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