Area preserving isotopies of self transverse immersions of S^1 in R^2

TL;DR

This paper proves that two smooth self transverse immersions of S^1 in R^2 can be connected by an area-preserving isotopy if and only if their subdivided disks have equal areas, establishing a complete criterion.

Contribution

It establishes that the necessary condition of equal disk areas is also sufficient for area-preserving isotopies between such immersions.

Findings

Equal disk areas are both necessary and sufficient for area-preserving isotopies.

The paper provides a complete characterization of area-preserving isotopies for self transverse immersions of S^1.

The result applies to smooth self transverse immersions subdividing the plane into disks.

Abstract

Let C and C' be two smooth self transverse immersions of S^1 into R^2. Both C and C' subdivide the plane into a number of disks and one unbounded component. An isotopy of the plane which takes C to C' induces a 1-1 correspondence between the disks of C and C'. An obvious necessary condition for there to exist an area-preserving isotopy of the plane taking C to C' is that there exists an isotopy for which the area of every disk of C equals that of the corresponding disk of C'. In this paper we show that this is also a sufficient condition.