation Theorem that yields controlled space-time fillings $S_j$ with vanishing variation, ensuring a time-resolved deformation from $T_j$ to $T$. A key contribution is the Space-Time Connectivity Theorem, together with an Equality Theorem and a Lipschitz deformation-distance framework that establish bi-Lipschitz relations between deformation distance and flat norms. The results have significant implications for modeling dislocation dynamics and elastoplastic evolutions, enabling a scalable, time-resolved description of convergence and transport in systems with large numbers of defect lines or currents.

Abstract

This work establishes a Space-Time Connectivity Theorem for normal currents. In analogy to classical results by Federer and Fleming as well as a recent theorem for integral currents by the second author, this result allows one to witness the weak* convergence of a uniformly bounded sequence of boundaryless normal currents with a space-time normal current that connects the elements of the sequence with their limit. The space-time setting is distinguished from the classical case in that this connecting current has a time coordinate and thus constitutes a progressive-in-time way to deform an element of the sequence to the limit.

The Space-Time Connectivity Theorem for Normal Currents

TL;DR

This work addresses witnessing weak* convergence of boundaryless normal currents by embedding them in space-time normal currents, extending the classical Federer–Fleming Connectivity to the normal-current setting. The authors develop a comprehensive framework of space-time currents, slices, variation, and a Dynamical Deformation Theorem that yields controlled space-time fillings with vanishing variation, ensuring a time-resolved deformation from to . A key contribution is the Space-Time Connectivity Theorem, together with an Equality Theorem and a Lipschitz deformation-distance framework that establish bi-Lipschitz relations between deformation distance and flat norms. The results have significant implications for modeling dislocation dynamics and elastoplastic evolutions, enabling a scalable, time-resolved description of convergence and transport in systems with large numbers of defect lines or currents.

Abstract

This work establishes a Space-Time Connectivity Theorem for normal currents. In analogy to classical results by Federer and Fleming as well as a recent theorem for integral currents by the second author, this result allows one to witness the weak* convergence of a uniformly bounded sequence of boundaryless normal currents with a space-time normal current that connects the elements of the sequence with their limit. The space-time setting is distinguished from the classical case in that this connecting current has a time coordinate and thus constitutes a progressive-in-time way to deform an element of the sequence to the limit.

Paper Structure

This paper contains 21 sections, 38 theorems, 273 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Currents
  4. Polyhedral Approximation of Normal Currents
  5. Bi-Lipschitz Equivalence of Normal Flat Norms
  6. Good Directions
  7. Space-Time Normal Currents
  8. Slices of Space-Time Normal Currents
  9. Space-Time Variation and Rate-Independence
  10. Estimates for the Variation
  11. A "Pythagoras" Lemma
  12. Compactness and Continuity
  13. Space-Time Trajectories and the Dynamical Deformation Theorem
  14. Space-Time Trajectories
  15. Cones over Normal Currents
  16. ...and 6 more sections

Key Result

Theorem 1

Let $K$ be a compact Lipschitz neighbourhood retract and let $M > 0$. Let $(T_j)_j \subset \mathrm{I}_{k}(K)$ be a sequence of boundaryless integral currents with $\mathbf{M}(T_{j}) \leq M$ for $j \in \mathbb{N}$ and for some $T \in \mathrm{I}_{k}(K)$. Then there is a sequence $(R_j)_j \subset \mathrm{I}_{k+1}(K)$ with

Figures (3)

  • Figure 1: The lattice $\mathbb{Z}^2$ and the standard dual cubical subdivisions of $\mathbb{R}^2$.
  • Figure 2: The retraction $\sigma: \mathbb{R}^{2} \backslash \mathrm{W}"_{0} \to \mathrm{W}'_{1}$.
  • Figure 3: $B((t,t),Lt) \cap B((-t,t),Lt) \cap \overline{\Omega} = \varnothing$

Theorems & Definitions (74)

  • Theorem
  • Theorem : Normal Connectivity Theorem
  • Theorem 1.1: Space-Time Connectivity Theorem
  • Proposition 2.1
  • Proposition 2.2
  • proof : Proof of Proposition \ref{['pp:BoundarylessDeformation']}
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • Corollary 2.5
  • ...and 64 more