The heat equation shrinks embedded plane curves to round points. (English) Zbl 0667.53001

Grayson, Matthew A.

J. Differ. Geom. 26, 285-314 (1987).

This paper contains the final solution of the long-standing “curve- shortening problem” for plane curves: Let \(\gamma_ 0: S^ 1\to {\mathbb{R}}^ 2\) be a regular embedded closed plane curve. Then the evolution equation \({\dot \gamma}=k\cdot N\) (N a unit normal field, k the curvature) with initial condition \(\gamma (0,s)=\gamma_ 0(s)\) always has a solution \(\gamma: {\mathbb{R}}^+\times S^ 1\to {\mathbb{R}}^ 2,\) \(S\mapsto \gamma_ t(s)=\gamma (t,s)\) is an embedded curve for all t and \(\gamma_ t\) approaches a (shrinking) round circle as \(t\to \infty\).

Reviewer: U.Pinkall

MSC:

53A04	Curves in Euclidean and related spaces
35G10	Initial value problems for linear higher-order PDEs
35K25	Higher-order parabolic equations

Keywords:

curve-shortening problem; closed plane curve; evolution equation