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

TL;DR

This paper investigates how rectifiable currents and differential forms behave under $C^0$-limits of diffeomorphisms on compact Riemannian manifolds, establishing convergence results in the flat norm and uniform norms.

Contribution

It provides new convergence theorems for pushforwards of rectifiable currents and pullbacks of forms under $C^0$-limits of diffeomorphisms, with applications to geometric stability.

Findings

Pushforwards of rectifiable currents converge in the flat norm.

Pullbacks of differential forms converge uniformly under certain conditions.

Applications to stability of geometric transformation groups.

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.