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Writhe - Four Vertex Theorem.14 June, 2006 Writhe - Four Vertex Theorem.14 June, 2006 THE FOUR VERTEX THEOREM AND ITS CONVERSE in honor of Björn Dahlberg The Four Vertex Theorem says that a simple closed curve in the plane, other than a circle, must have at least four "vertices", that is, at least four points where the curvature has a local maximum or local minimum. There must be at least two local maxima, separated by two local minima. 1 The Converse to the Four Vertex Theorem says that any continuous real-valued function on the circle which has at least two local maxima and two local minima is the curvature function of a simple closed curve in the plane. Preassign curvature ..... then find the curve That is, given such a function κ defined on the circle S1 , 1 2 there is an embedding α: S → R whose image has 1 curvature κ(t) at the point α(t) for all t ∊ S . 2 History. In 1909, the Indian mathematician S. Mukhopadhyaya proved the Four Vertex Theorem for strictly convex curves in the plane. Syamadas Mukhopadhyaya (1866 - 1937) 3 In 1912, the German mathematician Adolf Kneser proved it for all simple closed curves in the plane, not just the strictly convex ones. Adolf Kneser (1862 - 1930) In 1971, we proved the converse for strictly positive preassigned curvature, as a special case of a general result about the existence of embeddings of Sn into Rn+1 with strictly positive preassigned Gaussian curvature. 4 In 1997, the Swedish mathematician Björn Dahlberg proved the full converse to the Four Vertex Theorem without the restriction that the curvature be strictly positive. Björn E. J. Dahlberg (1949 - 1998) Publication was delayed by Dahlberg's death in January 1998, but his paper was edited afterwards by his student Vilhelm Adolfsson in collaboration with Peter Kumlin, and appeared in 2005 in the Proceedings of the AMS. The work of Dahlberg completes the almost hundred year long thread of ideas begun by Mukhopadhyaya. 5 I. Why is the Four Vertex Theorem true? A counter-example would have a nonconstant curvature function with one max and one min, and be weakly monotonic on the two arcs between them. Let's try to build such a counter-example from a few arcs of circles. Start with the largest circle at the bottom, giving the minimum curvature there. Then go part way up to the right, switch to a smaller circle as shown, go up a little more and switch to a yet smaller circle, and go to its top. As we increase curvature by switching to smaller circles, their vertical diameters move to the right. Hence the top point lies to the right of the vertical diameter of the original circle. 6 When we try the same construction, but this time move up to the left from the bottom of the original circle, the top of the third circular arc will lie to the left of the vertical diameter of the original circle. Thus the convex curve that we are trying to build does not close up at the top...and so does not exist. 7 If we permit one self-intersection, then monotonicity of curvature is easy to realize. r = — 1 — 2 sin θ 8 Recommended reading. Robert Osserman "The Four-or-More Vertex Theorem" American Mathematical Monthly Vol. 92, No. 5 (May 1985), 332 - 337. 9 II. The Converse to the Four Vertex Theorem for strictly positive curvature. The Generalized Minkowski Problem. Does there exist an embedding of the n-sphere Sn into Euclidean space Rn+1 whose curvature has been preassigned as a strictly positive continuous function? In the early 1970's, we proved the following result. GENERALIZED MINKOWSKI THEOREM. Let K: Sn → R , for n ≥ 2 , be a continuous, strictly positive function. Then there exists an embedding α: Sn → Rn+1 onto a closed convex hypersurface whose Gaussian curvature at the point α(p) is K(p) for all n p ∊ S . When n = 1, the Four Vertex Theorem places a constraint on the curvature, so the above theorem says that by contrast, there are no constraints for n ≥ 2 . The proof of this theorem was by a degree argument in the diffeomorphism group of the n-sphere Sn . 10 The Converse to the Four Vertex Theorem. The proof of the Generalized Minkowski Theorem, when rewritten for n = 1 , shows that the required embedding α exists if and only if the curvature is either a nonzero constant or else has at least two local maxima and two local minima. This gives us the converse to the Four Vertex Theorem in the case of strictly positive curvature. The spirit of the argument is shown in the following picture. A snake searches unsuccessfully for its tail 11 What story does this picture tell? Each of the eight little figures shows a map of an interval into the plane, which begins at the origin, moves off in the positive x-direction, ends up pointing the same way, and is positively curved throughout. We imagine that the snake is following a concealed loop of instructions, from which the above pictures are samples. The error vector in each picture extends from the snake's tail to its head, and the pictures are arranged so that the one at location θ on the circle has an error vector pointing in the direction θ . The resulting loop of error vectors circles once around the origin. If the loop of instructions is contractible in the space of all instructions, then some instruction in that space will have a corresponding error vector which is zero, and hence will tell the snake how to close up. 12 A specific example of the snake dance. The most elementary curvature function which can illustrate these ideas is a step-function, and the simplest case among these leads to curves built from four circular arcs, two cut from a circle of one size, alternating with two cut from a circle of another size, as shown below. Bicircle Since our curvatures are strictly positive, we can express the preassigned curvature as a function of the angle of inclination ϕ of the outward pointing normal vector. This has the advantage of making total curvature 2π automatic. Given a curvature function κ(ϕ) , there will then be a unique map of the interval [0, 2π] into the plane which realizes this curvature function, subject to the initial condition that it begin at the origin, and start moving in the positive x-direction. 13 The figures on the left report the curvature κ as a step function of the angle ϕ with the heavier markings indicating the larger circle. The figures on the right show the resulting curve in the plane, built from arcs of these two circles. Preassign κ0(ϕ) Get this bicircle Preassign κ(ϕ) Get this curve 14 A two-curvature-snake searches unsuccessfully for its tail 15 Explanation of this picture. The inner group of eight circles reports the preassigned curvature step-functions κ(ϕ) , while the outer group shows the corrresponding curves. Each curvature step function κ(ϕ) is a distortion, via a diffeomorphism of the circle, of the step function κ0(ϕ) which corresponds to the bicircle. The eight diffeomorphisms are a sample from a loop of diffeomorphisms, which is easily seen to be contractible in the space of all diffeomorphisms of the circle. Each error vector is parallel to the position of the curve in the group of eight, and therefore the error vectors turn once around the origin as we go once around the loop of diffeomorphisms. It follows that there is some diffeomorphism whose corresponding error vector is zero, and so the corres- ponding curve closes up. Of course we knew this, since a bicircle closes up. But this argument is robust, and so will apply to curvature functions which are close to κ0(ϕ) . 16 Analogy with standard topological proof of the Fundamental Theorem of Algebra. This is analogous to proving that the polynomial n p0(z) = an z has a zero by noting that it takes any circle in the complex plane to a curve which winds n times around the origin, even though we already know that p0(z) has a zero at the origin. Again the point is that this argument is robust, and will apply just as well to the polynomial n n—1 p(z) = an z + an—1 z + ... + a1 z + a0 , once we pay attention to a suitably large circle. 17 Proving the converse to the Four Vertex Theorem for strictly positive curvature. Given a preassigned strictly positive curvature funtion κ: S1 → R having at least two relative maxima and two relative minima, we first agree to parametrize our proposed curve by the inverse of the Gauss map. Then we find a diffeomorphism h: S1 → S1 so that the curvature function κ ◦ h: S1 → R is ε-close in measure to the two-valued step function of some bicircle, meaning that the function κ ◦ h is within ε of this step function on almost all of S1 , and violates this only on a subset of measure less than ε . The robustness of the winding number argument for the two-valued step function then implies that, for sufficiently small ε > 0 , there is another diffeomorphism h1 of the circle so that the curve built with curvature function κ ◦ h ◦ h1 closes up smoothly. Since the curvature is positive, this closed curve is convex, and since its normal vector rotates by just 2π as we go once around, the curve is simple. Reparametrizing this curve, it realizes the preassigned curvature function κ .
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