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Solitons of geometric flows and their applications
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Helmensdorfer, Sebastian (2012) Solitons of geometric flows and their applications. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b2623806~S1
Abstract
In this thesis we construct solitons of geometric
flows with applications in
three different settings.
The first setting is related to nonuniqueness for geometric heat
flows. We
show that certain double cones in Euclidean space have several self-expanding evolutions
under mean curvature
flow. The construction of the associated self-expanding
solitons leads to an application in
fluid dynamics. We present a new model for
the behaviour of oppositely charged droplets of
fluid, based on the mean curvature
flow of double cones. If two oppositely charged droplets of
fluid are close to each
other, they start attracting each other and touch eventually. Surprisingly, experiments
have shown, that if the strength of the charges is high enough, then the
droplets are repelled from each other, after making short contact. The constructed
self-expanders can be used to correctly predict the experimental results, using our
theoretical model.
Secondly we employ space-time solitons of the mean curvature
flow to give
a geometric proof of Hamilton's Harnack estimate for the mean curvature
flow.
This proof is based on the observation that the associated Harnack quantity is the
second fundamental form of a space-time self-expander. Moreover the self-expander
is asymptotic to a cone over the convex initial hypersurface. Hence the self-expander
can be seen as the mean curvature evolution of a convex cone, which we exploit to
show that preservation of convexity directly implies the Harnack estimate.
In the last chapter we study solutions of the mean curvature
flow in a Ricci
flow backgound. We show that the space-time track of such a solution can be seen
as a soliton. Moreover the second fundamental form of this soliton matches the
evolution of a functional, which is the analogue of G. Perelman's F-functional for
the Ricci
flow on a manifold with boundary and which also has relations to quantum
gravity. Furthermore our construction provides a link between the Harnack estimate
for the mean curvature
flow and the Harnack estimate for the Ricci
flow.
Item Type: | Thesis (PhD) | ||||
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Subjects: | Q Science > QA Mathematics Q Science > QC Physics |
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Library of Congress Subject Headings (LCSH): | Geometry, Differential, Solitons, Ricci flow | ||||
Official Date: | August 2012 | ||||
Dates: |
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Institution: | University of Warwick | ||||
Theses Department: | Mathematics Institute | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Supervisor(s)/Advisor: | Topping, Peter, 1971- | ||||
Extent: | vii, 71 leaves : illustrations. | ||||
Language: | eng |
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