Moser's Method and Conservative Extensions of Diffeomorphisms

TL;DR

This paper explores Moser's method and conservative extensions in symplectic geometry, proving a special case of the Strong Darboux Theorem and extending diffeomorphisms with various techniques, including generating functions.

Contribution

It introduces new conservative extension results for circle diffeomorphisms and boundary-defined diffeomorphisms using Moser's homotopy and generating functions methods.

Findings

Proved a special case of the Strong Darboux Theorem.

Established conservative extension results for circle diffeomorphisms.

Extended techniques to boundary-defined diffeomorphisms and proved an ambient Dacarogna-Moser Theorem.

Abstract

This paper shall be concerned with three main results. After a brief recollection of basic symplectic geometry, we prove using Moser's homotopy method a special case of the Strong Darboux Theorem found, for instance, in Theorem 21.1.6 of [Hor]. Next, we'll prove two conservative extension results for a diffeomorphism on a circle. One uses Moser's homotopy method but loses a degree of regularity. The other uses the method of generating functions as found in [BCW] and [BGV]. Finally, we'll prove a conservative extension result for a "diffeomorphism" defined on the boundary of $(0, 1)^2$ and use the techniques developed there and by [M] to prove an ambient Dacarogna-Moser Theorem.