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Celebratio Mathematica

Dusa McDuff

Symplectic embedding problems

by Leonid Polterovich and Felix Schlenk

In this es­say we de­scribe some of Dusa’s con­tri­bu­tions to sym­plect­ic em­bed­ding prob­lems. Let us first say what these prob­lems are about: A sym­plect­ic man­i­fold (M,ω) is a smooth man­i­fold M to­geth­er with a closed and nonde­gen­er­ate dif­fer­en­tial two-form ω. The di­men­sion must then be even. Ex­amples are Kähler man­i­folds and co­tan­gent bundles TQ with their ca­non­ic­al sym­plect­ic form i=1ndqidpi, in which Hamilto­ni­an mech­an­ics takes place. The time-1 map φ of a Hamilto­ni­an flow on (M,ω) pre­serves the sym­plect­ic form: φω=ω. Any map with this prop­erty is called sym­plect­ic. The study of sym­plect­ic map­pings thus gives in­sight in­to the pos­sible dy­nam­ics of a clas­sic­al mech­an­ic­al sys­tem.

Every sym­plect­ic man­i­fold loc­ally looks like R2n with the con­stant sym­plect­ic form ω0=i=1ndxidyi. It is thus already in­ter­est­ing to study sym­plect­ic map­pings between open sub­sets U, VR2n. Every sym­plect­ic map­ping is a loc­al em­bed­ding; but look­ing at all sym­plect­ic map­pings is not in­ter­est­ing, for in­stance all of R2n can be sym­plect­ic­ally im­mersed in­to a tiny ball. On the oth­er hand, de­cid­ing when two open sub­sets are sym­plect­ic­ally dif­feo­morph­ic turns out to be too hard in gen­er­al. To learn something in­ter­est­ing, one looks at the in­ter­me­di­ate prob­lem of sym­plect­ic em­bed­dings. We write UsV if there ex­ists a sym­plect­ic em­bed­ding UV. In this case, Vol(U)Vol(V), since sym­plect­ic dif­feo­morph­isms pre­serve the volume Vol(U)=1n!Uωn. We as­sume throughout that 2n4, since two-di­men­sion­al sym­plect­ic map­pings are just those that pre­serve the area and the ori­ent­a­tion. Also note that UsV if and only if λUsλV for every λ>0.

De­note by B2n(a) the open ball in R2n of “area” a=πr2, where r is the ra­di­us, and by Z2n the full cyl­in­der B2(1)×R2n2inR2(x1,y1)×R2n2(x2,y2,,xn,yn). The start­ing point of the sym­plect­ic em­bed­ding story is Gro­mov’s non­squeez­ing the­or­em [e1]: B2n(a)sZ2nonly ifa1. In oth­er words, the in­clu­sion B2n(1)Z2n is already “the best” sym­plect­ic em­bed­ding! There are (even lin­ear) volume pre­serving em­bed­dings of B2n(a) in­to Z2n for any a>1. Sym­plect­ic map­pings are thus much more ri­gid than volume pre­serving map­pings, i.e., the evol­u­tions in Hamilto­ni­an dy­nam­ics are much more con­straint than those in er­god­ic the­ory. The key in Gro­mov’s proof is the ex­ist­ence of a suit­able J-holo­morph­ic sphere in the par­tial com­pac­ti­fic­a­tion S2×R2n2 of Z2n.

The start­ing point of our dis­cus­sion is the fol­low­ing prob­lem, which demon­strates both ri­gid and flex­ible be­ha­viour of sym­plect­ic map­pings: fill as much as pos­sible of B4(1) by k sym­plect­ic­ally em­bed­ded equal balls B4(a).

Let pk be the per­cent­age of the volume of B4(1) that can be filled by k sym­plect­ic­ally em­bed­ded equal balls. Then pk has the fol­low­ing val­ues: k123456789pk1123412025242563642882891

The in­equal­it­ies p21/2 and p54/5 were found by Gro­mov [e1] as an ap­plic­a­tion of his the­ory of pseudo­holo­morph­ic curves in sym­plect­ic man­i­folds. Gro­mov’s dis­cov­ery was that the al­geb­ra­ic geo­metry of curves in com­plex man­i­folds to a high ex­tent ex­tends to curves in al­most-com­plex man­i­folds provided that the al­most com­plex struc­ture is com­pat­ible with a sym­plect­ic form. In par­tic­u­lar, the Gro­mov–Eu­c­lid the­or­em states that, un­der the com­pat­ib­il­ity as­sump­tion, through every pair of points in any al­most-com­plex pro­ject­ive plane passes a unique line. Giv­en a sym­plect­ic pack­ing of B4(1) by two balls, B1 and B2 of ra­di­us r, com­pac­ti­fy B4(1) to CP2 and change the com­plex struc­ture on CP2 to a com­pat­ible al­most com­plex struc­ture J which is stand­ard on B1 and B2. Take a J-holo­morph­ic line L passing through the cen­ters of the balls. By Lel­ong’s in­equal­ity, Area(LBi)πr2. On the oth­er hand, the total area of L equals π, which read­ily yields 2πr2π and hence p21/2.

In the sum­mer of 1991, both Dusa and Le­onid vis­ited Helmut Hofer in Bo­chum. Helmut sug­ges­ted Le­onid to work on the concept of su­per-re­cur­rence in Hamilto­ni­an dy­nam­ics: in­deed, sym­plect­ic maps tend to have more peri­od­ic or­bits than gen­er­al volume pre­serving maps. Le­onid’s take on the prob­lem was as fol­lows: as­sume that for some k we have pk<1. This means that CP2 can­not be fully packed by k balls of volume Vol(CP2)/k. In par­tic­u­lar, if B is such a ball and φ is any sym­plec­to­morph­ism of CP2, one of the sets φj(B), j=1,, k1 must in­ter­sect B. This con­straint, which is spe­cif­ic to sym­plect­ic maps, can be in­ter­preted as a short term su­per-re­cur­rence. To have a long term su­per-re­cur­rence, one wants to have in­equal­it­ies pk<1 for an in­fin­ite num­ber of val­ues of k (a few years later Paul Biran showed that this is not the case, see be­low). This gave rise to the fol­low­ing ques­tion: why did Gro­mov stop at k=5 balls, the case based on study­ing con­ics passing through 5 points? An ana­lys­is showed that for large k, the up­per bound for pk based on Gro­mov’s J-holo­morph­ic curves of high­er de­gree is use­less: it ex­ceeded 1! (Later on, Misha Gro­mov re­membered that it was Dav­id Kazh­dan who warned him about the large k case). What hap­pens in between, i.e., for which val­ues of k do pseudo­holo­morph­ic curves beat the trivi­al volume-based es­tim­ate? Dusa and Le­onid were in­trigued and de­cided to ex­plore. At that time Dusa had already in­tro­duced the tech­nique of sym­plect­ic blow­ing up, a fun­da­ment­al con­struc­tion go­ing back to Gro­mov and Guille­minStern­berg, in­to sym­plect­ic to­po­logy [1]: Giv­en an em­bed­ding φ:B4(a)sB4(1), one re­moves φ(B4(a)) and col­lapses the re­main­ing bound­ary along the char­ac­ter­ist­ic fo­li­ation, which is giv­en by the or­bits of the Hopf-flow on the 3-sphere. This tech­nique be­came the main tool in the pa­per of Mc­Duff and Pol­ter­ovich [2], which settled the above table for k8. Let us re­tell Gro­mov’s proof of p21/2 in the lan­guage of blow-ups. Giv­en a pack­ing of CP2 by balls of ra­di­us r, one can blow them up and get the del Pezzo sur­face M such that the area of the gen­er­al line L is still π, while the areas of the ex­cep­tion­al di­visors E1 and E2 equal πr2. One checks that the class LE1E2 cor­res­pond­ing to the prop­er trans­form of the Gro­mov–Eu­c­lid line is ex­cep­tion­al, i.e., has self-in­ter­sec­tion 1, and hence, by an earli­er res­ult of Dusa, is rep­res­en­ted by a pseudo­holo­morph­ic curve. Its area π2πr2 is thus pos­it­ive, and so we get p2<1/2. The ad­vant­age of this ap­proach is that it gen­er­al­izes to k8, and the pack­ing in­equal­it­ies for k8 balls fol­low from the beau­ti­ful clas­sic­al clas­si­fic­a­tion of ex­cep­tion­al curves on del Pezzo sur­faces.

On a more per­son­al note, I (Le­onid) com­peted for Dusa’s at­ten­tion with her son Thomas, at that time a young child who ac­com­pan­ied Dusa on her trip to Bo­chum. Dusa skill­fully handled the case. Thus I got two private les­sons for the price of one: not only in sym­plect­ic to­po­logy, but also in wise and very pa­tient par­ent­ing. I use this oc­ca­sion to ex­press my grat­it­ude and ad­mir­a­tion to Dusa for both of them.

J-holo­morph­ic curves do not yield non­trivi­al pack­ing con­straints for k9 balls. In fact it was ob­served in [2] that no such con­straints ex­ist (i.e., pk=1) mod­ulo an old con­jec­ture by Nagata in enu­mer­at­ive al­geb­ra­ic geo­metry, which in turn was mo­tiv­ated by Hil­bert’s 14th prob­lem. While Nagata’s con­jec­ture is, to the best of our know­ledge, still open, the pack­ing puzzle was re­solved in the PhD thes­is by Paul Biran [e4], who suc­ceeded to prove pk=1 for k10 by a dif­fer­ent meth­od. Biran’s in­geni­ous ar­gu­ment in­volved the sym­plect­ic in­fla­tion con­struc­tion of Lalonde and Mc­Duff [4] com­bined with TaubesSeibergWit­ten the­ory, which he learned from an early ver­sion of Dusa’s pa­per [5]. In fact, flex­ib­il­ity of pack­ings by a suf­fi­ciently large num­ber of equal balls holds for every closed sym­plect­ic four man­i­fold, as was shown by Biran [e5] for ra­tion­al sym­plect­ic forms and in gen­er­al very re­cently by Olga Buse, Richard Hind and Em­manuel Op­shtein [e14].

The above pack­ing ob­struc­tions for k=2,3, 5,6, 7,8 balls are sharp. Dusa and Le­onid proved this in a highly non­con­struct­ive way by us­ing the Na­kaiMoishezon cri­terion in al­geb­ra­ic geo­metry. This gave rise to the prob­lem of ex­pli­cit pack­ing con­struc­tions. For k=2,3 balls they were found by Yael Kars­hon in her ap­pendix [e2] to [2]. Later on Lisa Traynor [e3] con­struc­ted op­tim­al pack­ings for 5 and 6 balls, and Ingo Wieck [e7] in his PhD thes­is writ­ten un­der the su­per­vi­sion of Hansjörg Geiges settled the re­main­ing case of 7 and 8 balls. The case of k=m2 balls is easy: use sym­plect­ic po­lar co­ordin­ates to rep­res­ent a sym­plect­ic ball as the product of the square and the sim­plex, and chop the square in­to m2 equal squares. This ar­gu­ment im­me­di­ately ex­tends to any di­men­sion 2n for k=mn.

For sym­plect­ic pack­ings of a ball by equal balls in high­er di­men­sions 2n, Gro­mov’s two ball pack­ing the­or­em still works and yields sharp an­swers for k2n balls, but the rest of the meth­ods in [2] fails for 2n6. To make pro­gress, we need to make a de­tour through pack­ings by more gen­er­al col­lec­tions of balls and by el­lips­oids.

The pa­pers [1], [2] also con­tained the first step to the fol­low­ing al­geb­ra­ic re­for­mu­la­tion of the gen­er­al ball pack­ing prob­lem (1)i=1kB4(ai)sB4(A) that in its full form was es­tab­lished by Biran, LiLi and Li–Liu: An em­bed­ding (1) ex­ists if and only if

  1. (Volume con­straint) iai2A2;

  2. (Con­straint from ex­cep­tion­al spheres) iaimiAd for every vec­tor of non­neg­at­ive in­tegers (d; m1,, mk) that solves the Di­o­phant­ine sys­tem (2)imi=3d1,imi2=d2+1 and can be re­duced to (0; 1,0,, 0) by re­peated Cre­mona moves.

Here, a Cre­mona move takes a vec­tor (d; m1,, mk) with m1mk to the vec­tor (d;m)=(d+δ;m1+δ,m2+δ,m3+δ,m4,,mk), where δ=d(m1+m2+m3), and then re­orders m.

After a break of ten years, Dusa re­turned to sym­plect­ic em­bed­dings in [6], in which she stud­ied the prob­lem E(a,b)sE(c,d), where E(a,b)={(z1,z2)C2|π|z1|2a+π|z2|2b<1} is the el­lips­oid in C2 whose pro­jec­tions to the co­ordin­ate planes are discs of area a and b. Back then is was not quite clear why this is an in­ter­est­ing prob­lem, but like many of Dusa’s works this pa­per ini­ti­ated much pro­gress on sym­plect­ic em­bed­dings. We here as­sume that the tar­get el­lips­oid is a ball, that we again com­pac­ti­fy to CP2. Giv­en an em­bed­ding φ:E(a,b)sB4CP2, one can still blow-up its im­age, namely re­move φ(E(a,b)) and col­lapse the re­main­ing bound­ary along the char­ac­ter­ist­ic fo­li­ation, but one then ob­tains an or­bi­fold in­stead of a man­i­fold. It looks dif­fi­cult to use J-holo­morph­ic curves in such a space. But Dusa went around all sin­gu­lar­it­ies by us­ing a ver­sion of the Hirzebruch–Jung res­ol­u­tion of sin­gu­lar­it­ies, and in­flated along chains of J-spheres to re­duce the prob­lem E(1,a)sB4(A) to the prob­lem (1): (3)E(1,a)sB4(A)i=1kB4(ai)sB4(A) where for ra­tion­al a the ai are giv­en by (a1,,ak)=:w(a)=(1,,10,w1,,w11,,wN,,wNN) with the weights wi>0 such that w1=a0<1, w2=11w1<w1, and so on. For in­stance, w(3)=(1, 1,1) and w(114)=(1,1,34,14,14,14). In par­tic­u­lar, E(1,k)sB4(A)if and only ifkB4(1)sB4(A). The ball pack­ing prob­lem kB4(1)B4(A) is thus in­cluded in the 1-para­met­ric prob­lem E(1,a)sB4(A)!

This lat­ter prob­lem is en­coded in the func­tion c(a):[1,)[1,) defined by c(a)=inf{A|E(1,a)sB4(A)}. For the de­scrip­tion of c(a), re­call that the Fibon­acci num­bers are re­curs­ively defined by f1=1,f0=0,fn+1=fn+fn1. De­note by gn:=f2n1 the odd-in­dex Fibon­acci num­bers, (g0,g1,g2,g3,g4,)=(1,1,2,5,13,). The se­quence γn:=gn+1/gn, whose first terms are (γ0,γ1,γ2,γ3,)=(1,2,52,135,), con­verges to τ2, where τ:=(1+5)/2 is the golden ra­tio.

The Fibonacci stairs: the graph of c(a) on [1,τ4].

Define the Fibon­acci stairs as the graph on [1,τ4] al­tern­at­ingly formed by ho­ri­zont­al seg­ments {a=γn} and slanted seg­ments that ex­tend to a line through the ori­gin and meet the pre­vi­ous ho­ri­zont­al seg­ment on the graph a of the volume con­straint (the first ho­ri­zont­al seg­ment has zero length), see the fig­ure. The co­ordin­ates of all the nonsmooth points of the Fibon­acci stairs can be writ­ten in terms of the num­bers gn.

The­or­em 2 (Fibon­acci stairs, [8]):

  1. On the in­ter­val [1,τ4] the func­tion c(a) is giv­en by the Fibon­acci stairs.

  2. On the in­ter­val [τ2,(176)2] we have c(a)=a ex­cept on nine dis­joint in­ter­vals where c is a step made from two seg­ments.

  3. c(a)=a for all a(176)2.

Thus c(a) starts with an in­fin­ite com­pletely reg­u­lar stair­case, then has a few more steps, but for a(176)2=8136 is giv­en by the volume con­straint. The­or­em 2 bet­ter ex­plains the pack­ing num­bers in the table from The­or­em 1 since c2(k)=k/pk in view of (3).

I (Fe­lix) met Dusa for the first time twenty years ago at ETH Zürich, at the very be­gin­ning of my PhD Thes­is. An year later I still did not know what to work on, but I had read a few pa­pers, among them [3], where sym­plect­ic fold­ing was in­ven­ted. In Re­mark 2.4 therein the au­thors ex­plained how this em­bed­ding meth­od yields in­ter­est­ing lower bounds for c(a), and that the de­tails will be pub­lished else­where. When Dusa vis­ited ETH again, she said that I may work on this if I like. She gave me this prob­lem just like a pebble, but for me it was a gem. My res­ults looked safe, since it looked dan­ger­ous to use J-curves in the or­bi­fold ob­tained by re­mov­ing the im­age of an el­lips­oid in CP2 and col­lapsing the bound­ary. But in 2007, shortly after I had got a per­man­ent po­s­i­tion at ULB in Brus­sels, Dusa sent me a pre­print of [6]. My thes­is had gone up in smoke for a good part! I knew from scal­ing prop­er­ties of sym­plect­ic ca­pa­cit­ies that her res­ults im­ply the graph of c(a) for a5. Dusa im­me­di­ately replied “Let’s work out c(a) to­geth­er!”. The next weeks she pa­tiently spent with ex­plain­ing to me the meth­ods. It then be­came clear that we need many com­pu­ta­tions, to see which solu­tions in (2) are rel­ev­ant. I was very happy about this, since com­pu­ta­tions I thought I could do. But Dusa com­puted about three times faster than me for sev­er­al days. So I de­cided to use a com­puter code, something she may not be able to write. But at Brus­sels there was no Math­em­at­ica avail­able (the only lan­guage I was fa­mil­i­ar with), so after some days I took a train to Zurich, re­act­iv­ated my old ac­count, and then a day later sent Dusa a few pages of ex­amples. I think it was the first and last time she was im­pressed by me, un­til she un­der­stood I had a code. We then soon guessed the right an­swer. Next Dusa gen­er­ated a fire­work of ideas and meth­ods; some failed, some worked but did not help, but some were just per­fect, and so pro­gress was steady and thriv­ing. Show­ing that c(a)=a+13 on the in­ter­val [τ4,7] was the last hurdle, on which we got stuck for sev­er­al months. One day Dusa sent me an in­tric­ate and over­whelm­ing chain of es­tim­ates es­sen­tially set­tling the prob­lem. When I asked her how she came up with this, she just wrote: “in sheer des­per­a­tion!”

In con­junc­tion with Mi­chael Hutch­ings’s ca­pa­cit­ies de­rived from his em­bed­ded con­tact ho­mo­logy, as­so­ci­at­ing with every open sub­set UR4 a se­quence of num­bers ck(U) that are mono­tone with re­spect to sym­plect­ic em­bed­dings [e9], the above meth­ods led to many oth­er res­ults on the “fine struc­ture” of sym­plect­ic ri­gid­ity in di­men­sion four. An ex­ample is Dusa’s solu­tion of a con­jec­ture by Hofer [7]: Giv­en a,b, let (Nk(a,b)) be the nonin­creas­ing se­quence ob­tained by or­der­ing the set {ma+nb|m,nN{0}}. Then (4)E(a,b)sE(c,d)Nk(a,b)Nk(c,d)for all k. Since Nk(a,b)=ck(E(a,b)), this also shows that ECH-ca­pa­cit­ies are a com­plete set of in­vari­ants for the prob­lem of em­bed­ding one four-di­men­sion­al el­lips­oid in­to an­oth­er.

More sur­pris­ingly, these meth­ods also led to pack­ing sta­bil­ity in high­er di­men­sions: Let p(B2n) be the smal­lest num­ber (or in­fin­ity) such that the ball B2n can be fully filled by k sym­plect­ic­ally em­bed­ded equal balls for every kp(B2n). We have seen earli­er that p(B4)=9. Buse and Hind ob­served in [e10] that an em­bed­ding E(a1,a2)sE(b1,b2) can be sus­pen­ded to an em­bed­ding E(a1,a2,c)sE(b1,b2,c) for any c>0. It­er­at­ing this and us­ing meth­ods from Mc­Duff–Schlenk [8], they found that p(B2n) is fi­nite for all n3. Us­ing also (4) these bounds be­come reas­on­ably low, [e11]: p(B2n)[2n,3n]for all n, and p(B6) [8,21]. Is it true that p(B6)=8? If not, a really new ob­struc­tion must be found.

The el­lips­oid em­bed­ding prob­lem in di­men­sions 6 is wide open: We in gen­er­al have no clue when E(a1,,an)sE(b1,,bn) for n3. By work of Larry Guth [e6] there is no com­bin­at­or­i­al ob­struc­tion as in (4); in fact, E(1,,)sE(3,3,). Fur­ther, the case where ai=bi= for i3 is partly un­der­stood: For a[1,τ4] one has E(1,a)×Cn2sB4(A)×Cn2E(1,a)sB4(A), that is, the Fibon­acci stairs sur­vives sta­bil­iz­a­tion [e12].

There are sev­er­al reas­ons why we know much less about sym­plect­ic em­bed­dings in di­men­sion 6:

  1. pos­it­iv­ity of in­ter­sec­tions of J-holo­morph­ic curves is an im­port­ant tool in di­men­sion four;

  2. J-holo­morph­ic curves in di­men­sion four are much bet­ter un­der­stood, since of­ten their ex­ist­ence comes from Taubes–Seiberg–Wit­ten the­ory;

  3. sym­plect­ic in­fla­tion, that of­ten can be used to show that the ob­struc­tions are sharp, can be gen­er­al­ized from four to high­er di­men­sions, but to use this for con­struct­ing sym­plect­ic pack­ings one now needs to show the ex­ist­ence of cer­tain sym­plect­ic hy­per­sur­faces, which for the time be­ing seems out of reach.

Since ECH-ca­pa­cit­ies lead to much pro­gress in di­men­sion four, one can hope that oth­er ver­sions of Flo­er ho­mo­logy will provide in­ter­est­ing em­bed­ding ob­struc­tions also in high­er di­men­sions (see ([e8], Sec­tion 1.8.1) for a dis­cus­sion). While ECH is iso­morph­ic to a Seiberg–Wit­ten Flo­er ho­mo­logy and thus in­trins­ic­ally 4-di­men­sion­al, mim­ick­ing the con­struc­tion of ECH-ca­pa­cit­ies in terms of S1-equivari­ant sym­plect­ic ho­mo­logy may be a suc­cess­ful ap­proach. First steps in this dir­ec­tion were re­cently made by Jean Gutt and Mi­chael Hutch­ings.

Fi­nally, let us men­tion that, in con­trast to com­plex pro­ject­ive spaces, for cer­tain tar­get man­i­folds pack­ings by (not ne­ces­sar­ily equal!) balls are com­pletely flex­ible: a pack­ing ex­ists provided it is not pro­hib­ited by the volume con­straint. This was es­tab­lished by Janko Latschev, Dusa Mc­Duff and Fe­lix Schlenk [9] for all 4-di­men­sion­al lin­ear tori oth­er than R4/Z4, and later by Mi­chael Entov and Misha Ver­bit­sky [e13] for all lin­ear tori of ar­bit­rary di­men­sion and for cer­tain hy­perKähler man­i­folds. The bor­der­line between ri­gid­ity and flex­ib­il­ity is still far from be­ing un­der­stood.

Works

[1] D. Mc­Duff: “Blow ups and sym­plect­ic em­bed­dings in di­men­sion 4,” To­po­logy 30 : 3 (November 1991), pp. 409–​421. MR 1113685 Zbl 0731.​53035 article

[2] D. Mc­Duff and L. Pol­ter­ovich: “Sym­plect­ic pack­ings and al­geb­ra­ic geo­metry,” In­vent. Math. 115 : 1 (1994), pp. 405–​429. With an ap­pendix by Yael Kars­hon. MR 1262938 Zbl 0833.​53028 article

[3] F. Lalonde and D. Mc­Duff: “The geo­metry of sym­plect­ic en­ergy,” Ann. Math. (2) 141 : 2 (March 1995), pp. 349–​371. MR 1324138 Zbl 0829.​53025 article

[4] F. Lalonde and D. Mc­Duff: “The clas­si­fic­a­tion of ruled sym­plect­ic 4-man­i­folds,” Math. Res. Lett. 3 : 6 (1996), pp. 769–​778. MR 1426534 Zbl 0874.​57019 article

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