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Flat Convergence of Pushforwards of Rectifiable Currents Under $C^0-$Diffeomorphism Limits

Steéphane Tchuiaga

TL;DR

The paper addresses the stability of pullback and pushforward operations on currents and differential forms under $C^0$-limits of diffeomorphisms on a compact manifold. It develops and combines a polyhedral-approximation approach with a dual flat-norm argument to establish that, if ${\\phi_k}^*\omega\to{\\psi}^*\omega$ uniformly for all smooth forms, then ${\\phi_k}_*T\to{\\psi}_*T$ in the weak-* sense for order-0 currents and, for rectifiable currents, in the flat norm. These results yield $C^0$-rigidity conclusions for groups of geometric diffeomorphisms such as symplectomorphisms, cosymplectomorphisms, contactomorphisms, volume-preserving diffeomorphisms, and isometries, with broader implications for measure theory and dynamical systems. The framework provides a robust toolset for analyzing stability of geometric structures under $C^0$ perturbations and clarifies how convergence of pullbacks governs the behavior of pushforwards in the singular setting of currents.

Abstract

We study the action on currents and differential forms on compact Riemannian manifolds under $C^0$-limits of diffeomorphisms. Using tools from geometric analysis, measure theory, and homotopy theory, we establish several convergence theorems. We give conditions under which pullbacks of differential forms by a sequence of smooth diffeomorphisms converge uniformly (in the $C^0$ norm), and pushforwards of currents by smooth diffeomorphisms exhibit weak-* convergence. We prove that pushforwards of rectifiable currents are convergent in the flat norm, a property of particular interest in the study of singular geometric objects. These stability findings offer tools for the study of geometric structures, including applications to the stability of groups of symplectomorphisms, cosymplectomorphisms, volume-preserving transformations, and contact transformations under $C^0$ perturbations. We highlight applicability in measure theory and dynamical systems.

Flat Convergence of Pushforwards of Rectifiable Currents Under $C^0-$Diffeomorphism Limits

TL;DR

The paper addresses the stability of pullback and pushforward operations on currents and differential forms under -limits of diffeomorphisms on a compact manifold. It develops and combines a polyhedral-approximation approach with a dual flat-norm argument to establish that, if uniformly for all smooth forms, then in the weak-* sense for order-0 currents and, for rectifiable currents, in the flat norm. These results yield -rigidity conclusions for groups of geometric diffeomorphisms such as symplectomorphisms, cosymplectomorphisms, contactomorphisms, volume-preserving diffeomorphisms, and isometries, with broader implications for measure theory and dynamical systems. The framework provides a robust toolset for analyzing stability of geometric structures under perturbations and clarifies how convergence of pullbacks governs the behavior of pushforwards in the singular setting of currents.

Abstract

We study the action on currents and differential forms on compact Riemannian manifolds under -limits of diffeomorphisms. Using tools from geometric analysis, measure theory, and homotopy theory, we establish several convergence theorems. We give conditions under which pullbacks of differential forms by a sequence of smooth diffeomorphisms converge uniformly (in the norm), and pushforwards of currents by smooth diffeomorphisms exhibit weak-* convergence. We prove that pushforwards of rectifiable currents are convergent in the flat norm, a property of particular interest in the study of singular geometric objects. These stability findings offer tools for the study of geometric structures, including applications to the stability of groups of symplectomorphisms, cosymplectomorphisms, volume-preserving transformations, and contact transformations under perturbations. We highlight applicability in measure theory and dynamical systems.

Paper Structure

This paper contains 19 sections, 12 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Basic Concepts
  3. The $C^r$-topology on $\mathrm{Diff}^{\infty}(M)$
  4. Differential forms
  5. Inner product of differential forms
  6. Brief overview of currents
  7. The action of diffeomorphisms on currents
  8. Linking pullbacks and currents
  9. Illustrative example
  10. Hausdorff Distance
  11. Proofs of the Main Statements
  12. Proof of Theorem \ref{['TC']}
  13. Applications
  14. Illustrative Example (A)
  15. Illustrations and Further Discussions
  16. ...and 4 more sections

Key Result

Lemma 1

Let $M$ be a compact smooth manifold with a Riemannian metric $g$. Let $\{\phi_k\} \subseteq \mathrm{Diff}^{\infty}(M)$ be a sequence of smooth diffeomorphisms with $\phi_k \xrightarrow{C^0} \psi \in \mathrm{Diff}^{\infty}(M)$. Then:

Theorems & Definitions (33)

  • Lemma 1
  • Remark 1
  • Lemma 2
  • Theorem 1
  • Definition 1: Hirsch76
  • Definition 2: Hirsch76
  • Definition 3: Lee13
  • Definition 4
  • Definition 5: Hausdorff Distance
  • Theorem 2
  • ...and 23 more