Higher Symplectic Capacities and the Stabilized Embedding Problem for Integral Ellipsoids
HHIGHER SYMPLECTIC CAPACITIES AND THE STABILIZEDEMBEDDING PROBLEM FOR INTEGRAL ELLLIPSOIDS
DAN CRISTOFARO-GARDINER, RICHARD HIND AND KYLER SIEGEL
Abstract.
The third named author has been developing a theory of “higher” sym-plectic capacities. These capacities are invariant under taking products, and so arewell-suited for studying the stabilized embedding problem. The aim of this note is toapply this theory, assuming its expected properties, to solve the stabilized embeddingproblem for integral ellipsoids, when the eccentricity of the domain is the oppositeparity of the eccentricity of the target and the target is not a ball. For the other parity,the embedding we construct is definitely not always optimal; also, in the ball case,our methods recover previous results of McDuff, and of the second named author andKerman. There is a similar story, with no condition on the eccentricity of the target,when the target is a polydisc: a special case of this implies a conjecture of the firstnamed author, Frenkel, and Schlenk concerning the rescaled polydisc limit function.Some related aspects of the stabilized embedding problem and some open questionsare also discussed.
Contents
1. Introduction 21.1. The main results 21.2. Applications and remarks 4Structure of the note 6Acknowledgments 72. New capacities 72.1. The first approximation 72.2. Behavior under stabilization 82.3. The naive chain complex 82.4. Input from symplectic field theory 92.5. From spectral invariants to capacities 92.6. The case of ellipsoids 103. Optimal embeddings 103.1. The main theorems 103.2. The rescaled embedding function 153.3. The first step 163.4. The other parity 174. Discussion 184.1. Beyond the rescaled function 194.2. The opposite parity 194.3. The region from b = 1 to b = 2 a r X i v : . [ m a t h . S G ] F e b eferences 201. Introduction
The main results.
Let X and X be four-dimensional symplectic manifolds. Therehas recently been considerable interest in understanding the stabilized symplecticembedding problem , namely the question of whether or not there exists a symplecticembedding(1) X × C n s (cid:44) → X × C n . Indeed, certain techniques which are available for studying four-dimensional embeddingproblems do not have a clear analogue in higher dimensions, and so it is interesting tounderstand how different the stabilized problem is from the four-dimensional one. Formore about the stabilized problem, we refer the reader to [HK1, HK2, CGH, CGHM,McD2], the references therein, and the discussion below.The embedding problem (1) is already quite subtle when X and X are simple shapes,like ellipsoids E ( a, b ) := (cid:26) π | z | a + π | z | b ≤ (cid:27) ⊂ C n , balls B ( c ) := E ( c, c ) , polydiscs P ( a, b ) := (cid:26) π | z | a ≤ , π | z | b ≤ (cid:27) ⊂ C n , and cubes C ( c ) := P ( c, c ) . (Here, C n is equipped with its standard symplectic form andwe will write λ · E ( a, b ) for E ( λa, λb ) .) For example, what is known about the stabilizedellipsoid-into-ball problem has a curious mix of rigidity and flexibility: much aboutthis question remains unknown. In contrast, the stabilized polydisc-into-ball problem iscompletely solved [Sie2, Thm. 1.3.5] (for another approach see [Hin2]) and the answer isdescribed by a very simple function, namely a piecewise linear function with two pieces.The starting point for our investigations here is the stabilized ellipsoid-into-ellipsoidproblem. This is a special case of Problem in the influential problem list [MS1, Ch.14] by McDuff and Salamon, which asks for a solution to the symplectic embeddingproblem for n -dimensional symplectic ellipsoids: we can view stabilized ellipsoids as n dimensional ellipsoids with most arguments infinite. Consider the function c Nb,ell ( a ) ,defined to be the infimum, over λ , such that there exists an embedding(2) E (1 , a ) × C N s (cid:44) → λ · E (1 , b ) × C N . This function for a, b ≥ completely determines the stabilized ellipsoid-into-ellipsoidproblem, and we would ideally like to compute it. At present, this looks out of reach.As mentioned above, even the case b = 1 seems quite subtle; in fact it is the focus ofa conjecture by McDuff [McD2]. And, when b > , almost nothing is currently known.However, it turns out that when a and b are integers, there is a lot more traction. heorem 1.1. Assume that b > is an integer, and let a ≥ b + 1 be any integer withparity the opposite of b . Then for N ≥ ,c Nb,ell ( a ) = 2 aa + b − . We discuss the hypothesis a ≥ b + 1 here in Section 1.2.2 below, where we show thatit is essentially necessary.A key aspect of our proof of the above theorem, which is one of the motivations forwriting this note, involves the obstructions required to prove it. Symplectic embeddingproblems are profitably studied by symplectic capacities , see eg [CHLS]. The thirdnamed author has recently defined a new sequence of symplectic capacities g k which playa starring role here. These capacities g k are invariant under taking products with C andso give obstructions to the stabilized problem. As we will see in the proof of Theorem 1.1,the g k are very well-adapted to proving Theorem 1.1, and the obstructive side of theproof follows quite quickly once we can marshal them to our benefit. The constructiveside of the proof comes from a variant of the stabilized folding construction pioneered bythe second named author. Disclaimer 1.2.
Our high level discussion of symplectic capacities in §2 follows [Sie1],which in turn assumes the existence of rational symplectic theory with its expectedfunctoriality properties as outlined in [EGH]. Apart from simple special cases, such aformalism is known to require a virtual perturbation framework such as the theory ofpolyfolds; for the current status of this and related projects we refer the reader to e.g.[HWZ, FH, Par, HN, BH, Ish] and the references therein.The proofs of our main results on embedding obstructions in §3 take the propertiesof the capacities g k summarized in Theorem 2.1 as a black box, together with somecomputations from [Sie1] which we recall in §3.1.2. Our proof of Theorem 1.1 furthermorerequires the formula for g k ( E (1 , a )) which will appear in the forthcoming work [MS4].The latter reference also constructs an ersatz version of these capacities in the special caseof ellipsoids without appealing to virtual perturbations; these give equivalent obstructionsfor stabilized embeddings between four-dimensional ellipsoids, and the method also readilyadapts to the case of ellipsoid domain and polydisk target. Our proof of Proposition 1.7further depends on the formalism of [Sie2], which is based on [Sie1] and the forthcoming[Sie3].In dimension four, when b is integral there is an equivalence of embeddings(3) E (1 , a ) s (cid:44) → λP (1 , b ) , E (1 , a ) s (cid:44) → λE (1 , b ) , that is, one of these embeddings exists if and only if the other does, see for example[CGFS, Rmk. 1.2.1]. So, it is natural to compare Theorem 1.1 with the stabilizedellipsoid-into-polydisc problem. Here we get a somewhat parallel, but in fact strongerresult. Define c Nb,poly ( a ) to be the infimum, over λ , such that an embedding(4) E (1 , a ) × C N s (cid:44) → λ · P (1 , b ) × C N exists. heorem 1.3. Let a ≥ b − be any odd integer. Then for N ≥ c Nb,poly ( a ) = 2 aa + 2 b − . We remark that, in contrast to Theorem 1.1, there is no requirement here that b isan integer. As with the previous theorem, the hypothesis a ≥ b − is discussed inSection 1.2.2, where it is shown to be necessary.1.2. Applications and remarks.
Steps and the rescaled embedding function.
One of our motivations for studyingTheorem 1.3 is that it readily implies a conjecture of the first author, Frenkel, andSchlenk about the stabilized ellipsoid-into-polydisc function, namely Conjecture 1.4 in[CGFS], which we now explain.First we explain the motivation behind that conjecture. As alluded to above, atpresent, fully computing the function c Nb,poly ( a ) for N ≥ seems quite difficult. However,there is a related function, called the rescaled limit function ˆ c Nb,poly , see (5) below,that looks more tractable and in particular could be computed given a resolution of theaforementioned Conjecture 1.4.To elaborate, the function c b,poly ( a ) for b ∈ Z ≥ was previously computed by the firstauthor, Frenkel and Schlenk in [CGFS]. It was shown that the function c b,poly ( a ) is givenby the volume constraint a b , except on finitely many intervals. On all but one of theseintervals, the function c b,poly ( a ) is given by a “linear step”: it is piecewise linear, with asingle nonsmooth point, called its corner, where its graph changes from lying on a linethrough the origin to being horizontal. On the remaining interval, it is also piecewiselinear with a single nonsmooth point, but the linear piece does not lie on a line throughthe origin – it has an intercept, and so we call it the “affine step”. For more detail, see[CGFS].Conjecture . asserts that the linear steps from above are “stable”. In other words forany a , we have c Nb,poly ( a ) ≤ c b,poly ( a ) , by taking the product with the identity mapping.The conjecture, then, is that for a in the domain of the linear steps, we have c Nb,poly ( a ) = c b,poly ( a ) . To state that conjecture precisely, we define, for k ∈ { , , , . . . , (cid:98)√ b (cid:99)} , thenumbers u b ( k ) = (2 b + k ) b , v b ( k ) = 2 b (cid:18) b + 2 k + 12 b + k (cid:19) . We always have u b ( k ) < v b ( k ) ; for u b ( k ) < a < v b ( k ) , the graph of c Nb,poly ( a ) is preciselythe linear steps mentioned above. Corollary 1.4 (Conj. 1.4 of [CGFS]) . Assume that b is an integer and u b ( k ) ≤ a ≤ v b ( k ) . Then c b,poly ( a ) = c Nb,poly ( a ) = c b,ell ( a ) = c N b,ell ( a ) . The final two equalities here, concerning the ellipsoid-into-ellipsoid function, were notactually part of Conjecture 1.4; however, they fall out immediately from our proof. e now state the relevance of this to the rescaled limit function. The background isthat [CGFS] defined the rescaled functions(5) ˆ c Nb,poly ( a ) := 2 bc Nb,poly ( a + 2 b ) − b, a ≥ , in order to capture the qualitative behavior of the obstructive part of the embeddingfunction c b,poly that goes beyond Gromov’s nonsqueezing theorem. It was shown in[CGFS, Eq. 1.3] that the functions ˆ c b,poly ( a ) converge uniformly on bounded sets toa pleasing answer, namely the “infinite regular staircase” described by the function c ∞ ( a ) : [0 , ∞ ) → R whose graph consists of infinitely many linear steps of width 2, see[CGFS, Fig. 1.7] and Figure 1 below. For more about the motivation for studying therescaled function, we refer the reader to the discussion in [CGFS, Sec. 1.2]. Figure 1.
The rescaled limit function. Each step has width two, andconsists of a line of slope one and a horizontal line.
Corollary 1.5.
The rescaled limit function is stable. That is, for any N ∈ Z ≥ andintegral b , we have lim b →∞ ˆ c Nb,poly ( a ) = c ∞ ( a ) , a ∈ [0 , ∞ ) uniformly on bounded sets. We will explain the proofs of these corollaries in 3.2.1.2.2.
The first step.
We next remark that, in the context of Theorem 1.1, the lowerbound on a is essentially necessary. Indeed, if a ≤ b then inclusion gives an embeddingwhich Gromov’s non-squeezing theorem shows is optimal. That is, c Nb,ell ( a ) = 1 for all N ≥ . There is a similar story for Theorem 1.3 for a ≤ b − , but it requires a moreinteresting embedding. With a little more work, we can extend the range of a to workout the “first step” of the embedding functions considered in this note. Proposition 1.6.
Let b ∈ R ≥ . Then: • The function c Nb,ell starts as follows: – We have c Nb,ell ( a ) = 1 , ≤ a ≤ b. Actually, only the N = 0 case of these functions was defined, but the definition extends verbatim togeneral N , and that will be our working definition here. We have c Nb,ell ( a ) = ab , b ≤ a ≤ b + 1 . • The function c Nb,poly starts as follows. Let a be the smallest odd integer that isno less than b − . – We have c Nb,poly ( a ) = 1 , ≤ a ≤ a − + b. – We have c Nb,poly ( a ) = a +2 b − a, a − + b ≤ a ≤ a Note that there is no restriction above that b be integral, in contrast to the theoremsin the previous section.1.2.3. The case b = 1 . In view of Theorem 1.1, it is natural to ask about the case b = 1 .This was previously studied by McDuff [McD2], who proved an analogous result for anyinteger congruent to two, modulo three; we can recover this result with our methodsas well, see Example 1. Comparing our result to McDuff’s, it is interesting to note theswitch from three periodicity to two periodicity as b increases from one. There is asubstantial mystery about the structure as b ranges from to , see §4.3, which we planto investigate in follow-up work.1.2.4. The other parity.
In view of the above results, it is natural to ask: what happensfor a an integer of a parity not covered by our theorems. We certainly do not have asatisfactory answer to this at present. However, using the more general calculus of [Sie2],together with the aid of the computer, we can show for example: Proposition 1.7.
For ≤ a ≤ an even integer, the conclusion of Theorem 1.1 holdsfor b = 2 , that is for N ≥ we have c N ,ell ( a ) = 2 aa + 1 . Similarly, for ≤ a ≤ an even integer, the conclusion of Theorem 1.3 holds for b = 1 ,that is for N ≥ we have c N ,poly ( a ) = 2 aa + 1 . Remark 1.8.
The assumption a ≥ in Proposition 1.7 is necessary. Indeed, for a less than the squared silver ratio σ ≈ . , c ,poly ( a ) is an infinite staircase [FM]. Inparticular, we have c N ,poly ( a ) ≤ c ,poly ( a ) , and c ,poly ( a ) is strictly less than aa +1 for a = 2 , . The same applies for c N ,ell , since we have c ,ell = c ,poly .For more examples, suppose that a = 2 b +2 k +2 is an even integer. Referring to section§1.2.1 we see that v b ( k ) ≤ a ≤ u b ( k + 1) which for k ≥ implies that c b,poly ( a ) = (cid:112) a b ,that is, there is a volume filling embedding from E (1 , a ) into a scaling of P (1 , b ) (thepoint a = 2 b + 4 lies in the affine step). By (3) this is equivalent to the existence of avolume filling embedding from E (1 , a ) into a scaling of E (1 , b ) . Now, volume fillingembeddings in dimension improve on the folding construction giving Theorem 1.1 when a < b + 1 + 2 √ b . Hence the conclusion of Theorem 1.1 is false when a and b are evenand b + 4 < a < b + 1 + 2 √ b . Structure of the note.
In §2 we review the construction of the higher symplecticcapacities of the third named author; our discussion here includes some informal elementsto help convey the intution. Then in §3 we give the proofs of our results. The finalsection §4 discusses some natural follow-up questions to this work. cknowledgments. We thank Felix Schlenk for his encouragement, and for helping thefirst and third named authors better understand constructions of embeddings betweenstabilized ellipsoids. Our paper is dedicated to Claude Viterbo on the occasion of his th birthday. We are immensely grateful to Claude for his visionary leadership of ourfield.This research was completed while the first named author was on a von Neumannfellowship at the Institute for Advanced Study; he thanks the Institute for their support.The first named author is partially supported by NSF grant DMS-1711976 and the secondnamed author by Simons Foundation Grant no. 633715.2. New capacities
We first briefly review the capacities g k defined for k ∈ Z ≥ in [Sie1]. These are partof a more general family of capacities g b indexed by elements in the symmetric tensoralgebra S Q [ t ] = (cid:76) ∞ k =1 ( ⊗ k Q [ t ]) / Σ k . We give here only an impressionistic sketch, omittingsome of the more technical details. In addition to the computations described in §3.1.2,the key structural properties we will need are summarized in the following: Theorem 2.1. [Sie1]
For any Liouville domain X and k ∈ Z ≥ , we have g k ( X ) ∈ R > with the following properties:(1) symplectomorphism invariance : if X (cid:48) is another Liouville domain which is sym-plectomorphic to X , we have g k ( X ) = g k ( X (cid:48) ) (2) scaling : if X (cid:48) is the Liouville domain obtained by scaling the Liouville form of X by a constant c ∈ R > , we have g k ( X (cid:48) ) = c g k ( X ) (3) monotonicity : if X (cid:48) is another Liouville domain and there exists a symplecticembedding X s (cid:44) → X (cid:48) , then we have g k ( X ) ≤ g ( X (cid:48) ) (4) stabilization : we have g k ( X × B ( S )) = g k ( X ) , provided that S > g k ( X ) . The first approximation.
Suppose that X is a Liouville domain. We work withalmost complex structures J on the symplectic completion (cid:98) X which are admissiblein the sense of symplectic field theory (SFT). Fix a point p ∈ X along with a local J -holomorphic divisor D passing through p . To first approximation, g k ( X ) is simplythe minimal energy of a punctured J -holomorphic sphere u : Σ → (cid:98) X with some number l ≥ of positive ends asymptotic to Reeb orbits in ∂X , such that u passes through p and is tangent to D to order k − . We denote this tangency constraint by < T k − p > (see [MS3] and the references therein for more details).To see why this should be monotone with respect to symplectic embeddings, thebasic point is that given such a curve u in (cid:98) X and a symplectic embedding X (cid:48) s (cid:44) → X , wecan neck-stretch along ∂X (cid:48) . This forces u to break into a pseudoholomorphic buildingconsisting of • a curve u top (possibly disconnected) in the completed symplectic cobordism (cid:92) X \ X (cid:48) with the same positive asymptotics as u • a curve u bot in (cid:99) X (cid:48) which inherits the tangency constraint < T k − p > . Strictly speaking, X × B ( S ) is not a Liouville domain since it has corners, although these can beremoved by an arbitrarily small smoothing. See [Sie1, §5.4] for a more precise formulation. ince u bot is a candidate minimizer for g k ( X (cid:48) ) and it has energy at most that of u , thisshows that g k ( X (cid:48) ) ≤ g k ( X ) .2.2. Behavior under stabilization.
One role of the local tangency constraint in thedefinition of g k is to cut down the dimension of familes of curves, thereby giving access tocurves of higher Fredholm index. There are certainly other natural geometric constraintswhich lower the index, the most obvious being to impose k distinct point constraints.In fact, doing so leads to the “rational symplectic field theory capacities” (RSFT) firstconsidered in [Hut].However, point constraints behave in a rather complicated way under dimensionalstabilization. The RSFT capacities are therefore perhaps not well-suited for stabilizedproblems (although they may have other applications yet to be discovered). For example,note that each point constraint is codimension when dim X = 4 , but is generallycodimension n − when dim X = 2 n . This means that the same curve with the samepoint constraints has negative total index after stabilizing.By contrast, local tangency constraints behave quite well with respect to stabilization.This is closely related to the observation of Hind and Kerman from [HK1] that punc-tured rational curves with exactly one negative end have stable Fredholm index. Thestabilization property in Theorem 2.1 is also closely related to the stabilization theoremsappearing in the works [CGH, CGHM, McD2].2.3. The naive chain complex.
Unfortunately, the definition given in §2.1 is notparticularly robust, since it might depend on the choice of almost complex structure J .Indeed, if we try to deform J to some other almost complex structure J (cid:48) , somewherealong the way the curve u might degenerate into a pseudoholomorphic building and thendisappear. Therefore, in order to get something which is truly a symplectomorphisminvariant, we have to be a bit more “homological”. This is where the chain complexescoming from Floer theory or symplectic field theory become essential.The idea is to associate to X a filtered chain complex C ( X ) , where • as a vector space, C ( X ) is the (graded) polynomial algebra on the (not necessarilyprimitive) Reeb orbits of ∂X • the differential is defined by counting rigid-up-to-translation connected rationalcurves in R × ∂X with several positive ends and one negative end • the filtration is by the symplectic action functional, or equivalently by the periodsof Reeb orbits.Similarly, given an exact symplectic cobordism W with positive end ∂ + W = ∂X andnegative end ∂ − W = ∂X (cid:48) , we define a chain map from C ( X ) to C ( X (cid:48) ) by countingrigid possibly disconnected rational curves in W , such that each component has severalpositive ends and one negative end. By Stokes’ theorem, both the differential and thecobordism map are action-nondecreasing and hence preserve the filtrations.However, the above prescription does not work on face value due to transversalityissues. Namely, in order to show that the differential squares to zero and that thecobordism map is a chain map, the typical strategy is to analyze analogous moduli spacesof dimension one and show that (after compactifying) their boundaries give precisely the There is also a nice story extending the theory to non-exact symplectic cobordisms, but we willignore this for simplicity. esired relations. But it is well-known that the relevant SFT moduli spaces are rarelytransversely cut out for any choice of generic J . Multiply covered curves tend to appearwith higher-than-expected dimension, and this spoils our strategy.2.4. Input from symplectic field theory.
One way is get around this issue is tocount curves in a “virtual” sense, by introducing suitable abstract perturbations whichallow more room to achieve transversality. This is the basic strategy being pursued todefine SFT in full generality by various groups, with much recent progress but consensusnot yet achieved (see e.g. [HWZ, FH, Par, HN, BH, Ish] and the references therein).In the setting of SFT, the desired invariant C ( X ) can be written as B CH lin ( X ) . Here CH lin ( X ) is the linearized contact homology of X , which is roughly the chain complexgenerated by Reeb orbits of ∂X with differential counting cylinders in the symplectization R × ∂X . Linearized contact homology only involves curves with one positive end, butby incorporating curves with several positive ends we get an L ∞ structure, consistingof l -to- operations for all l ≥ satisfying various compatibility conditions. We canconveniently package this L ∞ structure into one large chain complex B CH lin ( X ) , the bar complex .2.5. From spectral invariants to capacities.
Getting back to the high level viewpoint,we have a filtered chain complex C ( X ) for each Liouville domain X , and filtration-preserving chain maps Ξ : C ( X ) → C ( X (cid:48) ) for any (exact) symplectic embedding X (cid:48) s (cid:44) → X . Now for any class α in the homology of C ( X ) , define c α ( X ) to be the minimalaction of any closed element of C ( X ) which represents α . By a simple diagram chase,we have c [Ξ]( α ) ( X (cid:48) ) ≤ c α ( X ) , where [Ξ] denotes the homology-level map induced by Ξ .At first glance, this construction appears to give a new family of symplectic capacitiesindexed by homology classes of C ( X ) . But there is still one issue, which is that we needa canonical way to reference these homology classes. Indeed, in principle the homologylevel map [Ξ] might be quite nontrivial, so how do we know when two numbers c α ( X ) and c β ( X (cid:48) ) can be compared to each other?This is where the tangency constraints come in. The claim is that by counting possiblydisconnected curves in (cid:98) X with each component u i satisfying a < T k i − p > constraint forsome k i ∈ Z > , we get a chain map (cid:15) X < T • > : C ( X ) → S Q [ t ] . For example, a term t (cid:12) t (cid:12) t in S Q [ t ] corresponds to counting curves with threecomponents which satisfy constraints < T p > , < T p > , and < T p > respectively. More-over, these maps are natural in the sense that the composition (cid:15) X (cid:48) < T • > ◦ Ξ agrees with (cid:15) X < T • > up to filtered chain homotopy.Now for any b ∈ S Q [ t ] , we define the capacity g b ( X ) ∈ R > by g b ( X ) := inf { c α ( X ) : [ (cid:15) X < T • > ]( α ) = b } . This defines a symplectomorphism invariant which scales like symplectic area, and forany symplectic embedding X (cid:48) s (cid:44) → X we have g b ( X (cid:48) ) ≤ g b ( X ) . In the case that X isLiouville deformation equivalent to a ball, one can show that (cid:15) X < T • > is actually a More precisely, we only allow “good” Reeb orbits, and we count cylinders which are additionally“anchored” in X . hain homotopy equivalence, so every spectral invariant of C ( X ) corresponds to somechoice of b .Finally, to define the simplified capacities g k , let π : S Q [ t ] → Q [ t ] denote the projectionto tensors of length (e.g. t + t (cid:12) t (cid:12) t maps to t ). We define g k ( X ) := inf b : π ( b )= t k − g b ( X ) . In essence, this means we look for the collection of Reeb orbits in ∂X of minimal actionwhich is closed with respect to the differential of C ( X ) , and which bounds a connectedrational curve in (cid:98) X satisfying a < T k − p > constraint (but disregarding any disconnectedcurves bounded by the same collection).2.6. The case of ellipsoids.
To get some intuition for g b ( X ) , we note that when X isan irrational ellipsoid E ( a , . . . , a n ) , the differential on C ( X ) vanishes for degree parityreasons. This means that C ( X ) already agrees with its homology, and the map (cid:15) X < T • > : C ( X ) → S Q [ t ] is in fact an isomorphism. Then g b ( X ) is simply the action of the unique element ( (cid:15) X < T • > ) − ( b ) ∈ C ( X ) which corresponds to b . However, recall that the map (cid:15) X < T • > is defined by counting curves in E ( a , . . . , a n ) satisfying local tangency constraints, soit could be quite nontrivial even in the case n = 2 . Indeed, in the very special case ofthe nearly round ball E (1 , (cid:15) ) , a closely related problem is to count rational curvesin CP satisfying local tangency constraints, which was recently solved in [MS3]. Forother ellipsoids, including those in higher dimensions, and for more general Liouvilledomains, computing g b seems to involve some very interesting and challenging enumerativeproblems.We discuss the computation of the capacities g k for four-dimensional ellipsoids in§3.1.2 below, based on the forthcoming work [MS4]. As for the larger family of capacities g b , a general recursive algorithm for their computation is given in [Sie2], and this will beutilized in the proof of Proposition 1.7.3. Optimal embeddings
The main theorems.
We now prove our main results. To prove Theorem 1.1, weneed a new construction and new obstructions. These two parts of our argument arelogically independent of each other and can be done in either order. To prove Theorem 1.3,we can use an existing construction and so we just need the obstructions.3.1.1.
The construction.
We begin with the construction.
Proposition 3.1.
For all a > and S > , let aa +1 ≤ µ ≤ a and λ = 1 − µa . Thereexists a symplectic embedding of E ( a, , S ) into an arbitrary neighborhood of { ( z , z ) | π | z | ≤ λ + µ, π | z | ≤ f ( π | z | ) } × C where f ( t ) = (cid:40) λ − t/ t ≤ µ λ − λ + µ − ;1 − (1 − λ )( t − λ +1)1 − λ + µ when 2 µ λ − λ + µ − ≤ t ≤ λ + µ. emark 3.2. Using the work of Pelayo-V˜u Ngo. c [PVuN, Theorem 4.4] we can extendto S = ∞ and embed the interior of the ellipsoid into the domain itself, rather than intoa neighborhood.We defer the proof for a moment, first stating some key corollaries we will need. Corollary 3.3.
For any N ≥ and a ≥ , ≤ b ≤ there exists a symplectic embedding int E ( a, × C N s (cid:44) → a ( b + 2)( a + 1) b · (cid:0) E ( b, × C N (cid:1) Here, “int” denotes the interior.
Proof of Corollary 3.3.
It clearly suffices to prove this when N = 1 . In Proposition 3.1,set µ = aa +1 so λ = 1 − µa = µ . In this case f ( t ) = 2 λ − t/ for all ≤ t ≤ λ = λ + µ and we see that the domain { ( z , z ) | π | z | ≤ λ + µ, π | z | ≤ f ( π | z | ) } is simply P (2 λ, λ ) ∩ E (4 λ, λ ) . This sits inside E ( cb, c ) when c ≥ a ( b +2)( a +1) b .This deals with the case when a > . When a = 1 we still have an embedding into anarbitrarily small neighborhood, and so can still apply [PVuN] for the precise result. (cid:3) Corollary 3.4.
Let b ∈ R ≥ . Then for any N ≥ and a ≥ b − there exists a symplecticembedding int E ( a, × C N s (cid:44) → aa + b − · (cid:0) E ( b, × C N (cid:1) Proof of Corollary 3.4.
Note that when a > we have − λ − λ + µ < , and so the graph of f ( t ) is convex. Hence f ( t ) is bounded above by the linear function between (0 , λ ) and ( λ + µ, λ ) and our domain is a subset of P ( λ + µ, λ ) ∩ E (2( λ + µ ) , λ ) .In the context of Proposition 3.1, set µ = a ( b − a + b − . We note that aa +1 ≤ µ ≤ a exactlywhen ≤ b ≤ a + 1 . Then λ = aa + b − and we find a symplectic embedding E ( a, × C s (cid:44) → (cid:18) P (cid:18) aba + b − , aa + b − (cid:19) ∩ E (cid:18) aba + b − , aa + b − (cid:19)(cid:19) × C ⊂ aa + b − E ( b, × C . (cid:3) We now give the promised proof of the Proposition.
Proof of Proposition 3.1.
Before the proof we fix some notation.Write A ⊂ ε B to mean that the set A lies in an ε neighborhood of B , or z ∈ ε B tomean that a point z lies ε close to B .Let π : C → C be the projection onto the z plane.In the z plane we fix sets W = [0 , × [0 , µ ] and W i = [2 i, i + 1] × [0 , λ ] for i ≥ .Finally, D ( a ) denotes the round disk in the plane centered at the origin of area a , and A i are the subsets of the z plane given by A = D ( S + ε ) and A i = D ( i ( S + ε )) \ D (( i − S + ε )) for i ≥ . Proof.
The condition µ ≥ aa +1 is equivalent to µ ≥ − µa = λ , and the condition µ ≤ a is equivalent to λ ≥ . Both of these inequalities will be used in our construction. e apply a slightly generalized version of Lemma 2.2 from [Hin1]. This says that,given ε , there exists a large K and a symplectomorphism φ from E ( a, , S ) to a set F K with the following properties. For z ∈ C we write F z = π − ( z ) ∩ F K .(1) π ( F K ) ⊂ ε (cid:83) Ki =1 ([2 i − , i ] × { } ) (cid:83) Ki =0 W i ;(2) if z = ( u, v ) ∈ ε W then F z ⊂ ε D (1 − uµa ) × A ;(3) if z ∈ ε [2 i − , i ] × { } and i is odd, then F z ⊂ ε D ( λ ) × A i ;(4) if z ∈ ε [2 i − , i ] × { } and i is even, then F z ⊂ ε ( D (2 λ ) \ D ( λ )) × A i ;(5) if z = (2 i + u, v ) ∈ ε W i and i is odd, then F z ⊂ ε D ((1 + u ) λ ) × ( A i ∪ A i +1 ) ;(6) if z = (2 i + u, v ) ∈ ε W i and i ≥ is even, then F z ⊂ ε D ((2 − u ) λ ) × ( A i ∪ A i +1 ) .Apart from slight changes of notation, the modification from Lemma 2.2 consistsin increasing the area of W (the original lemma fixed µ = λ = xx +1 ) and a refineddescription of the fibers over W . The estimate in item (2) follows easily because π − ( W ) is the set { π | z | ≤ µ } ⊂ E ( a, , S ) and restricted to this set φ takes theform φ ( z , z , z ) = ( ψ ( z ) , z , z ) where we may assume for all ≤ u ≤ that ψ mapspoints with π | z | ≤ µu (outside of which the fiber lies in π | z | < − uµx ) to an ε neighborhood of the set [0 , u ] × [0 , µ ] . Then if ψ ( z ) = ( u, v ) we have π | z | ≥ µu − ε and so π | z | ≤ − uµa + ε .The next step is to follow Step 3 of the proof from [Hin1, page 880] and apply asymplectic immersion τ : π ( F K ) → C . This can be arranged to restrict to an embeddingon each of the W i and each of the intervals [2 i − , i ] × { } , so that the W i with i oddmap into a neighborhood of [ − , × [0 , λ ] , the W i with i even map into [0 , × [0 , µ ] , andthe ε neighborhoods of the intervals [2 i − , i ] × { } map close to the origin, remainingdisjoint from the W i . The condition on W i with i even is possible since λ ≤ µ .Let ι be the identity map on the ( z , z ) -plane. Then we note that ( τ × ι ) : F K → C is an embedding. Indeed, the fibers of π over W i and W j intersect only if | i − j | ≤ (since otherwise by items (5) and (6) their z coordinates lie in different A k ), and inparticular are disjoint if i and j have the same parity. Also the fibers over neighborhoodsof different intervals [2 i − , i ] × { } are disjoint by items (3) and (4).We refine the immersion τ slightly to also satisfy the following. • if z = (2 i + u, v ) ∈ W i and i is odd, then τ ( z ) ∈ ε [ − u, × [0 , λ ] ; • if z = ( u, v ) ∈ W , then τ ( z ) ∈ ε [0 , u ] × [0 , µ ] • if z = (2 i + u, v ) ∈ W i and i ≥ is even, then τ ( z ) ∈ ε [0 , uλµ ] × [0 , µ ] .The following describes the fibers of the image of τ × ι . Lemma 3.5.
Let ( z , z , z ) lie in the image of τ × ι and z = ( u, v ) .If − ≤ u ≤ then F z ⊂ ε D ((2 + u ) λ ) × C ;if ≤ u ≤ λ − λ + µ − then F z ⊂ ε D (2 λ − uµ ) × C ;if λ − λ + µ − ≤ u ≤ then F z ⊂ ε D (1 − uµa ) × C .Proof. The description of the fibers when u ≤ follows directly from item (5) in thedescription of F K and the properties of τ . Also, if λµ ≤ u ≤ then the property followsfrom item (2).If < u ≤ λµ then either ( u, v ) = τ ( u (cid:48) , v (cid:48) ) where ( u (cid:48) , v (cid:48) ) ∈ W and u (cid:48) ≥ u , or ( u, v ) = τ (2 i + u (cid:48) , v (cid:48) ) where (2 i + u (cid:48) , v (cid:48) ) ∈ W i for i ≥ even and u (cid:48) ≥ uµλ . In the firstcase, by item (2), the z coordinate of the fiber lies in D (1 − uµx ) and in the second case, y (6), the z coordinate of the fiber lies in D (2 λ − uµ ) . Thus the lemma follows fromthe fact that λ − uµ ≥ − uµx exactly when u ≤ λ − λ + µ − (using the assumption that λ ≥ ). (cid:3) Finally we apply the map σ × ι , where σ is an embedding of a neighborhood of ([ − , × [0 , λ ]) ∪ ([0 , × [0 , µ ]) in the z plane to a neighborhood of the disk D ( λ + µ ) .We can choose σ to satisfy the following. • if u ∈ [ − µλ t, t ] and ≤ t ≤ λ − λ + µ − then σ ( u, v ) ∈ ε D (2 tµ ) for all v ; • if u ∈ [ − λ − − λ ) tλ , t ] and λ − λ + µ − ≤ t ≤ then σ ( u, v ) ∈ ε D ((2 λ − − λ + µ ) t ) for all v .Such a map σ exists because the intersection of ([ − , × [0 , λ ]) ∪ ([0 , × [0 , µ ]) , theimage of τ , with { u ∈ [ − µλ t, t ] } has area µt and the intersection of the image of τ with { u ∈ [ − λ − − λ ) tλ , t ] } has area (2 λ −
1) + (1 − λ + µ ) t . Claim.
The image of σ × ι lies in an ε neighborhood of { ( z , z ) | π | z | ≤ λ + µ, π | z | ≤ f ( π | z | ) } × C , concluding the proof. Proof of the claim.
We check the fibers of π over points w ∈ D ( λ + µ ) . First, if w isin the image of a point in one of the segments [2 i − , i ] × { } then w is close to andthe z coordinate of the fiber lies in D (2 λ ) .Next suppose that π | w | = s + ε where s ≤ µ λ − λ + µ − . Then w = σ ( u, v ) where either u > s µ or u < − s λ (since by our conditions on σ points with u ∈ [ − s λ , s µ ] are mappedinto D ( s ) ). By Lemma 3.5, in the first case the z coordinate of the fiber lies ε closeto D (2 λ − s ) and in the second case the z coordinate of the fiber also lies in an ε neighborhood of D ((2 − s λ ) λ ) . Hence π | z | ≤ λ − π | z | / .Finally suppose that π | w | = s + ε where µ λ − λ + µ − ≤ s ≤ λ + µ . Then w = σ ( u, v ) where either u > s − (2 λ − − λ + µ or u < − (2 λ − µ +(1 − λ ) sλ (1 − λ + µ ) . By Lemma 3.5, in the first case thethe z coordinate of the fiber lies ε close to D (1 − s − (2 λ − − λ + µ µa ) = D (1 − (1 − λ )( s − λ +1)1 − λ + µ ) recalling that λ = 1 − µa . In the second case the z coordinate of the fiber lies ε close to D (2 λ − (2 λ − µ +(1 − λ ) s − λ + µ ) which we check is also D (1 − (1 − λ )( s − λ +1)1 − λ + µ ) . Hence π | z | ≤ − (1 − λ )( π | z | − λ +1)1 − λ + µ + ε . (cid:3) With the claim proven, we have completed the proof of the proposition. (cid:3)
Some obstructions.
We now turn our attention to the obstructive side. Notably,this will be quite short, because we can cite work on these higher capacities that haspreviously been done or is forthcoming. Namely, here we only recall the followingcomputations for the capacities of ellipsoids and polydisks from [Sie1, §6.3]: g k ( P (1 , a )) = min( k, a + (cid:100) k − (cid:101) ) for a ≥ , k ≥ odd(6) g k ( E (1 , a )) = k for a ≥ , ≤ k ≤ a. (7)It seems plausible that the computation for P (1 , a ) is also valid for k even. This wouldfollow if we knew that the capacities g k are nondecreasing with k , although this is notyet clear. e will also need the following more general expected formula for ellipsoids, whichwill be proved in [MS4]. For ≤ a ≤ / , we have g k ( E (1 , a )) = ia for k = 1 + 3 i with i ≥ a + ia for k = 2 + 3 i with i ≥
02 + ia for k = 3 + 3 i with i ≥ . (8)For a > / , we have g k ( E (1 , a )) = k for ≤ k ≤ (cid:98) a (cid:99) a + i for k = (cid:100) a (cid:101) + 2 i with i ≥ (cid:100) a (cid:101) + i for k = (cid:100) a (cid:101) + 2 i + 1 with i ≥ . (9)3.1.3. The proofs.
We now give the promised proofs.
Proof of Theorem 1.1.
Let a, b and N be as in the statement of the theorem. Then, byCorollary 3.4, we have c Nb,ell ( a ) ≤ aa + b − . To prove the opposite inequality, we use the higher capacities g k . That is, take k = a .Then, by (7) and (9), we have, g k ( E (1 , a )) = a, g k ( E (1 , b )) = a + b − . Hence, by the scaling, monotonicity, and stabilization properties of the g k in Theorem 2.1,we have c Nb,ell ( a ) ≥ aa + b − , hence the Theorem. (cid:3) Remark 3.6.
Note that in the above proof we only need the inequality g a ( E (1 , b )) ≤ a + b − ,and in the case that b is even (and hence a ≥ b + 1 is odd) this can be deduced directlyfrom (6). Indeed, by (3) there is an embedding E (1 , b ) s (cid:44) → P (1 , b/ , whence we have g a ( E (1 , b )) ≤ g a ( P (1 , b/ b/ (cid:100) ( a − / (cid:101) = a + b − . Proof of Theorem 1.3.
The proof is similar to the previous one. Let a, b and N be as inthe statement of the Theorem.The bound c Nb,poly ( a ) ≤ aa + 2 b − follows from the existence of a variant of the embedding from above, which was previouslyshown to exist in [CGFS, Lem. 1.3].To show that no better embedding exists, we use the above capacities. Namely, let k = a . Then, by (6) and (7) above, we have g k ( E (1 , a )) = a, g k ( P (1 , b )) = b + a − . The theorem now follows by the same argument as above. (cid:3) xample 1. It is interesting to compare the above methods with the case b = 1 . Forthis, we recall for the convenience of the reader an argument from [Sie1, §1.4]. There, avariant of the embedding used in the previous theorems, constructed in [Hin1], gives c Nb,ell ( a ) ≤ aa + 1 . On the other hand, if a is an integer congruent to two, modulo three, then taking k = a as above yields g k ( E (1 , a ) × C n ) = a, g k ( E (1 , × C n ) = 1 + a . Hence, combining these inequalities, we get that for a congruent to two modulo three, c Nb,ell ( a ) = 3 aa + 1 . This recovers the result of McDuff [McD2, Thm. 1.1].3.2.
The rescaled embedding function.
We now provide the proofs of the promisedcorollaries regarding the conjecture of the second named author, Frenkel, and Schlenk.
Proof of Corollary 1.4.
We will first prove the statement about c Nb,poly , after which theresult about c Nb,ell will follow easily.The function c Nb,poly ( a ) is nonincreasing in N . We want to show that it is in factconstant in N for a in the intervals given by the theorem. The computation of c b,poly ( a ) from [CGFS], together with Theorem 1.3 from above, shows that it does not depend on N for the exterior corners of each linear step.Now note that if an embedding E (1 , a ) × C n s (cid:44) → λP (1 , b ) × C n exists, then for any a (cid:48) > a , by scaling there is an embedding E (1 , a (cid:48) ) × C n s (cid:44) → a (cid:48) a λP (1 , b ) × C n . Thus, c Nb,poly ( a (cid:48) ) ≤ a (cid:48) a c Nb,poly ( a ) . So, given y = c N ( a ) , the graph of c N ( a (cid:48) ) for a (cid:48) > a cannot lie above the line through ( a, y ) and the origin. For future reference, we call thisthe subscaling property . We can now prove the corollary.Consider any linear step for c b,poly ( a ) . Recall that this consists of a linear part,then an exterior corner, and then a horizontal part. Consider the linear part. Wewant to show that this stabilizes. We know that c Nb,poly ( a ) ≤ c b,poly ( a ) . If there wereany a value corresponding to the first step for which strict inequality held, then bythe linearity property above, at the exterior corner a of the step, we would have c Nb,poly ( a ) < c b,poly ( a ) . However, above we saw in Theorem 1.3 that the exterior corneris stable. Hence, the whole linear part must stabilize. As for the horizontal part, weknow that we must have c Nb,poly ≤ c b,poly , but on the other hand the function c Nb,poly isnondecreasing, and so must be constant here. Thus, the whole step stabilizes, so all thelinear steps do.In view of Theorem 1.1, the exact same argument implies the result about c N b,ell , sincefor N = 0 there is an equivalence of embeddings (3). Proof of Corollary 1.5.
Corollary 1.4 shows that, after the initial part of the graph, where c Nb,poly ( a ) = 1 , the graph has (cid:98)√ b (cid:99) + 1 linear steps that are all stable. The length ofthese steps is given by the formula (cid:96) b ( k ) from [CGFS, p. 6]. In particular, as explainedthere, the length of the k th step converges to as b tends to infinity. Since the steps arecentered at the odd numbers, increase in number without bound as b increases, and ourrescaled function is centered so that the initial part of the graph with height one, thatis, the part determined by Gromov’s nonsqueezing theorem, does not appear, the resultfollows. (cid:3) The first step.
We now prove Proposition 1.6.
Proof of Proposition 1.6.
The key is the following lemma.
Lemma 3.7.
Let a be the smallest odd integer that is no less than b − . There is asymplectic embedding (10) int (cid:18) E (cid:18) , a −
12 + b (cid:19)(cid:19) s (cid:44) → P (1 , b ) . Proof.
By, for example, [CG], it is equivalent to find an embedding(11) int ( E (1 , b )) ∪ int (cid:18) E (cid:18) , a − (cid:19)(cid:19) s (cid:44) → P (1 , b ) . Indeed, the argument for [CG, Thm. 2.1] implies that both (10) and (11) are equivalentto ball packing problems of the P (1 , b ) , where in the first case, the size of the balls isgiven by the weight sequence defined in [CG, §2] for ( a − / b , and in the second casethe size of the balls is given by the union of the weight sequence for b and for ( a − / .Since ( a − / is an integer, the first ( a − / of the weights for ( a − / b willbe , so (10) and (11) are equivalent to the same ball packing problem.We know that a ≤ b + 1 , hence(12) a − ≤ b. We can therefore find an embedding as in (11) as follows. We think of the momentimage of P (1 , b ) as a union of two triangles, joined along the diagonal that does notcontain the origin. The triangle with legs on the axes contains an E (1 , b ) factor byinclusion. As for the other triangle, it is affine equivalent to the first, via multiplicationby − I , where I is the two-by-two identity matrix. Hence, by the Traynor trick, see forexample [Tra] and [CCGF + , Lem. 1.8] it also contains a copy of an E (1 , b ) ; the interiorsof these two E (1 , b ) are disjoint. Now, by (12) this latter int( E (1 , b )) contains a copy of int ( E (1 , ( a − / . (cid:3) We can now prove the proposition. We first prove the second bullet point. ByLemma 3.7, we know that c Nb,poly ≤ , for a in the given range. However, by Gromov’snon-squeezing theorem, we also know that c Nb,poly ≥ , for a in this range. As for the restof the first bullet point, this follows from the subscaling property of c Nb,poly , as in the proofof Corollary 1.4 above, given the lower bound on c Nb,poly ( a ) coming from Theorem 1.1. e now prove the first bullet point. The result for ≤ a ≤ b follows because inclusiongives an embedding for a in this range, which is optimal by Gromov’s nonsqueezingtheorem. Similarly, for b ≤ a ≤ b + 1 , scaling gives an embedding as in the subscalingproperty, which is optimal by the second Ekeland-Hofer capacity, see eg [CHLS, §2.3.1,§4.1.1] for the relevant formula. (cid:3) The other parity.
The proof of the remaining proposition, Proposition 1.7, requiresthe g b and computer assistance as well. It turns out that the simplified capacities g k donot suffice in these cases. For example, for E (1 , × C N s (cid:44) → λ · P (1 , × C N , one can checkthat the simplified capacities give only λ ≥ / , whereas we have in fact c N (6) = 12 / for N ∈ Z ≥ .On the other hand, we have the more general capacities g b , which could in principlegive sharp obstructions for all a ∈ R ≥ and b ∈ Z ≥ in (2) and (4). This is related todiscussion at the end of [Sie1, §6.3], where it is observed that the simplified capacities g k do not generally give sharp obstructions for E (1 , a ) × C N s (cid:44) → λ · E (1 , × C N , but thecapacities g b necessarily give sharp obstructions. Moreover, the formalism from [Sie2]gives an explicit recursive algorithm to compute the capacities g b for all convex toricdomains, although unfortunately it appears to be somewhat difficult to compute with“by hand”. Proof of Proposition 1.7.
We begin with the computation of c N ,poly ( a ) for a = 6 , , . . . , .By [Hin1], we have the upper bound c N ,poly ( a ) ≤ aa +1 , so it suffices to establish the lowerbound c N ,poly ( a ) ≥ aa +1 . Suppose that we have a symplectic embedding E (1 , a ) × C N s (cid:44) → λ · P (1 , × C N .Following the notation and exposition of [Sie2], the idea is as follows. By [Sie2, Cor.1.2.3], there is a filtered L ∞ homomorphism Q : V P ( λ,λ ) → V E (1 ,a ) which is unfiltered L ∞ homotopic to the identity. Here V is an explicit DGLA with generators α i,j for i, j ∈ Z ≥ and β i,j for i, j ∈ Z ≥ not both zero. The filtered DGLA V P ( λ,λ ) is just V asan unfiltered DGLA, and its filtration is specified by A P ( λ,λ ) ( α i,j ) = A P ( λ,λ ) ( β i,j ) = λi + λj. Similarly, the filtered DGLA V E (1 ,a ) is just V as an unfiltered DGLA, with filtrationspecified by A E (1 ,a ) ( α i,j ) = A E (1 ,a ) ( β i,j ) = max( i, aj ) . Recall that an L ∞ homomorphism Q : V P ( λ,λ ) → V E (1 ,a ) consists of a sequence of maps Q l : (cid:12) l V P ( λ,λ ) → V E (1 ,a ) for l = 1 , , , . . . , and these must satisfy an infinite sequence ofquadratic relations.Any element of the form β i ,j (cid:12) · · · (cid:12) β i k ,j k defines a cycle in the bar complex B V P ( λ,λ ) .In particular, (cid:98) Q ( β i ,j (cid:12) · · · (cid:12) β i k ,j k ) must be homologous to β i ,j (cid:12) · · · (cid:12) β i k ,j k in B V E (1 ,a ) . Moreover, there is a filtered L ∞ homomorphism Φ ,a : V E (1 ,a ) → V can E (1 ,a ) ,where V can E (1 ,a ) denotes the homology of V E (1 ,a ) (viewed as a filtered L ∞ algebra withtrivial L ∞ operations), and hence ( (cid:98) Φ ,a ◦ (cid:98) Q )( β i ,j (cid:12) · · · (cid:12) β i k ,j k ) is homologous to (cid:98) Φ ,a ( β i ,j (cid:12) · · · (cid:12) β i k ,j k ) in B V can E (1 ,a ) . ow suppose that we have a = p/q with p + q = 2 d for some p, q, d ∈ Z ≥ . Considersome d , d ∈ Z ≥ satisfying d + d = d , and suppose that we have Φ d ,a ( (cid:12) d β , (cid:12) (cid:12) d β , ) (cid:54) = 0 . (13)Then we claim that we have λ ≥ aa +1 , which gives the desired lower bound. Indeed, fora general input of the form β i ,j (cid:12) · · · (cid:12) β i k ,j k , it follows by degree considerations that Φ k ,a ( β i ,j (cid:12) · · · (cid:12) β i k ,j k ) is either trivial, or else it is the unique element up to scaling in V can E (1 ,a ) of its given degree. In the latter case, its action is given by the l th Ekeland–Hofercapacity of E (1 , a ) , i.e. c EH q ( E (1 , a )) , for l = (cid:80) km =1 ( i m + j m ) + k − . Also, the actionof the input is given by A P ( λ,λ ) ( β i ,j (cid:12) · · · (cid:12) β i k ,j k ) = k (cid:88) m =1 A P ( λ,λ ) ( β i m ,j m ) = k (cid:88) m =1 ( λi m + λj m ) . Specializing to the case of input (cid:12) d β , (cid:12) (cid:12) d β , and l = 2 d − , using a = p/q and p + q = 2 d it is straightforward to check that we have c EH l ( E (1 , a )) = p . Since (cid:98) Φ ,a ◦ (cid:98) Q is filtration-preserving and Φ d ( (cid:12) d β , (cid:12) (cid:12) d β , ) is a summand of the image of [ (cid:12) d β , (cid:12) (cid:12) d β , ] under [ (cid:98) Φ ,a ◦ (cid:98) Q ] , we must have λ ( d + d ) ≥ p , and hence λ ≥ pd = 2 pp + q = 2 aa + 1 , as claimed.Let us now specialize to the case that a is an even integer. Then we have a = p/q for p = 2 a and q = 2 , and hence p + q = 2 d for d = a + 1 . By computer calculations,(13) holds for d = 3 and d = d − d = a − for a = 6 , . . . , . Geometrically, thiscorresponds to a nonvanishing count of rational curves in CP × CP \ λ · E (1 , a ) ofbidegree ( d , d ) with one negative puncture asymptotic to the p = 2 a fold cover of theshort simple Reeb orbit. Curiously, the analogous counts for d = 1 , vanish.The computation of c N ,poly ( a ) for a = 6 , , . . . , is similar. In this case, we supposethat we have a symplectic embedding E (1 , a ) × C N s (cid:44) → λ · E (1 , × C N , and we take ourinput cycle to be of the form (cid:12) β , (cid:12) (cid:12) d − β , , for d = a − . By computer calculationwe have Φ d ,a ( (cid:12) β , (cid:12) (cid:12) d − β , ) (cid:54) = 0 (14)for a = 6 , , . . . , . The action of the output is that of the l th Ekeland–Hofer capacityof E (1 , a ) for l = 5 + 2 d , and we have c EH l ( E (1 , a )) = 2 a . Meanwhile, the action of theinput is A E ( λ, λ ) ( (cid:12) β , (cid:12) (cid:12) d − β , ) = 6 + ( d −
3) = d + 3 , whence the lower bound a ≥ aa +1 readily follows. (cid:3) Discussion
We close by discussing some natural follow-up questions to our work. .1. Beyond the rescaled function.
One can of course ask whether the function c Nb,poly ( a ) can in any sense be computed completely. As explained in [CGFS, Lem. 1.3],and mentioned previously here, a previous folding construction of the second namedauthor gives the bound c Nb,poly ( a ) ≤ aa + 2 b − . This bound can not be optimal for all a . For example, as we have seen in this paper,there are sometimes four-dimensional embeddings beating this bound, and these can bestabilized by taking the product with the identity. For a sufficiently large with respectto b , though, in particular for(15) a ≥ ( √ b + 1) , the above folding bound beats the four-dimensional volume obstruction, and so mustgive a better construction than any stabilized four-dimensional one. The main questionat the moment here is as follows. Question 4.1.
Is it the case that either c Nb,poly ( a ) = c b,poly ( a ) , or c Nb,poly ( a ) = 2 aa + 2 b − If this is true, it looks hard to prove. For example, if a < ( √ b + 1) , then the volumebound is strictly below the folding bound from above. On the other hand, for b ∈ Z ≥ ,then it is known that there are entire intervals of the subset a < ( √ b + 1) for whichthe volume bound is optimal for c b,poly : for example, for b = 2 , [CGFS, Thm. 1.1] statesthat there is an interval on which c b,poly is given by the volume starting at a = 7 . , buton the other hand by (15) the folding curve is above the volume curve up until a = 9 .Finding the holomorphic curves needed to show that this volume bound stabilizes wouldbe a completely new phenomenon.The same question, but concerning c Nb,ell is also open and just as interesting.4.2.
The opposite parity.
It is also natural to ask what happens for the stabilizedembedding problem for ellipsoids, when the parity of the domain and target are thesame. For example, one might hope that an analogue of our Proposition 1.7 holdsin the case b > . If this is true, however, it is not so clear how to prove it: ourpreliminary computer search to generalize the method required to prove it has not turnedup promising candidates. It would be very interesting to find a candidate of curves tosolve this problem, or to find another embedding.4.3. The region from b = 1 to b = 2 . For b ≥ , our Corollary 3.4 produces anembedding such that c Nb,ell ( a ) ≤ aa + b − . Meanwhile for ≤ b ≤ , Corollary 3.3 shows(16) c Nb,ell ( a ) ≤ a ( b + 2)( a + 1) b . t is interesting to ask when this bound is sharp, for instance whether there are sequencesof a where this holds. We now list some facts suggesting the answer may not bestraightforward.Note that when b = 1 the bound gives c N ,ell ( a ) ≤ aa + 1 , which as mentioned above is sharp when a ≡ modulo , [McD2]. There is anothersequence starting at a = 2 where (16) is an equality. By work of the first and secondnamed authors, [CGH], we have c N ,ell ( a ) = c ,ell ( a ) for all ≤ a ≤ τ . This region ofthe graph is an infinite staircase, that is, piecewise linear with infinitely many singularpoints accumulating at τ , see [MS2]. Between these singular points the graph alternatesbetween being constant and sitting on a line through the origin. One can check thecorners of the stairs, the left endpoints of the constant intervals, lie on the folding graph aa +1 .When b = 2 our bound gives c N ,ell ( a ) ≤ aa + 1 . The graph of c ,ell also begins with an infinite staircase, see [CGK, FM], and again thetips of the stairs lie on the graph aa +1 . It seems extremely likely that at such a we have c N ,ell ( a ) = c ,ell ( a ) for all N so the bound (16) is again sharp.However when b = 3 / the situation is mysterious. Now our bound gives c N / ,ell ( a ) ≤ aa + 1 . Here again work of the first named author and Kleinman shows that c / ,ell ( a ) has aninfinite staircase [CGK], but now the tips of the stairs lie on the graph aa +1 . Moreover the g k show that c N / ,ell ( a ) ≥ aa +1 at integer a . It is unclear whether an improved constructioncan show this lower bound is indeed sharp, or whether enhanced obstructions can be usedto show that even though the folding graph (16) lies strictly above the infinite staircaseit is still asymptotically sharp.4.4. A combinatorial rule?
While the functions c b,ell and c b,poly themselves are knownto be quite complicated. see for example [MS2, Ush], they are governed by simple tostate combinatorial rules. For example, McDuff shows in [McD1] that c b,ell is completelydetermined by the combinatorics of the sequence N ( a, b ) , whose k th term is the ( k + 1) st smallest entry among the nonnegative integer linear combinations of a and b . It wouldbe extremely interesting if the functions c Nb,ell and c Nb,poly are also governed by some kindof relatively simple to state combinatorial rule. It might be easier to find such a rulethan to actually compute these functions explicitly.
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