The Four Vertex Theorem and Its Converse Dennis DeTurck, Herman Gluck, Daniel Pomerleano, and David Shea Vick
Dedicated to the memory of Björn Dahlberg.
he Four Vertex Theorem,oneoftheear- Why Is the Four Vertex Theorem True? liest results in global differential geome- A Simple Construction try, says that a simple closed curve in the Acounterexamplewouldbeasimpleclosedcurvein plane, other than a circle, must have at the plane whose curvature is nonconstant, has one least four “vertices”, that is, at least four minimum and one maximum, and is weakly mono- Tpoints where the curvature has a local maximum tonic on the two arcs between them. We will try to or local minimum. In 1909 Syamadas Mukhopad- build such a curve from a few arcs of circles, fail, hyaya proved this for strictly convex curves in the and see why. plane,andin1912AdolfKneserproveditforallsim- ple closed curves in the plane, not just the strictly convex ones. The Converse to the Four Vertex Theorem says that any continuous real-valued function on 3 the circle that has at least two local maxima and 2 two local minima is the curvature function of a 3' simple closed curve in the plane. In 1971 Herman Gluck proved this for strictly positive preassigned 2' curvature, and in 1997 Björn Dahlberg proved the full converse, without the restriction that the curva- ture be strictly positive. Publication was delayed by Dahlberg’s untimely death in January 1998, but his paper was edited afterwards by Vilhelm Adolfsson and Peter Kumlin, and finally appeared in 2005. The work of Dahlberg completes the almost 1' = 1 hundred-year-long thread of ideas begun by Mukhopadhyaya, and we take this opportunity to provide a self-contained exposition. Figure 1: A failed counterexample.
The authors of this article are all affiliated with the Uni- ThinkofthecirclesinFigure1astraintracks.Our versity of Pennsylvania. Dennis DeTurck is the Evan C. curve will be traced out by a train that travels along Thompson Professor in the department of mathemat- these circular tracks, switching from one to another ics. His email addresss is [email protected]. at the labeled junction points. Herman Gluck is professor of mathematics. His email address is [email protected]. Daniel Pomerleano Letthe trainstartat point1 on the bottom ofthe and David Shea Vick are graduate students in the largest circle, giving the minimum curvature there. department of mathematics. Their email addresses It then moves off to the right along this large circle, are [email protected] and dvick@math. comes to the point2, and switches there to a smaller upenn.edu. circle, increasing its curvature.
192 Notices of the AMS Volume 54, Number 2 The importantthing to note is thatthe verticaldi- Let E be a fixed nonempty compact set in the ameterofthissmallercircleisdisplacedtotheright plane. Among all circles C that enclose E, there is a of the verticaldiameterofthe originalcircle. unique smallest one, the circumscribed circle, as To keep things brief, we let the train stay on this in Figure3. second circle until it comes to the top at the point 3, which is to the right of the vertical diameter of the original circle. Then we go back to the beginning, start the train once again at the bottom of the original circle, re- label this point 1′, andnow letthe trainmoveoff to the left. Itswitches to a smaller circle atpoint2′,and comes to the top of this circle at 3′, which is now to theleftoftheverticaldiameteroftheoriginalcircle. Ifthecurvedpathofthe trainisto formaconvex simple closed curve, then the top points 3 and 3′ should coincide...but they don’t. So the curve does not exist. No matter how many circles we use, we get the Figure 3: The compact set E and its same contradiction: the curve cannotclose up while circumscribed circle. staying simple, and therefore does not exist. But if we permita self-intersection,then it’seasy What can C ∩ E look like? to get just one maximum and one minimum for the (1) C must meet E, otherwise C could be made curvature. smaller and still enclose E. (2) C ∩ E cannot lie in an open semi-circle. Oth- −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 erwise we could shift C a bit, withoutchang- 0.0 ing its size, so that the new C still encloses E
−0.5 but does not meet it. But this would contra- dict (1).
−1.0 (3) Thus C ∩ E has at least two points, and if only these, then they must be antipodal. −1.5 This happens when E is anellipse.
−2.0 An Overview of Osserman’s Proof Osserman’s Theorem. Let α be a simple closed −2.5 curve of class C2 in the plane and C the circum- scribed circle. If α ∩ C has at least n components, −3.0 then α has at least 2n vertices. Herearethemajorstepsinthe proof. Figure 2: A curve with just two vertices. (1) Let R be the radius of the circumscribed circle C, and K = 1/R its curvature. If α ∩ C has at least n components, then α has at least n vertices where Figure 2 shows a curve whose equation in polar the curvature satisfies κ ≥ K, and at least n more coordinates is r = 1 − 2sin θ. The curvature takes vertices where κ February 2007 Notices of the AMS 193 κ ≥ K C' C α P2 κ < K C κ < K Q1 κ ≥ K κ < K P κ ≥ K 3 Figure 4: The curve α and its circumscribed circle. P1 Now focus on the arc C1 of C from P1 to P2, and on ′ the corresponding arc α1 of the curve α.Wehavear- Figure 6: C translated to the left. ranged for the convenience of description that the line connecting P1 and P2 is vertical. The arc α1 of α is tangent to C at P1 and P2 and (2) Bonus clause. The components of α ∩ C are does not lie entirely along C. It therefore contains either single points or closed arcs. For each com- a point Q that is not on C1, yet lies to the right of ponent that is an arc rather than a single point, we the vertical line connecting P1 and P2. Consider the can get two extra vertices, over and above the 2n ′ circle C through P1, Q and P2,asshowninFigure5. vertices promisedby Osserman’s theorem. The picture of α in Figure 7 is roughly the same as before, but we have distorted it so that an entire C' arc of α about the point P2 coincides with an arc of the circle C. The curvature of α at P2 is now exactly P2 κ. But in addition, there must be points of α to ei- ther sideofthis arc,arbitrarilyclose to it, wherethe C α curvature κ of α is > K. Two such points are shown Q ′ in Figure 7 and marked R2 and R2. As a result, the point P2, where the curvature of α used to be larger than the curvature at the two neighboring marked α C 1 1 points, has been demoted, and the curvature of α at P2 is now less than that at the two neighboring markedpoints.Inthisway,wehavegainedtwoextra P3 vertices for α. P1 Figure 5: The circle C′. Since Q is in the interior of C, and since C1 is no larger than a semicircle, it follows that the circle C′ is larger than C. Hence its curvature K′ 194 Notices of the AMS Volume 54, Number 2 2 (3) It can happen that α ∩ C has only one compo- αh : [0, 2π] → R whose curvature at the point nent, which must then be a closed arc on C greater αh(θ) is κ ◦ h(θ), the corresponding error vectors than or equal to a semi-circle. Osserman’s theorem E(h) willwind once around the origin. Furthermore, promises that α has at least two vertices, and the we will do this so that the loop is contractible in bonus clause promisestwo more. the group Diff(S1) of diffeomorphisms of the circle (4) Thus Osserman’s theorem, together with the and conclude that for some h “inside” this loop, the bonus clause, yields the Four Vertex Theorem. curve αh will have error vector E(h) = 0 and hence close upto formasmoothsimpleclosed curve. The Converse in the Case of Strictly Posi- Its curvature at the point αh(θ) is κ ◦ h(θ), soif −1 tive Curvature we write αh(θ) = αh ◦ h ◦ h(θ), and let h(θ) = t, −1 Full Converse to the Four Vertex Theorem. Let then its curvature at the point αh ◦ h (t) is κ(t). −1 κ : S1 → R be a continuous function that is either Therefore α = αh ◦ h is a reparametrization of a nonzero constant or else has at least two local the same curve, whose curvature at the point α(t) is maxima and two local minima. Then there is an κ(t). embedding α : S1 → R2 whose curvature at the point α(t) is κ(t) for all t ∈ S1. How Do We Find Such a Loop of Diffeomor- phisms? Basic Idea of the Proof for Strictly Positive The first step is to replace κ by a simpler curvature Curvature function. We use a winding number argument in the group of Using the hypothesis that κ has at least two lo- diffeomorphismsof the circle, as follows. cal maxima and two local minima, we find positive Let κ : S1 → R be any continuous strictly positive numbers 0 ε/4 Figure 9: κ ◦ h1 is ǫ-close in measure to the α(2π) step function κ0. α(θ) θ If the curvature functions κ ◦ h and κ are E 1 0 ǫ-close in measure, then the corresponding curves α(0) are easily seento be C1-close. With this in mind, we will focus attention on the curvature step function Figure 8: The error vector E. κ0 and ignore the originalcurvature function κ until almost the end of the argument. Given this shift of attention, we next find a con- If the curvature function κ has at least two lo- tractible loop of diffeomorphisms h of the circle so cal maxima and two local minima, we will show that the error vectors E(h) for the curvature step how to find a loop of diffeomorphisms h of the functions κ0 ◦ h wind once around the origin, as in circle so that when we construct, as above, curves Figure 10. February 2007 Notices of the AMS 195 Figure 10: A curve tries unsuccessfully to close up. How Do We Read This Picture? The eight diffeomorphisms h aresamplesfroma The picture has eight portions, located at the eight loop of diffeomorphisms of the circle, and the cor- points of the compass. Each contains a small circle responding error vectors E(h) wind once around the origin. The loop is contractible in Diff(S1) that shows a curvature step function κ ◦ h, and a 0 because each diffeomorphism leaves the bottom larger curve that realizes it, with common param- point of the circle fixed. The precise description of eter the angle of inclination θ. This larger curve is such a family of diffeomorphisms of the circle is built from four circular arcs, two cut from a circle given in Gluck (1971), but we think there is enough of curvature a and two from a circle of curvature b. information in Figure 10 for the reader to work out Heaviermarkingsindicate the largercircle. the details. This picture can be seen in animated The curves fail to close up, and we show their form at our website, http://www.math.upenn. error vectors. The figures are arranged so that the edu/∼deturck/fourvertex. oneintheeasthasanerrorvectorpointingeast,the Sincetheloopofdiffeomorphismsiscontractible one in the northeast has an error vector pointing in Diff(S1), it follows that for some other diffeo- northeast, and so on. morphism h of the circle, the corresponding curve 196 Notices of the AMS Volume 54, Number 2 n with curvature κ0 ◦ h has error vector zero and S . The hypothesis that n ≥ 2 is used to guaran- therefore closes up. This is hardly news: if h is the tee the existenceof a diffeomorphism that arbitrar- identity, then the curve with curvature κ0 closes up, ily permutes any finite number of points. This hy- asshowninFigure11. pothesis fails for n = 1, but when we rewrite the argument in this case, we get the winding number proof of the converse to the Four Vertex Theorem for strictly positive curvature describedabove. Building Curves From Arcs of Circles Before we leave this section, we record the following observationof Dahlberg,andsaveit forlater use. Proposition 1. A plane curve, built with four arcs cut in alternation from two different size circles, and assembled so that the tangent line turns con- Figure 11: Preassign κ0 . . . and get this tinuously through an angle of 2π, will close up if “bicircle”. and only if opposite arcs are equal in length. Evidence for this proposition is provided by the figure of a bicircle, and the large figure containing But the point is that the above argument is ro- eight separatecurvesthat fail tocloseup. bust and hence applies equally well to the curvature Proof. We parametrize such a curve by the angle of function κ ◦ h which is ǫ-close in measure to κ , 1 0 inclination of its tangent line, suppose that 0 < a < since the corresponding curves are C1-close. Hence b arethecurvaturesofthetwocirclestobeused,and there mustalso be a diffeomorphism h of the circle, record the curvature function κ(θ) in Figure 12. so that the curve with curvature κ ◦ h1 ◦ h has error vectorzeroandthereforeclosesuptoformastrictly convex simple closed curve. Reparametrizing this θ3 κ = b curve as discussed earlier shows that it realizes the θ preassigned curvature function κ, completing the 2 proof of the converse to the Four Vertex Theorem for strictly positive curvature. κ = a κ = a Why Does This Argument Sound Familiar? Provingsomethingthatisobviousbyusingarobust θ argument reminds us of the proof that the polyno- 1 n mial p0(z) = anz has a zero bynoting that it takes any circle in the complex plane to a curve that winds θ4 n timesaroundtheorigin.Afterall,itisobviousthat κ = b p0(z) hasa zeroatthe origin. But this argument is also robust and applies Figure 12: The curvature function κ(θ). equally well to the polynomial p(z) = a zn + a zn−1 +···+ a z + a , n n−1 1 0 If α(θ) is the resulting curve, thenits error vector when z moves around a suitably large circle, giving isgivenby us the usual topological proof of the Fundamental 2π iθ Theoremof Algebra. E = α(2π) − α(0) = e ds = Z0 2π ds 2π eiθ Where Does Our Winding Number Argument eiθ dθ = dθ. Come From? Z0 dθ Z0 κ(θ) Here is a higher-dimensionalresultproved inGluck We evaluate this explicitly as a sum of four inte- (1972) and surveyed in Gluck (1975). grals, andget n Generalized Minkowski Theorem. Let K : S → R, E = (1/ia − 1/ib)([ǫiθ2 − ǫiθ1 ]+ for n ≥ 2, be a continuous, strictly positive func- [ǫiθ4 − ǫiθ3 ]). tion. Then there exists an embedding α : Sn → Rn+1 onto a closed convex hypersurface whose Gaussian We are left with the question of whether the two curvature at the point α(p) is K(p) for all p ∈ Sn. vectorsshown in Figure13 addup to zero.A glance The proof of this theorem is by a degree argu- at the figure makes the answer obvious and proves ment in the diffeomorphism group of the n-sphere the proposition. February 2007 Notices of the AMS 197 i i e 2 which has an almost constant tangent direction, e 3 but largely varying curvature, including negative curvature. This curve “marginalizes” its negative curvature and emphasizes its positive curvature, andfromadistancelookslike abicircle. Dahlberg’s Key Idea. You can construct such a curve with any preassigned curvature that has at least two local maxima and two local minima. You can use the winding number argument to get i e 1 it to close up smoothly, and you can also make it C1-close to a fixed convex curve, which will imply that it is simple. i e 4 Dahlberg’s Proof Plan Figure 13: When do these two vectors add up The overall plan is to follow the winding number to zero? argument from the strictly positive curvature case, preservingtheprominentroleplayedbytwo-valued curvature step functions, but mindful of the need Dahlberg’s Proof of the Full Converse to for the following changes. the Four Vertex Theorem (1) A smooth simple closed curve with strictly Dahlberg’s Key Idea positive curvature has a distinguished parametrization by the angle of inclina- When asked to draw a simple closed curve in the tion of its tangent line. This does not work plane with strictly positive curvature, and another in general, so another parametrization one with mixed positive and negative curvature, a schemeis needed. typicalresponsemightbeasshown inFigure14. (2) Asmoothclosedcurvewithstrictlypositive curvature whose tangentline turns through anangleof2π is automaticallysimple.This is false without the curvature assumption, and part of Dahlberg’s Key Idea is to force the curve to be C1 close to a fixed convex curveinorderto makeitsimple. To accomplish this, he exhibits a 2-cell D in Diff(S1), centered at the identity, and satisfying a certain transversality condition that guarantees that the winding number argument will work using arbitrarilysmall loops in D about its center. Figure 14: Typical Curves. Finding the Right Parametrization Choose, as common domain for all our curves, the But for mixed positive and negative curvature, unit circle S1 with arc length s as parameter. Since Dahlberg envisioned the curve shown in Figure 15. most of the curves will fail to close up, cut the cir- cle open at the point (1, 0) and think of the inter- val [0, 2π] as the domain. All the curves will have length 2π, and at the end will be scaled up or down to modify their curvature. We make sure the total curvature is always 2π as follows. Given a preassigned curvature function κ : S1 → R which is not identically zero, evaluate 2π 0 κ(s) ds. If this is zero, precede κ by a prelimi- naryR diffeomorphism of S1 so as to make the inte- gral nonzero, but still call the composition κ. Then rescale this new κ by a constant c so that Figure 15: Dahlberg’s Vision. 2π c κ(s) ds = 2π. Z0 Its four major subarcs are almost circular. They When we later modify this curvature function by are connected by four small wiggly arcs, each of anotherdiffeomorphismh : S1 → S1,wewillrescale 198 Notices of the AMS Volume 54, Number 2 the new curvature function κ ◦ h by a constant ch so p 2 p that the total curvature is again 2π: 1 2π ch κ ◦ h(s) ds = 2π. Z0 2 We then build a curve αh : [0, 2π] → R ,parame- trized by arc length, that begins at the origin, αh(0) = (0, 0), starts off in the direction of the ′ p3 positive x-axis, αh(0) = (1, 0), and whose curva- ture at the point αh(s) is ch κ ◦ h(s). The curve αh is uniquely determined by these requirements. The effect of these arrangements is that all our curves begin and end up pointing horizontally to the right. If any such curve closes up, it does so p4 smoothly. Thus, just as in the positive curvature case, the emphasis is on getting the curve to close Figure 16: An element of the configuration up. space CS. At the end, using ideas expressed above, we will makesurethe finalcurveissimple. We noted in Proposition 1 that a curve built with Configuration Space four arcs cut in alternation from two different size Defining a configuration space of four ordered circles will close up if and only if opposite arcs are points on a circle will help us visualize what is to equal in length. come and aid us in carrying out the transversality Thus a point (p1, p2, p3, p4) of CS lies in the core arguments mentioned in the proof plan. if and only if p1 and p3 are antipodal, and also p2 As in the positive curvature case, we use the hy- and p4 are antipodal. A nice exercise for the reader pothesis that κ has at least two local maxima and is to check thatthis conditionholds if and only if the two local minima to find positive numbers 0 < a < equation p1 − p2 + p3 − p4 = 0 holds in the complex b so that κ takes the values a,b,a,b at four points plane. 1 R1 in order around the circle. The only difference here The core CS0 is diffeomorphic to S × . is that we might first have to change the sign of κ to accomplish this. Reduced Configuration Space Then we find a preliminary diffeomorphism h1 To aid in visualization, and also to help with our of the circle so that the function κ ◦ h1 is ǫ-close proof of the required transversality results, we de- in measure to the step function κ0 with values fine the reduced configuration space RCS ⊂ CS to a,b,a,b on the arcs [0, π/2), [π/2, π), [π, 3π/2), be the subsetwhere p1 = (1, 0) = 1 + 0 i = 1. Then [3π/2, 2π). RCS is diffeomorphic to R3 and we use the group 1 As before, we focus on the step function κ0 and structure on S to express the diffeomorphism 1 1 its compositions κ0 ◦ h as h ranges over Diff(S ). S × RCS → CS by These are all step functions with the values a,b,a,b (eiθ , (1,p,q,r)) → (eiθ ,eiθ p,eiθ q,eiθ r). on the four arcs determined by some four points 1 For the purpose of drawing pictures, we change p1, p2, p3, p4 in order on S . For each such step function, we follow Dahlberg’s parametrization coordinates by writing scheme from the previous section to construct a p = e2πix, q = e2πiy and r = e2πiz . curve from circular arcs with curvatures propor- Then RCS ≅{(x, y, z) : 0 February 2007 Notices of the AMS 199 z (0,0,1) (0,1,1) (1,1,1) y (0,0,0) x Figure 19: A loop in the reduced configuration space. Figure 17: The reduced configuration space RCS appears as an open solid tetrahedron. the winding number phenomenon that makes the proof of the converse to the Four Vertex Theorem z (0,1,1) (0,0,1) actually work. (1/2,1/2,1) Proposition 2 is an immediate consequence of the transversality result below. (1,1,1) Proposition 3. The differential of the error map E : RCS → R2 is surjective at each point of the (0,1/2,1/2) core. Proof. We are given distinct positive numbers 0 Figure 18: The core of the reduced a(P)L1 + b(P)L2 + a(P)L3 + b(P)L4 = 2π. configuration space. Then construct a curve α : [0, 2π] → R2 accord- ing to Dahlberg’s plan: begin at the origin and head in the direction of the positive x-axis along circles division points for an element (1,p,q,r) in RCS. of curvatures a(P), b(P), a(P), b(P) for lengths L1, This simplifies the construction of the curve, and L2, L3, L4. The error vector E(P) = α(2π) − α(0) the computationofits error vector E(1,p,q,r),and indicates the failure to close up. we make use of this in the following section. Consider Figure 20, whichshows the points 1, p2 , p , p on the unit circle parametrized by arc length The Topology of the Error Map 3 4 s,andthecorrespondingdivisionpoints1, q2, q3, q4 R2 The error map E : CS → takes the core CS0 to on the unit circle parametrized by angle of inclina- the origin and the complement of the core to the tion θ. complement of the origin. Let λ be a loop in CS − CS0 which is null- p2 homologous in CS but links the core CS0 once. For b(P)L q2 L 2 convenience, we go down one dimension and show 2 L1 a(P)L1 this loop in RCS − RCS0. p q Any two such loops in CS −CS0 arehomotopicto 3 3 one another, up to sign. 1 1 Proposition 2. The image of the loop λ under the a(P)L L 3 ± 4 error map E has winding number 1 about the L3 b(P)L4 R2 q4 origin in . p4 What this proposition says in effect is that plane curves, built with four arcs cut in alternation from Figure 20: Arclength s, angle of inclination θ. two different size circles, are capable of exhibiting 200 Notices of the AMS Volume 54, Number 2 This momentary switch in loyalty from the arc length parameter s to the angle of inclination pa- β/|β| rameter θ will be rewarded by an explicit formula for the error map that makes the proof of the cur- β rent proposition easy. Themapthat takes 0 P = (1, p2, p3, p4) → Q = (1, q2, q3, q4) −β is a diffeomorphism from the s-parametrized ver- sionof RCS to a θ-parametrized version, taking the core1−p2+p3−p4 = 0tothecore1−q2 +q3−q4 = 0. Thisalternative versionof RCS helps us to calcu- late theerrormap.Firstnote that −β/|β| 2π = iθ(s) = E(P) e ds Figure 21: The action of gβ on the unit disk in Z0 the complex plane. 2π ds 2π eiθ eiθ dθ = dθ. Z0 dθ Z0 κ(θ) This can be computed explicitly as a sum of four thediskbyangle θ about its center. These rotations integrals, and we get interact with Dahlberg’s transformations gβ by the E(P) = (1/ib(P) − 1/ia(P))(1 − q2 + q3 − q4). intertwining formula iθ iθ Thedifferentialofthismapisparticularlyeasyto geiθ β (e z) = e gβ(z). compute along the core, since the vanishing of the expression1 − q2 + q3 − q4 = 0there freesus from The Second Transversality Result the need to consider the rate of change of the factor Let gβ be a point in Dahlberg’s disk D and P = (1/ib(P) − 1/ia(P)). (p1, p2, p3, p4) a point in the configuration space If we move q2 counterclockwise at unit speed CS. We define along the circle, we have dq2/dt = iq2, and hence gβ(P) = (gβ(p1), gβ(p2), gβ(p3), gβ(p4)), dE/dq2 = (1/ib(P) − 1/ia(P))(−iq2). so that now D acts on CS. Likewise, Proposition 4. The evaluation map CS × D → CS dE/dq3 = (1/ib(P) − 1/ia(P))(iq3). 0 defined by (P, g ) → g (P) is a diffeomorphism. These two rates of change are independent be- β β D cause q2 and q3 are always distinct, and along the Proof. Theevaluationmap CS0 × → CS issmooth. coretheycannotbeantipodal.Itfollowsthatthedif- To visualize it, startwith apoint P = (p1, p2, p3, p4) ferentialof the errormap issurjectiveateachpoint in the core CS0 and a transformation gβ in D which of the core of the θ-parametrized version of RCS. takes P to Q = (q1, q2, q3, q4). In the Poincaré disk, Since the map between the two versions of RCS is the geodesicwith ends at p1 and p3 is a straightline a diffeomorphism taking core to core, the same re- through the origin, and likewise for the geodesic sult holds for our original s-parametrized version with ends at p2 and p4. The isometry gβ takes them of RCS, completing the proofof the proposition. to geodesics with ends at q1 and q3, respectively q2 and q4, which appear to the Euclidean eye as circu- Dahlberg’s Disk D lar arcs meeting the unit circle orthogonally and Dahlberg’s choice of 2-cell D ⊂ Diff(S1) consists of intersecting one another at the point gβ(0) = −β, the special Möbius transformations asshowninFigure22. ¯ Vice versa, suppose we start with the point gβ(z) = (z − β)/(1 − βz), Q = (q1, q2, q3, q4) in CS. Draw the hyperbolic lines, where |β| < 1and β¯is the complex conjugate of β. that is, circles orthogonal to the boundary of the These special Möbius transformations are all disk, with ends at q1 and q3, likewise q2 and q4. The isometries of the Poincaré disk model of the hyper- intersection point of these two circular arcs varies bolic plane. The transformation g0 is the identity, smoothly with the four end points. and if β ≠ 0, then gβ is a hyperbolic translation Call this intersection point −β. The transforma- along the line through 0 and β that takes β to 0 and tion g−β takes the point −β to the origin, and hence 0 to −β. The point β/|β| and its antipode −β/|β| on takes the two hyperbolic lines through −β to lines 1 S (the circle at infinity) are the only fixed points of through the origin. It follows that P = g−β(Q) lies gβ. in the core CS0 and that gβ(P) = Q. The associa- iθ A unitcomplex number e acts by multiplication tion Q → (P, gβ) provides a smooth inverse to the on the Poincaré disk modelof the hyperbolic plane, evaluation map CS0 × D → CS, showing it to be a and as such is a hyperbolic isometry that rotates diffeomorphism. February 2007 Notices of the AMS 201 Corollary 5. For each fixed point P in the core CS0, the six edgesof the tetrahedron. This is a reflection the evaluation map gβ → gβ(P) is a smooth em- of the fact that as β approaches the boundary of bedding of Dahlberg’s disk D into CS that meets the unit disk, the special Möbius transformation the core transversally at the point P and nowhere gβ leaves the point β/|β| on the boundary circle else. fixed, but moves all other points on the boundary circle towards the antipodal point −β/|β|, which This follows directly from Proposition4. −1 is also fixed. Hence the four points 1, gβ(1) gβ(i), −1 −1 The Image of Dahlberg’s Disk in the Reduced gβ(1) gβ(−1), gβ(1) gβ(−i) converge to at most Configuration Space two points. This “double degeneracy” corresponds toedges,ratherthan faces,of the tetrahedron. We seek a concrete picture of Dahlberg’s disk D in action. Recall from the preceding section that for Dahlberg’s Proof any point P = (p1, p2, p3, p4) in CS, the correspon- dence We are given a continuous curvature function κ : S1 → R that has at least two local maxima and g → (g (p ), g (p ), g (p ), g (p )) β β 1 β 2 β 3 β 4 two local minima and must find an embedding 1 2 maps D into CS. If we start with the point P0 = α : S → R whose curvature at the point α(t) is (1,i, −1, −i) and then follow the above map by the κ(t) for all t ∈ S1. projectionof CS to RCS,wegetthecorrespondence Step 1. We temporarily replace κ by a curvature −1 −1 gβ → (1, gβ(1) gβ(i), gβ(1) gβ(−1), step function κ0, to which we apply the −1 winding number argument. gβ(1) gβ(−i)). Changingthesignofκ ifnecessary,therearecon- stants 0 embedding of D into CS which meets the core CS0 transversally at the point P0 and nowhere else. By Proposition 2, each loop |β| = constant in D is sent by the composition evaluation map error map D CS R2 0 0 1 1 R2 1 1 into a loop in −{origin} with winding number ±1 about the origin. Figure 23: Two views of Dahlberg’s disk D We translate this conclusion into more concrete mapped into the tetrahedron. terms as follows. Let c(β) κ0 ◦ gβ be the rescaling of the curvature step function κ ◦ g that has total curvature 2π, If we convert to xyz-coordinates so that RCS is 0 β and let α(β) : [0, 2π] → R2 be the corresponding the open solid tetrahedron previously introduced, arc-length parametrized curve with this curvature then the image of D is shown in Figure 23. In the pic- ture on the left, the tetrahedron has its usual posi- function. As β circles once around the origin, the tion;ontheright,itisrotatedsoastoclearlydisplay corresponding loop of error vectors E(α(β)) has the negativecurvature ofthe imagedisk. winding number ±1 aboutthe origin. These computer-drawn pictures of the image of Step 2. We transfer this winding number argu- D clearly show that its boundary lies along four of ment to the curvature function κ. 202 Notices of the AMS Volume 54, Number 2 Let c(h1,β)κ ◦ h1 ◦ gβ be the rescaling of the Ulrich Pinkall conjectured that if a curve bounds curvature function κ ◦ h1 ◦ gβ which has total cur- an immersed surface of genus g in the plane, then 2 vature 2π, and let α(h1, β) : [0, 2π] → R be the it must have 4g + 2 vertices. This was disproved corresponding arc-length parametrized curve with by Cairns, Ozdemir, and Tjaden (1992), who con- this curvature function. structed for any genus g ≥ 1 a planar curve with Fixing |β|, we can choose ǫ sufficiently small so only 6 vertices bounding an immersed surface of 1 that each curve α(h1, β) is C -close to the curve genus g. Later, Umehara (1994) showed that this re- α(β) constructed in Step 1. Then as β circles once sult is optimal. around the origin, the corresponding loop of error Nonsimple curves. Pinkall(1987) showed thatif vectors E(α(h1, β)) will also have winding number a closed curve bounds an immersed surface in the ±1aboutthe origin. Note thatwe only need C0-close plane, then it must have at least four vertices. He for this step. also pointed out that there exist closed curves of Itfollows thatthere is a diffeomorphism gβ′ with any rotation index that have only two vertices. For ′ ′ |β |≤|β| so that E(α(h1,β )) = 0, which tells us a study of such curves,seeKobayashiand Umehara ′ that the curve α(h1,β ) closes up smoothly. If |β| (1996). and ǫ are sufficiently small, then the closed curve The generalizations of Jackson and Pinkall may ′ 1 α(h1,β ) willbeas C -close as we like to the fixed bi- be combined to show that any closed curve that circle with curvature c0 κ0,andwillhencebesimple. bounds an immerseddisk on a complete orientable ′ The simple closed curve α(h1,β ) realizes the surface of constant curvature has at least four ′ curvature function c(h1,β ) κ ◦ h1 ◦ gβ′ . Rescal- vertices. See Maeda (1998) and Costa and Firer ing it realizes the curvature function κ ◦ h1 ◦ gβ′ , (2000). and then reparametrizing it realizes the curvature Space curves. In his (1912) paper, Adolf Kneser function κ. observed that stereographic projection maps the ThiscompletesDahlberg’sproofoftheConverse vertices of a plane curve to vertices (with respectto to the FourVertex Theorem. geodesic curvature) of its image on the sphere and used this to prove the four vertex theorem for all Extensions and Generalizations of the simple closed curves in the plane, not just the con- Four Vertex Theorem vex ones. Kneser’s observation is a special case of During its almost hundred- year life span, the four the more general phenomenon that any conformal vertex theorem has been extended and generalized transformation of a surface preserves vertices of inmanydifferentways:wegivethereaderaglimpse curveson it. ofthishere,mentionsomeoftheassociatedpapers, On a sphere, the points where the geodesic cur- and makea few suggestionsforfurther reading. vature vanishes are points of vanishing torsion. More general surfaces. Hans Mohrmann (1917) Thus Jackson’s spherical four vertex theorem im- stated that any simple closed curve on a closed plies that every simple closed curve on the sphere convex surface has at least four vertices, meaning hasatleastfourpointsofvanishingtorsion.Sedykh local maxima or minima of the geodesic curvature; (1994) showed that this result holds more generally the full proof was given by Barnerand Flohr (1958). for all spacecurvesthat lieon the boundaryoftheir Jackson (1945) proved a four vertex theorem for convex hull, thus proving a conjecture of Scherk. simple closed curves that are null-homotopic on The full generalization of this result, namely that surfaces of constant curvature in Euclidean space. every simple closed curve bounding a surface of See also Thorbergsson(1976). positive curvature has at least four points where Anewproofofthefourvertextheoremforclosed the torsion vanishes, is still unknown. convex curves in the hyperbolic plane was given Many more results connected with the four ver- by David Singer (2001). He associated with such a tex theorem can be found in the book by Ovsienko curve γ a diffeomorphism f of the circle, proved and Tabachnikov (2004) and in the two papers that the vertices of γ correspond to the zeros of Umehara (1999) and Thorbergsson and Umehara the Schwarzian derivative S(f ), and then applied a (1999). Theirbibliographies are excellentresources theorem of Ghys (1995), which says that S(f ) must and will give the reader a sense of the breadth of vanishin atleastfour distinct points. research spawned by Mukhopadhyaya’s original More than four vertices. Wilhelm Blaschke contribution. (1916) proved that a convex simple closed curve in the plane that crosses a circle 2n times has at least History 2n vertices. S. B. Jackson (1944) generalized this Syamadas Mukhopadhyaya was born on June 22, to nonconvex curves; see also Haupt and Künneth 1866, at Haripal in the district of Hooghly. After (1967). Osserman focused on the circumscribed graduating from Hooghly College, he took his M.A. circle to get a 2n-vertex theorem for simple closed degree from the Presidency College, Calcutta, and curves, convex or not, and we summarized his was later awarded the first Doctor of Philosophy in (1985) expositionatthe beginningof this paper. Mathematics from the Calcutta University, with a February 2007 Notices of the AMS 203 thesis titled, Parametric Coefficients in the Differ- pupils will remember him not only ential Geometry of Curves in an N-space, which as an able Professor and a success- earnedfor its author the Griffith MemorialPrize for ful researcher with a deep insight 1910. into the fundamental principles of synthetic geometry but also as a sincere guide, philosopher, and friend, ever ready to help the poor in distress. Mukhopadhyaya’s obituary appeared in the Bul- letin of the Calcutta Mathematical Society, Vol. 29 (1937), pages 115–120. The comments above were taken from this obituary, and the final sentence was copied verbatim. Adolf Kneser was born on March 19, 1862, in Grüssow, Germany. He received his Doctor of Phi- losophy from the University of Berlin in 1884, with a thesis Irreduktibilität und Monodromiegruppe algebraischer Gleichungen, written under the di- rection of Leopold Kronecker and Ernst Kummer and influenced by Karl Weierstrass. Kneser was appointed to the chair of mathematics in Dorpat and then later to the University of Breslau, where he spentthe restofhis career. Figure 24: Syamadas Mukhopadhyaya (1866–1937). Mukhopadhyaya began work as a professor in the Bangabasi College and some years later moved to the Bethune College, both in Calcutta. There he lectured not only on mathematics, but also on philosophy and English literature. He was later transferred to the Presidency College as professor of mathematics. When the Calcutta Mathematical Society was started in 1909, he was a founding member and in 1917 became vice president of the Society. All of Mukhopadhyaya’s mathematical works had a strong geometric flavor. He wrote many papers on the geometry of curves in the plane, and his proof of the Four Vertex Theorem in the case of strictly positive curvature came at the very beginning of his research career. His work on the geometry of curves in n-space, the early parts of which were submitted for his doctoral thesis, con- tinued afterwards and was published in six parts in Figure 25: Adolf Kneser (1862–1930). the Bulletin of the Calcutta Mathematical Society overaperiodof years. After his retirement in 1932, Mukhopadhyaya After his initial interest in algebraic functions went to Europe to study methods of education and and equations, Kneser turned his attention to the on returning to India wrote a series of memoirs geometry of space curves and in 1912 proved the about this. He was elected President of the Calcutta Four Vertex Theorem in the general case of mixed Mathematical Society and served in this capacity positive and negative curvatures. until his death from heart failure on May 8, 1937. Adolf’s son Hellmuth, born in 1898 while the His death has removed from the family was still in Dorpat, followed in his father’s world of science a man of out- mathematicalfootsteps andobtainedhis doctorate standing genius; but the intimate fromGöttingenin1921underthedirectionofDavid circle of his friends, admirers and Hilbert, with a thesis titled, Untersuchungen zur 204 Notices of the AMS Volume 54, Number 2 Quantentheorie. The next year, in 1922, Hellmuth at ETH (Zurich), Yale, Caltech, Chicago, and South published a new proof of the Four Vertex Theorem. Carolina. Hellmuth helped Wilhelm Süss to found the Mathe- While in Göteborg, Dahlberg started a project matical Research Institute at Oberwolfach in 1944 with Volvo on computer-aided geometric design and provided crucialsupport over the yearsto help of the surfaces of a car, leading to new theoret- maintain this world famous institution. ical results and a new design system. In his last Hellmuth’s son Martin continued in the family years, Dahlberg worked on questions of discrete business, receiving his Ph.D. from the Humboldt geometry connected to his work with Volvo and University in Berlin in 1950 under the direction of other companies, as well as to his older interest in Erhard Schmidt, with a thesis titled, Über den Rand non-smooth domains. von Parallelkörpern. He spent the early part of his mathematical career in Munich and then moved to Göttingen. Adolf died on January 24, 1930, in Breslau, Ger- many (now Wroclaw, Poland). Hellmuth died on Au- gust 23, 1973, in Tübingen, Germany. Martin died on February 16, 2004. The comments above were taken from the on- line mathematical biographies at the University of St. Andrews in Scotland, http://www.history. mcs.st-and.ac.uk. Björn E. J. Dahlberg was born on Novem- ber 3, 1949, in Sunne, Sweden. He received his Ph.D. in mathematics from Göteborg University in 1971 with a thesis, Growth properties of sub- harmonic functions and with Tord Ganelius as advisor. Dahlberg was 21 at the time, the youngest mathematics Ph.D. in Sweden. Dahlberg’s mathematical interests were shaped during his postdoctoral fellowship year at the Mittag-LefflerInstituteinStockholm,whichLennart Figure 26: Björn E. J. Dahlberg (1949–1998). Carleson had recently revitalized. Dahlberg was deeply influenced by Carleson and participated in the programs of harmonic analysis and quasi- Dahlberg died suddenly on January 30, 1998, conformal mappings at the Institute, while learning from meningitis, an inflammation of the lining that about partial differential equations. protects the brainandspinal cord. One issue of great importance to the mathemat- The comments above were taken mainly from ical community at that time was to extend known a letter written by Vilhelm Adolfsson, one of results in these areas to domains with complicated Dahlberg’s Ph.D. students who, along with Peter boundaries. In 1977 Dahlberg showed that, for Kumlin, found Dahlberg’s manuscript on the Con- bounded Lipschitz domains, surface measure on verse to the Four Vertex Theorem after his death, the boundary and harmonic measure are mutually edited it, and submitted it for publication. Ville absolutely continuous, which implies the solvabil- went onto addsomepersonalnotes. ity of the Dirichlet problem for the Laplacian on Björn was a very inspiring and such domains. charismatic person. Where other For this work, Dahlberg was awarded the Salem mathematicians tended to see Prize (Paris) in harmonic analysis in 1978 and the difficulties, Björn saw possibili- Edlund’s prize from the Royal Academy of Science ties: nothing was impossible! I in Stockholm in 1979, and was invited to give a remember leaving his office after 45-minute address at the International Congress of discussing current problems, I al- Mathematicians in Warsaw, Poland, in 1982. ways felt encouraged and full of In 1980 Dahlberg was appointed to a full pro- enthusiasm. He often cheerfully fessorship at Uppsala University and three years exclaimed: The sky is the limit! later to a full professorship at Göteborg, which One might then think that he was remained his home base for the rest of his life. He overly optimistic, but he was well traveled a lot and held visiting faculty positions at aware of the difficulties and the the Universities of Minnesota, Michigan, and Texas, uncertainties. One of his favorite at Purdue University and Washington University sayings was “Hindsight is the only in St. Louis, at the Université Paris-Sud, as well as certain science.” February 2007 Notices of the AMS 205 Acknowledgments 1933 Otto Haupt, Zur TheoriederOrdnung reeler The entire mathematical community is indebted to Kurven in der Ebene bezüglich vorgegebener Vilhelm Adolfsson and Peter Kumlin for rescuing Kurvenscharen, Monatsh. Math. Phys. 40, 1–53. Dahlberg’s proof of the Full Converse to the Four 1934 T. Takasu, Vierscheitelsatz für Raumkurven, Vertex Theorem, reading and editing it, and shep- Tohoku Math. J. 39, 292–298. herding it through to publication. In addition, we 1937 W. C. Graustein, Extensions of the four- are grateful to Ville for sending us a copy of the vertex theorem, Trans. Amer. Math. Soc. 41, proof, to both Ville and Peter for telling us about 9–23. Dahlberg’s life and work, and to Ville for the de- 1937 W. C. Graustein and S. B. Jackson, The four- tailed letter that supplied the information for the vertextheoremforacertaintypeofspacecurves, previous section. Bull. Amer. Math. Soc. 43, 737–741. We are enormously grateful to Benjamin Schak 1940 S. B. Jackson, The four-vertex theorem for and Clayton Shonkwiler, two graduate students in spherical curves, Amer. J. Math. 62, 795–812. mathematics at the University of Pennsylvania, for 1944 , Vertices of plane curves, Bull. Amer. readingtheentiremanuscriptandmakingmanyim- Math. Soc. 50, 564–578. provements, as well as for studying and explaining 1945 , The four-vertex theorem for surfaces to us the proofs of Mukhopadhyaya and Kneser. of constant curvature, Amer. J. 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