Great answer! Thanks!

I see, thanks for the answer! I'll still wait for other people's inputs before accepting yours though. Do you know of any topological results we can obtain using mean curvature flow?

I know that mean curvature isn't completely far from topology, since a sphere for example can't be embedded in a way such that the mean curvature is negative everywhere. But this is a pretty weak result and I can't think of anything else like that right now

@JonnyEvans My reasoning is that the mean curvature flow doesn't care about topology, only about mean curvature, and surfaces can be locally isometric (so that they have the same mean curvature) yet not homeomorphic. While sectional curvature, Ricci curvature and scalar curvature are all intrinsic, so in my head it would be much easier to obtain topological information from there rather than something that depends on ambient space. Of course, I don't trust my intuition and don't know whether this is right. So I'd really like to see some topological results using MCF.

@RBega2 that's true! Thanks for the patience, my doubts are all solved now. If you want to post a short answer I'll accept it.

@RBega2 what's the distance to the tip then?

I hadn't realized that, thanks. I don't want $\alpha \cup \beta$ to be necessarily simple, just smoothness will work well enough.

I thought that was implied, but clearly it's not. I'll correct my post.

@DeaneYang I'll try to work on that, thanks! But I can't quite see yet why the endpoints of the unrolled curve have to be connected by a horizontal line segment whose length is an integer multiple of $2\pi$, could you elaborate a little more on that?