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

Abstract: This article deals with the stability of the pushforward operation on currents with respect to $C^0$ limits of diffeomorphisms on compact Riemannian manifolds. We have established the uniform convergence of pullbacks of smooth forms and weak-* convergence of pushforwards of general currents. The key lemma brings convergence on closed 1-forms for the evaluation of 1-currents. The main theorem shows that the pushforward of rectifiable $k$-currents converges in the flat topology for $C^0$ convergent sequences of diffeomorphisms. We discuss implications in symplectic, cosymplectic, and contact geometry, making connections with the $C^0$ rigidity of certain geometric structures. We also consider applications to free boundary problems and stochastic currents. These results provide insight into the behavior of geometric objects under non-smooth perturbations, which have relevance in geometric measure theory, dynamical systems, and optimal transport. We highlight various open problems regarding weaker regularity and more involved geometric settings.