J. reine angew. Math. 50 6 (1999), 43-83 Journal fur die reine und angewandte Mathematik © W a l t e r d e G r u y t e r Berlin · New York 1999 The affine curve-lengthening flow B y Ben Andrews1) at Canberra A b s t r a c t .The motion o f any smooth closed convex curve in the plane i n the d i r e c ¬ tion o f steepest increase o f its a f f i n e arc length can be continued smoothly f o r all time. The evolving curve remains strictly convex w h i l e expanding to i n f i n i t e s i z e and approaching a homothetically expanding ellipse. 1. Introduction In this paper w e study an affine-geometric, fourth-order parabolic evolution equation f o r closed convex curves in the plane. This i s d e f i n e d by following the direction o f steepest ascent o f the a f f i n e arc length functional L, with respect to a natural affine-invariant inner product. The definitions o f these are g i v e n i n Section 2, along with a discussion o f other aspects o f the a f f i n e differential geometry o f curves. The evolution equation can be written as follows: ( 1 ) $ where $ is the affine normal vector, and $ is the affine curvature. In terms of Euclidean- geometric invariants, this can be written as follows: $ where is the Euclidean curvature of the curve, u is an anticlockwise Euclidean arc-length parameter, n is the outward unit normal, and t is the Euclidean unit tangent vector. This is a fifth-order system of partial differential equations which is equivalent to a scalar fourth- order parabolic equation (see Section 3). Equation (1) is the parabolic analogue of the affine-geometric maximal surface equation, which has been an important problem in affine differential geometry, and with regard to which many problems remain unsolved (see [NS] and [LSZ]). 1) Research partly supported by NSF grant DMS 9504456 and a Terman fellowship. Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM 44 Andrews, The affine curve-lengthening flow The analogous Euclidean-geometric evolution equation, the curve-shortening flow, has been studied extensively b y Gage ([Ga1], [Ga2]), Gage and Hamilton [GH], Grayson [Gr] and many others. For that evolution equation i t has been shown that any initial closed, smooth embedded curve produces a solution which exists f o r a f i n i t e time, contract- ing at the end o f that time to a single point, in such a w a y that the curves can be rescaled about the final point to converge to a circle. The curve-shortening flow i s a second-order partial differential equation, and many details of its analysis are based on the parabolic maximum principle, which i s not available to us i n the present problem. However, the result w e prove i n this paper i s somewhat analogous to that f o r the curve-shortening flow: T h e o r e m 1 . Let $ be a smooth, strictly locally convex embedding of a curve $ into the plane. Then there exists a unique, smooth solution {x t } of th e ev o l u t i o n eq u a t i o n (1), which exists for all positive times. There exists a smooth embedding $, a point $, and a real number $, such that the image of x∞ is an ellipse of enclosed area π with centre at the origin, and $ Any e l l i p s e ev o l v e s b y expanding homothetically about its centre. Some other evolution equations o f higher order f o r curves i n the plane have been considered before - Polden [Po1] has considered the L2 gradient flow o f the integral o f the square o f the curvature, which i s a fourth-order parabolic equation (note that Langer and Singer [LS] earlier introduced a d i f f e r e n t gradient flow, i n which the integral o f the squared curvature i s treated as a functional on the space o f W2,1 indicatrices o f curves o f fixed length - this produces a non-local flow o f zeroth order; W e n [Wl-2] has also considered the gradient flow o f the same functional on the space o f L2 indicatrices o f curves o f fixed length, which i s a second order parabolic equation). Higher order equations have also been considered f o r the evolution o f Riemannian metrics, b y Rendall [Re], Singleton [Si], Chruściel [Ch], and Polden [Po2]. The main resuit used in characterising the long-term behaviour of solutions (presented in Section 4) is closely related to integral estimates which the author developed in appli¬ cations to second-order evolution equations for curves and for hypersurfaces ([A1]-[A3]). The regularity estimates for the flow, given here in Section 7, are inspired by the work of Alexander Polden [Po1-2] on higher order evolution equations for curves and for metrics on surfaces. The equation under investigation here has some remarkable special features not shared by most other higher-order equations, however: First, we have the advantage of a large group of symmetries (namely, the special affine group); second, the equation has a certain degeneracy (see Eq. (19)) which has important consequences for the analysis - although in many situations such degeneracy may considered a disadvantage, it is vital to the argument of the present paper, because non-degenerate fourth-order equations do not preserve pointwise inequalities, and in particular could not keep arbitrary convex curves convex. Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM Andrews, The affine curve-lenghtening flow 45 The organisation o f the paper i s as follows: In Section 2 w e summarise various aspects o f the a f f i n e differential geometry o f curves, support functions and their use i n describing convex sets, and other results required i n the later analysis. In Section 3 w e introduce the evolution equation, establish the short-time existence o f solutions, and discuss the n o r m a ¬ lisation o f solution curves using scalings, translations, and area-preserving a f f i n e boosts. Section 4 accomplishes the k e y step i n the proof, using the Brunn-Minkowski theorem about concavity of the area functional to deduce an integral estimate with two important consequences: First, it implies that solutions which e x i s t f o r a l l time must evolve purely b y scaling i n the large-time limit; and secondly i t implies that the evolving curve remains strictly convex as long as it exists, as w e demonstrate i n Section 5 . Section 6 completes our characterisation o f the long-term behaviour b y showing that the only solutions which e v o l v e purely b y scaling are ellipses. This i s proved b y partially integrating the fourth-order ordinary differential equation characterising such solutions, and deriving a contradiction to the a f f i n e isoperimetric inequality i n a l l cases except ellipses. In particular w e make use o f an affine-geometric version o f the four-vertex theorem (Proposition 1 8 ) . Section 7 provides regularity estimates which imply that solutions e x i s t f o r a l l time, and Section 8 combines all these elements to complete the proof o f Theorem 1 . W e note that the classification o f homothetic solutions f o r this evolution equation has also been carried out b y Lima and Montenegro [LM], and their analysis also g i v e s a classification o f non-embedded homothetic solutions. A d i f f e r e n t affine-geometric evolution equation, the f l o w b y the a f f i n e normal vector, has been studied b y Tannenbaum and Sapiro [TS1-2] and the author [A4]. That evolution equation i s o f second order, and the methods associated with it are more c l o s e l y related to those applied to the curve- shortening f l o w . The author would like to thank Katsumi Nomizu f o r suggesting the equation con- sidered here, and Alexander Polden f o r some f r u i t f u l discussions concerning the regularity o f solutions. 2. Background a n d notation 2.1. Support functions. Let $ be a closed, smooth, convex embedded curve in the plane, and Ω the region enclosed by $. Th e su p p o r t fu n c t i o n $ o f $ ( o r Ω) is d e f i n e d b y ( 2 ) $. The support function contains all geometric information about the curve. In particular, there is a natural embedding $ with image equal to $, defined by (3) $ for all $, where we think of S 1 as na t u r a l l y in c l u d e d in $, an d θ is the usual angle coordinate on S1.