The embedding capacity of 4-dimensional symplectic ellipsoids
Dusa McDuff, Felix Schlenk · Annals of Mathematics · 2012
This paper calculates the function c(a) whose value at a is the infimum of the size of a ball that contains a symplectic image of the ellipsoid E(1, a).(Here a ≥ 1 is the ratio of the area of the large axis to that of the smaller axis.)The structure of the graph of c(a) is surprisingly rich.The volume constraint implies that c(a) is always greater than or equal to the square root of a, and it is not hard to see that this is equality for large a.However, for a less than the fourth power τ 4 of the golden ratio, c(a) is piecewise linear, with graph that alternately lies on a line through the origin and is horizontal.We prove this by showing that there are exceptional curves in blow ups of the complex projective plane whose homology classes are given by the continued fraction expansions of ratios of Fibonacci numbers.On the interval [τ 4 , 7] we find c(a) = (a + 1)/3.For a ≥ 7, the function c(a) coincides with the square root except on a finite number of intervals where it is again piecewise linear.The embedding constraints coming from embedded contact homology give rise to another capacity function cECH which may be computed by counting lattice points in appropriate right angled triangles.According to Hutchings and Taubes, the functorial properties of embedded contact homology imply that cECH(a) ≤ c(a) for all a.We show here that cECH(a) ≥ c(a) for all a. Contents DUSA MCDUFF and FELIX SCHLENK 2.3.The nature of the obstructions 1217 2.4.Connection to the lattice counting problem 1221 3. The Fibonacci stairs 1226 3.1.Main results 1226 3.2.Identities for Fibonacci numbers 1228 3.3.Reducing E(a n ) 1231 4. The interval [τ 4 , 7] 1240 4.1.Reduction to special points 1240 4.2.The classes E Ä b k (i) ä 1246 4.3.The ghost stairs 1250 5.The interval [7, 9] 1253 5.1.Preliminaries 1253 5.2.The interval [7, 8] 1255 5.3.The interval [8, 9] 1266 Appendix A. Weight expansions and Farey diagrams 1268 Appendix B. Computer programs 1272 B.1.Computing c(a) at a point a 1272 B.2. Computing c(a) on an interval 1275 References 1280• w 1 = 1, and w n ≥ w n+1 > 0 for all n;• if w i > w i+1 = • • • = w n (where we set w 0 := a), then• the sequence stops at w n if the above formula gives w n+1 = 0.The number M of entries in w(a) is called the length ℓ(a) of a.For example, a = 25/9 has weight expansion w(a) = (1, 1, 7 9 , 2 9 , 2 9 , 2 9 , 1 9 , 1 9 ), which we will abbreviate as (1 ×2 , 7 9 , 2 9