In this thesis, we study self-similar solutions to the mean curvature flow and parabolic inverse curvature flows by degree ‒1 homogeneous symmetric functions of principal curvatures in Euclidean spaces. There are four main results.

The first result is to prove that round spheres are the only compact self-expanders to these inverse curvature flows. Secondly, any complete non-compact mean convex self-expander to the mean curvature flow which has asymptotically polynomial end(s) must be rotationally symmetric about the axis of the rotation (one special case is the conical end(s)). Thirdly, any complete non-compact self-expander to these inverse curvature flows which is asymptotically cylindrical must be rotationally symmetric about the axis of the cylinder. Lastly, we prove...[ Read more ]

In this thesis, we study self-similar solutions to the mean curvature flow and parabolic inverse curvature flows by degree ‒1 homogeneous symmetric functions of principal curvatures in Euclidean spaces. There are four main results.

The first result is to prove that round spheres are the only compact self-expanders to these inverse curvature flows. Secondly, any complete non-compact mean convex self-expander to the mean curvature flow which has asymptotically polynomial end(s) must be rotationally symmetric about the axis of the rotation (one special case is the conical end(s)). Thirdly, any complete non-compact self-expander to these inverse curvature flows which is asymptotically cylindrical must be rotationally symmetric about the axis of the cylinder. Lastly, we prove the existence of complete non-compact self-expanders with the same topology as, but different geometry from round cylinders, to uniformly parabolic inverse curvature flows. They are C² asymptotic to two co-axial round cylinders with different radii, hence establishing non-uniqueness of non-compact self-expanders even in the same topological class.