Evolution of Locally Convex Closed Curves in Nonlocal Curvature Flows

Natasa Sesum, Dong-Ho Tsai, Xiao-Liu Wang

Published 2017-08-16Version 1

We provide sufficient conditions on an initial curve for the area preserving and the length preserving curvature flows of curves in a plane, to develop a singularity at some finite time or converge to an $m$-fold circle as time goes to infinity. For the area-preserving flow, the positivity of the enclosed algebraic area determines whether the curvature blows up in finite time or not, while for the length-preserving flow, it is the positivity of an energy associated with initial curve that plays such a role.