Comment by matheist
12 days ago
To be precise, the mean curvature and metric are the same but the immersions are different (they're not related by an isometry of the ambient space).
Topologically they're the same (the example found was different immersions of a torus).
Is it the case that 'they' are simply two ways of immersing the same two tori in R^3 such that the complements in R^3 of the two identical tori are topologically different?
If so, isn't this just a new flavor of higher-dimensional knot theory?
They don't appear to care about the images of the immersions or their complements, aside from them not being related by an isometry of R^3. They're not doing any topology with the image.
In other works, they have two immersions from the torus to R^3, whose induced metric and mean curvature are the same, and whose images are not related by an isometry of R^3. I didn't see anything about the topology of the images per se, that doesn't seem to be the point here.