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Singularity Analysis of Evolving Curves using the Distance Comparison Principle

In this paper, we extend Grayson’s theorem [Gra89] on curvature flow of embedded curves in a compact Riemannian surface. The main result is a new proof of a theorem of X. Zhu that, if a singularity develops in finite time, then the curve converges to a round point in a C∞ sense. The proof will extend Huisken’s distance comparison technique for curvature flow of embedded curves in the plane [Hui98].