A note on C^0 rigidity of Hamiltonian isotopies

TL;DR

This paper proves that a symplectic isotopy is Hamiltonian if it is a $C^0$ limit of Hamiltonian isotopies with converging generating Hamiltonians in the $L^{(1, abla)}$ topology, extending the understanding of $C^0$ rigidity.

Contribution

It establishes a new $C^0$ rigidity result for Hamiltonian isotopies under $L^{(1, abla)}$ convergence of generating Hamiltonians.

Findings

Symplectic isotopies are Hamiltonian if they are $C^0$ limits of Hamiltonian isotopies.

Convergence of generating Hamiltonians in $L^{(1, abla)}$ topology is sufficient.

The result extends the class of isotopies known to be Hamiltonian under $C^0$ limits.

Abstract

We show that a symplectic isotopy that is a $C^0$ limit of Hamiltonian isotopies is itself Hamiltonian, if the corresponding sequence of generating Hamiltonians converge in $L^{(1, \infty)}$ topology.