Evolving plane curves by curvature in relative geometries (Q1320598)

In this paper the motion of a plane curve in which the velocity vector field is determined at each point by the curvature of the curve and by the direction of the curves normal vector is studied. Let \(X\) be the position vector of the curve, the subscript \(t\) denote partial differentiation with respect to time, \(N = (-\cos \theta, -\sin \theta)\) be the normal vector to the curve, \(\nu\) be some given function of direction which is smooth and strictly positive. The main result is the following: Theorem. Let \(X_ t = \nu(\theta)kN\) be the equation describing the motion. If \(\nu\) can be written as \(\nu(\theta) = \widetilde{h}(\theta)/\widetilde{k}(\theta)\), where \(\widetilde{h}\) and \(\widetilde{k}\) are the support function and the curvature respectively of some smooth, symmetric strictly convex body \(\widetilde{K}\), then every convex curve converges to the shape of \(\partial \widetilde{K}\) as the curve shrinks to a point. More precisely, the laminae enclosed by the evolving curves, when renormalized to have the same area as \(\widetilde{K}\) and appropriately translated, will converge to \(\widetilde{K}\) in the Hausdorff metric.