Curve Shortening and Grayson’s Theorem

In this chapter and the next we discuss the curve shortening flow (CSF). A number of important techniques in the field of geometric flows exhibit themselves in the curve shortening flow in an elegant and less technical way. The CSF was proposed in 1956 by Mullins to model the motion of idealized grain boundaries. In 1978 Brakke studied the mean curvature flow, of which the CSF is the 1-dimensional case, in the context of geometric measure theory. Renewed interest in the CSF resulted from the works of Gage and Hamilton in 1986 on convex plane curves and Grayson in 1987 on embedded plane curves. In this chapter and throughout the rest of the book we shall assume that the objects we consider are smooth, i.e., C1. This assumption is made more out of convenience than necessity and we refer the reader to the notes and commentary at the end of this chapter for references to results with weaker regularity hypotheses.