THESIS
2017
ix, 79 pages : illustrations ; 30 cm
Abstract
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...[
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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 C
^{2} 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.
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