Singularities in crystalline curvature flows

Abstract

This paper discusses the behaviour of polygonal convex curves in the plane moving under crystalline curvature °ows, 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. Show more