Evolving plane curves by curvature in relative geometries

The Heat Equation Shrinks Ising Droplets to Points

“…The basic reason is that such ideas crucially relied on the fact that, in the convex case, the normal velocity is always directed inward (which is clearly false for non-convex droplets, at points where the curvature is negative). Secondly, proving existence and regularity of solution requires very different analytic and geometric arguments in the non-convex case with respect to the convex one (there, we were able to use ideas from [15,13,14]). At any rate, our previous result [20] is important in Section 6, where the evolution of the droplet boundary is controlled by locally comparing it with that of a suitable convex droplet.…”

“…The other crucial point in the proof is obtaining precise regularity estimates on the deterministic curve shortening flow. This builds on geometric and analytic ideas of Grayson [16], GageHamilton [15], Gage-Li [13,14], Chou-Zhu [6] and others. …”

Non-convergence and instability in the asymptotic behaviour of curves evolving by curvature

“…It has been considered both in the convex case ( [25], [3]), where solutions become elliptical in shape as they contract to points, and in the non-convex case [9], where closed embedded curves eventually become convex. Anisotropic evolutions (with ip non-constant) arise naturally in Finsler or Minkowski geometry on the plane [16,17], and in physical interface problems (see [6] and 409 [12]). These have also been considered for convex curves ( [1.6, 17], [5], [13]) and more generally ( [7,8], [23], [27]).…”

Finsler Active Contours

“…Flows relative to anisotropic metrics have been studied in the mathematics and physics literature; see [5], [6] and the references therein. A very simple directional flow was proposed in some of our earlier work; see [7].…”