| Recently, new proofs of Grayson's theorem [Gra87] for curvature flow of embedded curves in the plane have been given by Hamilton [Ham95b] and Huisken [Hui98]. Hamilton proved this using monotonicity of isoperimetric estimates, and Huisken proved it by obtaining a lower bound for the quotient of the extrinsic distance in the plane by the intrinsic distance along the curve.; In this thesis, we will extend Grayson's theorem [Gra89] for the curvature flow of embedded curves in a compact Riemannian surface, by showing, if a singularity develops in finite time, then the curve converges to a round point in the Cinfinity sense. We give two different proofs; one using Hamilton's isoperimetric estimates technique and the other one using Huisken's distance comparison technique. |
