Curve shortening makes convex curves circular

Abstract

In this paper we show that as a smooth convex plane curve is deformed along its normal vector field at a rate proportional to its curvature, the isoperimetric ratio LZ/A of the curve approaches 4g as the enclosed area approaches zero. If the one parameter family of curves evolving in this fashion is normalized by homothetic expansion of the plane so that each curve encloses area g, then the resulting family will converge to the unit circle.If the evolving curves should develop a singularity (for example, a corner) the evolution process is undefined and the enclosed area may not approach zero. A subsequent paper by Richard Hamilton and the author will show that such singularities do not occur for convex curves.I would like to thank E. Calabi and H. Gluck for many useful conversations and much encouragement during the preparation of this paper. Notation and preliminariesWe let 7(0 be a one parameter family of closed convex C 2 curves; G(t) is the lamina enclosed by 7(0, while L(t) and A(t) are the length of 7(0 and the area of G(t) respectively. The position vector X(t, q)) parameterizes the curve; N(t, q)) is the inward pointing unit normal; and the curvature is ~c(t, cp). The evolution equation can now be written as Xt(t, qo) = ~c(t, q)) N (t, q~)( 1) where the subscript denotes partial differentiation with respect to t.In general the parameter q) is not the arc length parameter for each curve. X(t,q~) dq~.The arc length form of a curve is given by ds--~ The support function of the curve ?(t) is p(t, ~o) = - (X(t, q~), N(t, q~)).The following formulae for area and length and their time derivatives under the evolution equation (1)

Non-convergence and instability in the asymptotic behaviour of curves evolving by curvature

“…Many authors have considered the motion of curves in the plane by speeds depending on curvature and normal direction: If 70 is a convex closed curve given by an embedding XQ : C -> M?, this motion is described by an equation of the form (1) J^&t) --^(n(e,t))(^t)rn(e,t), A well-known example of such an evolution equation is the curveshortening flow, in which a = 1 and ip = I. Gage [14,15] and Gage and Hamilton [18] proved that convex embedded curves become circular while contracting to points, and Grayson [19] extended this to arbitrary embedded closed curves. The case a = 1/3, ip = 1 is natural in affine geometry, and has been applied to image processing and related problems.…”

Remarks on some isoperimetric properties of thek− 1 flow

“…Thus the generic behavior of the k − 1 flow is either converging to a point or expanding to infinity. The asymptotic behavior of γ t as t → T max (or t → ∞) in the above three cases are known due to [Chou and Zhu 2001, Theorem 3.12;Gage 1984;Gage and Hamilton 1986;Chow and Tsai 1996]. Also see [Yagisita 2005] for a more refined estimate in the expanding case.…”

Shape Memory Wires in R3

“…By Gage [23], Gage and Hamilton [24] and Grayson [25], it is well known that the curve-shortening flow shrinks simple closed curves to a point in a finite time. Since the curve-shortening flow can be regarded as a one-dimensional case of mean curvature flow for surfaces, the flow is applicable to various mathematical analysis.…”