Curvature shortening flow for immersions

Question

I was wondering if for an immersed curve in the plane, is it true that if the singularity points are evolved appropriately, then the curve becomes more embedded. And if so, would it eventually become an embedded curve.

Answer 1

ummmm, depends. False for a figure 8

Answer 2

It is not true that an immersed curve will become embedded when evolved by curve shortening flow.

The easiest proof of this is that there are immersed closed curves which are self-shrinkers for curve shortening flow (i.e., under the flow, they only shrink, but do not otherwise change shape). Such a curve obviously will never become embedded. I think such curves were first studied systematically by Abresch--Langer in this paper. See Figure 4 here for an illustration.