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 an

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.