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On the affine heat equation for non-convex curves. (English) Zbl 0902.35048
Recently G. Sapiro and A. Tannenbaum [J. Funct. Anal. 119, 79-120 (1994; Zbl 0801.53008)] considered the evolution of convex plane curves moving with velocity given by their affine normal. Their result is that any smooth closed convex curve moving under this flow remains smooth and convex until it shrinks to a point. Moreover, the limiting shape is an ellipse rather than a circle as in the usual curve shortening flow studied by M. Gage and R. S. Hamilton [J. Differ. Geom. 23, 69-96 (1986; Zbl 0621.53001)] because this is an affine invariant flow, which makes it particularly interesting for applications to computer vision and image processing.
Here the authors continue this study by considering this affine invariant flow for non-convex curves. The first difficulty that needs to be addressed is finding a suitable extension of the flow to non-convex curves, because the basic invariants of affine differential geometry are defined only for convex curves. A suitable extension to non-convex curves is obtained by taking the normal velocity vector to be \(\kappa^{1/3}{\mathcal N}\), where \(\kappa\) is the curvature and \({\mathcal N}\) is the usual Euclidean normal to the curve. The basic result is that under this flow any simple closed curve shrinks to a point, and moreover, the total curvature of the curve tends to \(2\pi\).
L. Alvarez, F. Guichard, P. L. Lions and J. M. Morel [C. R. Acad. Sci., Paris, Sér. I 315, 135-138 (1992; Zbl 0792.68196) and 315, 265-268 (1992; Zbl 0805.68134)] have considered an equivalent flow from the point of view of viscosity solutions. In addition, a higher dimensional affine invariant flow for convex hypersurfaces has been studied by B. Andrews [J. Differ. Geom. 43, 207-230 (1996; Zbl 0858.53005)].
Reviewer: J.Urbas (Bonn)

MSC:
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
53A15 Affine differential geometry
58J35 Heat and other parabolic equation methods for PDEs on manifolds
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