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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)

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Curve shortening on surfaces Annales scientifiques de l'É.N.S. 4 e série, tome 23, n o 2 (1990), p. 229-256 © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1990, tous droits réservés. L'accès aux archives de la revue « Annales scientifiques de l'É.N.S. » (http://www. elsevier.com/locate/ansens) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Partially supported by NSF grant DMS-8604856.