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Research Report MRR01-011
Singularities in crystalline curvature flows
Ben Andrews
Abstract:
This paper discusses the behaviour of polygonal convex curves in the
plane moving under crystalline curvature flows, in which the speed of
motion of each edge is determined by a function of its length. The
behaviour depends on the rate of growth of the speed as the length of
the edge approaches zero: For slow growth - including the homogeneous
case where speed is inversely proportional to a power $\alpha\in (0,
1)$ of the length - there are always solutions for which the enclosed
area approaches zero while the length remains positive. If $\alpha
>1$, then all solutions are asymptotic to homothetically contracting
solutions, and if $\alpha = 1$ then there is a range of different
kinds of singularity that occur.
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