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. 


**EDIT:** To make things a little bit more objective, what I really wanna see are topological results obtained from mean curvature flow. Naturally I think results using mean curvature flow can only be as strong as results that relate topology and mean curvature, and as far as I know, those are kinda pretty weak.



Does the mean curvature flow naturally come with less applications than intrinsic curvature flows?
Here is a list of some topological and geometric applications:


1. Huisken-Ilmanen used inverse MCF to prove the Riemannian Penrose inequality: <https://projecteuclid.org/euclid.jdg/1090349447>
2. Huisken-Sinestrari used MCF with surgery to classify two-convex hypersurfaces:
<https://link.springer.com/article/10.1007/s00222-008-0148-4>
3. Buzano, Hershkovits and I used MCF with surgery to prove that the moduli space of two-convex embedded spheres is connected:
<https://arxiv.org/abs/1607.05604>
4. Ketover and I used MCF with surgery to prove existence of smooth mean convex foliations and minimal spheres:
<https://arxiv.org/abs/1708.06567>
5. Bernstein-Wang used MCF to classify hypersurfaces with low entropy: <https://arxiv.org/abs/1511.00387>
6. Ilmanen-White used MCF to prove sharp lower bounds on the density of minimal cones: <https://arxiv.org/abs/1010.5068>
7. Schulze proved the optimal isoperimetric inequality for surfaces in any codimension in Cartan-Hadamard manifolds: <https://arxiv.org/abs/1802.00226>


