Flow under Curvature: Singularity Formation, Minimal Surfaces, and Geodesics

We study hypersurfaces moving under flow that depends on the mean curvature. The approach is based on a numerical technique that embeds the evolving hypersurface as the zero level set of a family of evolving surfaces. In this setting, the resulting partial differential equation for the motion of the level set function φ may be solved by using numerical techniques borrowed from hyperbolic conservation laws. This technique is used to analyze a collection of problems. First we analyze the singularity produced by a dumbbell collapsing under its mean curvature and show that a multi-armed dumbbell develops a separate, residual closed interface at the center after the singularity forms. The level set approach is then used to generate a minimal surface attached to a one-dimensional wire frame in three space dimensions. The minimal surface technique is extended to construct a surface of any prescribed function of the curvature attached to a given bounding frame. Finally, the level set idea is used to study the flow of curves on 2-manifolds under geodesic curvature dependent speed.