Abstract

The curve shortening flow of smooth curves, also referred to as flow by curvature, has seen detailed study in the last four decades. In this paper, we go over the main existence theorems of the theory, with an focus on making the presentation clear and self-contained.
Starting with convex curves in the plane, we examine the Gage-Hamilton theorem, which proves the flow of these curves exists and converges to a point. We inspect the original proof by Gage and Hamilton. We next look at Grayson's Theorem which proves the same for non-convex curves. Instead of Grayson's original proof, we study a proof by Bryan and Andrews.
The aim is to highlight the core arguments and make the proofs more accessible. We seek to emphasize the intuition underlying the key ideas.
Finally, we turn to the flow of curves on Riemannian surfaces. We present some of the necessary differential geometry material which is otherwise typically assumed. We compare results on the surface to their analogues in the plane and examine how the curvature affects the respective proofs.