Asymptotic Analysis of Models for Geometric Motions

In Chapter 1, we introduce geometric motions from the general perspective of gradient flows. Here we develop the basic framework in which to pose the two main results of this thesis.

In Chapter 2, we examine the pinch-off phenomenon for a tubular surface moving by surface diffusion. We prove the existence of a one parameter family of pinching profiles obeying a long wavelength approximation of the dynamics.

In Chapter 3, we study a diffusion-based numerical scheme for curve shortening flow. We prove that the scheme is one time-step consistent.