On the Affine Heat Equation for Non-convex Curves

In the past several years, there has been much research devoted to the study of evolutions of plane curves where the velocity of the evolving curve is given by the Euclidean curvature vector. This evolution appears in a number of different pure and applied areas such as differential geometry, crystal growth, and computer vision. See for example [4, 5, 6, 15, 16, 17, 19, 20, 35] and the references therein. As is well known, this Euclidean curvature evolution is a “Euclidean curve shortening” process, precisely because the flow lines in the space of curves are tangent to the gradient of the length functional. Therefore, the curve perimeter is shrinking as fast as possible [17]. The behavior of an embedded curve evolving according to this flow has been well-studied. Gage and Hamilton prove that a convex embedded smooth initial curve converges to a round point under this evolution [13, 14, 15]. Grayson [16] has shown that a non-convex embedded curve converges to a convex one, and from there to a round point according to the Gage and Hamilton result. Since this evolution is based on Euclidean invariant concepts (Euclidean curvature vector), the solution is invariant only under rigid plane motions (i.e., the group of proper Euclidean motions in R generated by rotations and translations). This equation has also been called the geometric heat equation. Recently, the affine analogue of the Euclidean curve shortening flow was considered for convex curves [30]. In this case, the velocity of the evolving curve is given by the affine normal vector. The investigation of this type of evolution was motivated by the search for affine invariant flows in computer vision and image processing [32, 31]. Among the results proven in this work is that when a curve is evolving according to this flow, the area shrinks as fast as possible with respect to a certain affine metric [27]. Since the affine distance is based on area [8], in this sense, this evolution is an affine shortening flow. (See our discussion in Section 3 as well.) We have also shown that any convex plane curve converges to an elliptical point (defined relative to the corresponding family of normalized dilated curves) when the deformation is given by this affine shortening. These results make this evolution the affine analogue of the Euclidean curve shortening for convex curves.