Singularities in crystalline curvature flows (Q700537)

The crystalline curvature flows evolve polygonal planar curves such that the speed of motion of each edge is determined by a function of its length. The author discusses the asymptotic behavior of polygonal convex curves based on the rate of growth of the speed of edges as their length approaches zero. If the speed is inversely proportional to a power \(\alpha \in (0,1)\) of the length, the so-called slow growth, he shows that there always exist solutions for which the 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 singularities that can occur.