A Survey on Volume-preserving rigidity
Abstract
This paper employs $C^0-$arguments to study the action of the identity component, topologized with the $C^\infty$ Whitney topology, $Diff^{\Omega,\infty}_{id}(M) $ in the group of volume-pre\-serving diffeomorphisms on the space $\mathcal Z^1(M)$ of all closed $1-$forms on a compact connected oriented manifold $(M, \Omega)$. When $M$ is closed, we recover that $Diff^{\Omega,\infty}_{id}(M) $ is $C^0-$closed in the group $Diff^{\infty}(M) $ of all smooth diffeomorphisms of $M$. This implies that in two dimensions, the identity component in the group of symplectomorphisms is $C^0-$closed. We discuss several applications in the context of $C^0$ symplectic geometry for Lefschetz closed symplectic manifolds. This includes an attempt to solve the $C^0$ flux conjecture.
Key words: Rigidity, Continuum theory, Oriented manifolds, Topological dynamics, Homeomorphisms, Differential forms.
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