I know studying the mean curvature flow is a very interesting area of research, I've fooled around with it a bit myself. But it honestly doesn't look like it has much applications within mathematics itself (at least when I try to search for "mean curvature flow applications", everything I find is phyics related or something like that), while, say, the Ricci flow, besides being interesting by itself (like the mean curvature flow), eventually provides a classification of all closed 3-manifolds (even in a lower level, you can use it to prove the sphere theorem or the uniformization theorem in dimension 2).

My intuition tells me that the mean curvature flow, just like any other extrinsic curvature flow, can't ever be used to get topological results about our manifolds, for example. Is this correct? If not, I'd really like seeing some interesting applications like this for the mean curvature flow (or other extrinsic flows).

I don't know if this is too elementary for mathoverflow but I decided to risk it since I didn't really find any questions like this on MSE.