$C^0$-coerciveness of Moser's problem and smoothing area preserving homeomorphisms

Abstract

In this paper, we establish the $C^0$-coerciveness of Moser's problem of mapping one smooth volume form to another in terms of the weak topology of measures associated to the volume forms. The proof relies on our analysis of Dacorogna-Moser's solution to Moser's problem of mapping one volume form to the other with the same total mass. As an application, we give a proof of smoothing result of area preserving homeomorphisms and its parametric version in two dimension, (or more generally in any dimension in which the smoothing theorem of homeomorphisms is possible, e.g., in dimension 3 but not necessarily in dimension 4). This in turn results in coincidence of the area-preserving homeomorphism group and the symplectic homeomorphism group in two dimension.

The previously indicated flaw in the scheme of the proof on the cube has been corrected. Somewhat different scheme is used relying more heavily on Dacorogna-Moser's solution. Many details of the proofs on the cube are largely new

Cited by in corpus (6)

• Uniform approximation of homeomorphisms by diffeomorphisms
• Extension of Calabi homomorphism and non-simpleness of the area preserving homeomorphism group of $D^2$
• Continuous Hamiltonian dynamics and area-preserving homeomorphism group of $D^2$
• Fragmented Hofer's geometry on Hameomorphism groups and its application
• Topological extension of Calabi homomorphism and non-simpleness of the area-preserving homeomorphism group of two-disc
• Trivial and simple spectrum for SL(2,R) cocycles with free base and fiber dynamics