Michael E. Gage

Evolving Plane Curves by Curvature in Relative Geometries

In (0.1) X :S × [0, ω) → IR is the position vector of a family of closed convex plane curves, kN is the curvature vector, with k being the curvature and N the inward pointing normal given by N = −(cos θ, sin θ). The weight function γ(θ) = γ(N) is a function of the normal vector to the curve at each point but does not depend on position in the plane. Equation (0.1) has two significant interpretations. It can be seen as the generalization of the “curve shortening” problem ([Ga8]) to Minkowski geometry or as a simplified model of the motion of the interface of a metal crystal as it melts ([AnGu],[Ta1] and [Ga8]). The proof illustrates most of the techniques that have been used recently in understanding geometric evolution equations as described in [Ha3]. It is not hard to show that the self-similar solutions correspond to positive, 2π periodic solutions of the equation

The Curve Shortening Flow

This is an expository paper describing the recent progress in the study of the curve shortening equation

Curve shortening makes convex curves circular

{X_{{t\,}}} = \,kN

On an area-preserving evolution equation for plane curves

(0.1) Here X is an immersed curve in ℝ2, k the geodesic curvature and N the unit normal vector. We review the work of Gage on isoperimetric inequalities, the work of Gage and Hamilton on the associated heat equation and the work of Epstein and Weinstein on the stable manifold theorem for immersed curves. Finally we include a new proof of the Bonnesen inequality and a proof that highly symmetric immersed curves flow under (0.1) to points.