zbMATH — the first resource for mathematics

On an area-preserving evolution equation for plane curves. (English) Zbl 0608.53002
Nonlinear problems in geometry, Proc. AMS Spec. Sess., 820th Meet. AMS, Mobile/Ala. 1985, Contemp. Math. 51, 51-62 (1986).
[For the entire collection see Zbl 0579.00012.]
The paper continues investigations of the following problem: ”What happens to a plane curve \(X(u,t_ 0)\) if it evolves according to the ”area-preserving gradient flow” \(X_ t=(k-2\pi /L)\cdot N\) \((k=curvature\), \(L=length\), \(N=unit\)-normal vector of the curve)”. The main result is that a convex curve remains convex and converges to a circle in the \(C^{\infty}\) metric. The proof is based on estimates, which show that the curvature remains positive and that the curvature and its derivatives remain bounded.
Reviewer: H.Viesel

53A04 Curves in Euclidean and related spaces
51M25 Length, area and volume in real or complex geometry
Zbl 0579.00012