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The heat equation shrinking convex plane curves. (English) Zbl 0621.53001

Given a curve in the plane, let it deform in the direction of its curvature vector. It was conjectured that the curve would collapse in finite time to a point and that, as the curve collapsed, it would become asymptotically close to a shrinking standard circle. The authors prove this conjecture for a convex plane curve. In a more recent paper [J. Differ. Geom. 26, 285-314 (1987)], M. A. Grayson shows that any embedded curve eventually becomes convex, completely settling the conjecture.

Integral estimates (and the isoperimetric inequality) are used to obtain the necessary bounds for proving long time existence and convergence towards a shrinking circle. This is in contrast to G. Huisken’s work on deforming higher dimensional hypersurfaces by the mean curvature vector, where the Codazzi-Mainardi equations and the maximum principle are used.

Reviewer: D.Yang

MSC:
53A04Curves in Euclidean space
58J35Heat and other parabolic equation methods for PDEs on manifolds
58D10Spaces of embeddings and immersions