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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 \( (M,\omega) \) is a smooth man­i­fold \( M \) to­geth­er with a closed and nonde­gen­er­ate dif­fer­en­tial two-form \( \omega \). The di­men­sion must then be even. Ex­amples are Kähler man­i­folds and co­tan­gent bundles \( T^*Q \) with their ca­non­ic­al sym­plect­ic form \[ \sum_{i=1}^n dq_i \wedge dp_i ,\] in which Hamilto­ni­an mech­an­ics takes place. The time-1 map \( \varphi \) of a Hamilto­ni­an flow on \( (M,\omega) \) pre­serves the sym­plect­ic form: \( \varphi^*\omega = \omega \). 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 \( \mathbb{R}^{2n} \) with the con­stant sym­plect­ic form \[ \omega_0 = \sum_{i=1}^n dx_i \wedge dy_i .\] It is thus already in­ter­est­ing to study sym­plect­ic map­pings between open sub­sets \( U \), \( V \subset \mathbb{R}^{2n} \). Every sym­plect­ic map­ping is a loc­al em­bed­ding; but look­ing at all sym­plect­ic map­pings is not in­ter­est­ing, for in­stance all of \( \mathbb{R}^{2n} \) 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 \( U \stackrel{s}{\hookrightarrow} V \) if there ex­ists a sym­plect­ic em­bed­ding \( U \hookrightarrow V \). In this case, \( \operatorname{Vol} (U) \leqslant \operatorname{Vol} (V) \), since sym­plect­ic dif­feo­morph­isms pre­serve the volume \[ \operatorname{Vol} (U) = \frac{1}{n!} \int_U \omega^n .\] We as­sume throughout that \( 2n \geqslant 4 \), since two-di­men­sion­al sym­plect­ic map­pings are just those that pre­serve the area and the ori­ent­a­tion. Also note that \( U \stackrel{s}{\hookrightarrow} V \) if and only if \( \lambda U \stackrel{s}{\hookrightarrow} \lambda V \) for every \( \lambda > 0 \).

De­note by \( \operatorname{B}^{2n}(a) \) the open ball in \( \mathbb{R}^{2n} \) of “area” \( a = \pi r^2 \), where \( r \) is the ra­di­us, and by \( \operatorname{Z}^{2n} \) the full cyl­in­der \[ \operatorname{B}^2(1) \times \mathbb{R}^{2n-2} \quad\text{in}\quad \mathbb{R}^2(x_1,y_1) \times \mathbb{R}^{2n-2}(x_2, y_2, \dots, x_n,y_n) .\] The start­ing point of the sym­plect­ic em­bed­ding story is Gro­mov’s non­squeez­ing the­or­em [e1]: \[ \operatorname{B}^{2n}(a) \stackrel{s}{\hookrightarrow} \operatorname{Z}^{2n} \quad\mbox{only if}\quad a \leqslant 1. \] In oth­er words, the in­clu­sion \( \operatorname{B}^{2n}(1) \subset \operatorname{Z}^{2n} \) is already “the best” sym­plect­ic em­bed­ding! There are (even lin­ear) volume pre­serving em­bed­dings of \( \operatorname{B}^{2n}(a) \) in­to \( \operatorname{Z}^{2n} \) 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 \( S^2 \times \mathbb{R}^{2n-2} \) of \( \operatorname{Z}^{2n} \).

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 \( \operatorname{B}^4(1) \) by \( k \) sym­plect­ic­ally em­bed­ded equal balls \( \operatorname{B}^4(a) \).

Let \( p_k \) be the per­cent­age of the volume of \( \operatorname{B}^4(1) \) that can be filled by \( k \) sym­plect­ic­ally em­bed­ded equal balls. Then \( p_k \) has the fol­low­ing val­ues: \[ \begin{array}{c|ccccccccc} \hline k & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & \geqslant 9 \\ \hline p_k & 1 & \dfrac 12 & \dfrac{3}{4} & 1 & \dfrac{20}{25} & \dfrac{24}{25} & \dfrac{63}{64} & \dfrac{288}{289} & 1 \\ \hline \end{array} \]

The in­equal­it­ies \( p_2 \leqslant 1/2 \) and \( p_5 \leqslant 4/5 \) 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 \( \operatorname{B}^4(1) \) by two balls, \( B_1 \) and \( B_2 \) of ra­di­us \( r \), com­pac­ti­fy \( \operatorname{B}^4(1) \) to \( \operatorname{\mathbb{C}P}^2 \) and change the com­plex struc­ture on \( \operatorname{\mathbb{C}P}^2 \) to a com­pat­ible al­most com­plex struc­ture \( J \) which is stand­ard on \( B_1 \) and \( B_2 \). Take a \( J \)-holo­morph­ic line \( L \) passing through the cen­ters of the balls. By Lel­ong’s in­equal­ity, \[ \operatorname{Area}(L \cap B_i) \geqslant \pi r^2 .\] On the oth­er hand, the total area of \( L \) equals \( \pi \), which read­ily yields \( 2\pi r^2 \leqslant \pi \) and hence \( p_2 \leqslant 1/2 \).

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 \( p_k < 1 \). This means that \( \operatorname{\mathbb{C}P}^2 \) can­not be fully packed by \( k \) balls of volume \( \operatorname{Vol}(\operatorname{\mathbb{C}P}^2)/k \). In par­tic­u­lar, if \( B \) is such a ball and \( \varphi \) is any sym­plec­to­morph­ism of \( \operatorname{\mathbb{C}P}^2 \), one of the sets \( \varphi^j(B) \), \( j=1,\dots \), \( k-1 \) 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 \( p_k < 1 \) 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 \( p_k \) 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 \[ \varphi \colon \operatorname{B}^4(a)\stackrel{s}{\hookrightarrow} \operatorname{B}^4(1) ,\] one re­moves \( \varphi (\operatorname{B}^4(a)) \) 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 \( k \leqslant 8 \). Let us re­tell Gro­mov’s proof of \( p_2 \leqslant 1/2 \) in the lan­guage of blow-ups. Giv­en a pack­ing of \( \operatorname{\mathbb{C}P}^2 \) 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 \( \pi \), while the areas of the ex­cep­tion­al di­visors \( E_1 \) and \( E_2 \) equal \( \pi r^2 \). One checks that the class \( L-E_1-E_2 \) 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 \( \pi-2\pi r^2 \) is thus pos­it­ive, and so we get \( p_2 < 1/2 \). The ad­vant­age of this ap­proach is that it gen­er­al­izes to \( k \leqslant 8 \), and the pack­ing in­equal­it­ies for \( k \leqslant 8 \) 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 \( k \geqslant 9 \) balls. In fact it was ob­served in [2] that no such con­straints ex­ist (i.e., \( p_k=1 \)) 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 \( p_k=1 \) for \( k \geqslant 10 \) 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 \( k=m^2 \) 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 \( m^2 \) equal squares. This ar­gu­ment im­me­di­ately ex­tends to any di­men­sion \( 2n \) for \( k=m^n \).

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 \( k \leqslant 2^n \) balls, but the rest of the meth­ods in [2] fails for \( 2n \geqslant 6 \). 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 \begin{equation} \label{e:Bai} \bigsqcup_{i=1}^k \operatorname{B}^4(a_i) \stackrel{s}{\hookrightarrow} \operatorname{B}^4(A) \end{equation} that in its full form was es­tab­lished by Biran, LiLi and Li–Liu: An em­bed­ding \eqref{e:Bai} ex­ists if and only if

  1. (Volume con­straint) \( \sum_i a_i^2 \leqslant A^2 \);

  2. (Con­straint from ex­cep­tion­al spheres) \( \sum_i a_i m_i \leqslant A d \) for every vec­tor of non­neg­at­ive in­tegers \( (d \); \( m_1, \dots \), \( m_k) \) that solves the Di­o­phant­ine sys­tem \begin{equation} \label{eq:ee} \sum_i m_i = 3d-1, \qquad \sum_i m_i^2 = d^2+1 \end{equation} and can be re­duced to \( (0 \); \( -1,0,\dots \), \( 0) \) by re­peated Cre­mona moves.

Here, a Cre­mona move takes a vec­tor \( (d \); \( m_1, \dots \), \( m_k) \) with \( m_1 \geqslant \dots \geqslant m_k \) to the vec­tor \[ (d^{\prime}; \mathbf{m}^{\prime}) = (d + \delta; \, m_1+\delta, m_2+\delta, m_3+\delta, m_4, \dots, m_k) ,\] where \( \delta = d - (m_1+m_2+m_3) \), and then re­orders \( \mathbf{m}^{\prime} \).

After a break of ten years, Dusa re­turned to sym­plect­ic em­bed­dings in [6], in which she stud­ied the prob­lem \[ \operatorname{E} (a,b) \stackrel{s}{\hookrightarrow} \operatorname{E}(c,d) ,\] where \[ \operatorname{E} (a,b) = \biggl\{ (z_1, z_2) \in \mathbb{C}^2 \biggm| \frac{\pi |z_1|^2}{a} + \frac{\pi |z_2|^2}{b} < 1 \biggr\} \] is the el­lips­oid in \( \mathbb{C}^2 \) 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 \( \operatorname{\mathbb{C}P}^2 \). Giv­en an em­bed­ding \[ \varphi \colon \operatorname{E}(a,b) \stackrel{s}{\hookrightarrow} \operatorname{B}^4 \subset \operatorname{\mathbb{C}P}^2 ,\] one can still blow-up its im­age, namely re­move \( \varphi (\operatorname{E}(a,b)) \) 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 \[ \operatorname{E}(1,a) \stackrel{s}{\hookrightarrow} \operatorname{B}^4(A) \] to the prob­lem \eqref{e:Bai}: \begin{equation} \label{e:equiv} \operatorname{E}(1,a) \stackrel{s}{\hookrightarrow} \operatorname{B}^4(A) \quad\Longleftrightarrow\quad \bigsqcup_{i=1}^k \operatorname{B}^4(a_i) \stackrel{s}{\hookrightarrow} \operatorname{B}^4(A) \end{equation} where for ra­tion­al \( a \) the \( a_i \) are giv­en by \[ (a_1, \dots, a_k) =: \mathbf{w} (a) = (\underbrace{1,\dots,1}_{\ell_0}, \underbrace{w_1,\dots,w_1}_{\ell_1}, \dots, \underbrace{w_N,\dots,w_N}_{\ell_N} ) \] with the weights \( w_i > 0 \) such that \( w_1 = a-\ell_0 < 1 \), \( w_2 = 1-\ell_1 w_1 < w_1 \), and so on. For in­stance, \( \mathbf{w} (3) = (1 \), \( 1,1) \) and \[ \mathbf{w} \bigl(\tfrac{11}{4}\bigr) = \bigl( 1,1, \tfrac 34, \tfrac 14, \tfrac 14, \tfrac 14 \bigr). \] In par­tic­u­lar, \[ \operatorname{E}(1,k) \stackrel{s}{\hookrightarrow} \operatorname{B}^4(A) \quad\text{if and only if}\quad \bigsqcup_k \operatorname{B}^4(1) \stackrel{s}{\hookrightarrow}\operatorname{B}^4(A) .\] The ball pack­ing prob­lem \[ \bigsqcup_k \operatorname{B}^4(1) \to \operatorname{B}^4(A) \] is thus in­cluded in the 1-para­met­ric prob­lem \[ \operatorname{E}(1,a) \stackrel{s}{\hookrightarrow} \operatorname{B}^4(A) !\]

This lat­ter prob­lem is en­coded in the func­tion \[ c(a) \colon [1,\infty) \to [1, \infty) \] defined by \[ c(a) = \inf \bigl\{ A \bigm| \operatorname{E} (1,a) \stackrel{s}{\hookrightarrow} \operatorname{B}^4(A) \bigr\}. \] For the de­scrip­tion of \( c(a) \), re­call that the Fibon­acci num­bers are re­curs­ively defined by \[ f_{-1}=1,\quad f_0=0,\quad f_{n+1} = f_n + f_{n-1} .\] De­note by \( g_n: = f_{2n-1} \) the odd-in­dex Fibon­acci num­bers, \[ (g_0, g_1, g_2, g_3, g_4,\dots ) = (1,1,2,5,13,\dots) .\] The se­quence \( \gamma_n := g_{n+1}/{g_n} \), whose first terms are \[ ( \gamma_0, \gamma_1, \gamma_2, \gamma_3, \dots ) = \bigl( 1, 2, \tfrac 52, \tfrac{13}{5}, \dots \bigr) , \] con­verges to \( \tau^2 \), where \( \tau := (1+\sqrt{5})/2 \) is the golden ra­tio.

The Fibonacci stairs: the graph of \( c(a) \) on \( [ 1,\tau^4 ] \)

Define the Fibon­acci stairs as the graph on \( [1,\tau^4] \) al­tern­at­ingly formed by ho­ri­zont­al seg­ments \( \{ a = \gamma_n \} \) and slanted seg­ments that ex­tend to a line through the ori­gin and meet the pre­vi­ous ho­ri­zont­al seg­ment on the graph \( \sqrt a \) of the volume con­straint (the first ho­ri­zont­al seg­ment has zero length), see the fig­ure be­low. The co­ordin­ates of all the nonsmooth points of the Fibon­acci stairs can be writ­ten in terms of the num­bers \( g_n . \)

The­or­em 2 (Fibon­acci stairs, [8]):

  1. On the in­ter­val \( [1,\tau^4] \) the func­tion \( c(a) \) is giv­en by the Fibon­acci stairs.

  2. On the in­ter­val \( \bigl[\tau^{2},\bigl(\frac{17}{6}\bigr)^2 \bigr] \) we have \( c(a)=\sqrt a \) ex­cept on nine dis­joint in­ter­vals where \( c \) is a step made from two seg­ments.

  3. \( c(a)=\sqrt a \) for all \( a \geqslant \bigl(\frac{17}{6}\bigr)^2 \).

Thus \( c(a) \) starts with an in­fin­ite com­pletely reg­u­lar stair­case, then has a few more steps, but for \[ a \geqslant \bigl(\tfrac{17}{6}\bigr)^2 = 8 \tfrac{1}{36} \] is giv­en by the volume con­straint. The­or­em 2 bet­ter ex­plains the pack­ing num­bers in the table from The­or­em 1 since \( c^2(k) = k/p_k \) in view of \eqref{e:equiv}.

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 \( \operatorname{\mathbb{C}P}^2 \) 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 \( a \leqslant 5 \). 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 \eqref{eq:ee} 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 \( c(a) = \frac{a+1}{3} \) on the in­ter­val \( [\tau^4,7] \) 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 \( U \subset \mathbb{R}^4 \) a se­quence of num­bers \( c_k(U) \) 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 \( (N_k(a,b)) \) be the nonin­creas­ing se­quence ob­tained by or­der­ing the set \[ \bigr\{ ma +nb \bigm| m,n \in \mathbb{N} \cup \{0\} \bigr\} .\] Then \begin{equation} \label{e:Hofer} \operatorname{E}(a,b) \stackrel{s}{\hookrightarrow} \operatorname{E}(c,d) \quad\Longleftrightarrow\quad N_k(a,b) \leqslant N_k(c,d) \quad\mbox{for all } k . \end{equation} Since \( N_k(a,b) = c_k(\operatorname{E} (a,b)) \), 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 \( p(\operatorname{B}^{2n}) \) be the smal­lest num­ber (or in­fin­ity) such that the ball \( \operatorname{B}^{2n} \) can be fully filled by \( k \) sym­plect­ic­ally em­bed­ded equal balls for every \( k \geqslant p(\operatorname{B}^{2n}) \). We have seen earli­er that \( p(\operatorname{B}^4) = 9 \). Buse and Hind ob­served in [e10] that an em­bed­ding \[ \operatorname{E} (a_1,a_2) \stackrel{s}{\hookrightarrow} \operatorname{E}(b_1,b_2) \] can be sus­pen­ded to an em­bed­ding \[ \operatorname{E} (a_1,a_2,c) \stackrel{s}{\hookrightarrow} \operatorname{E}(b_1,b_2,c) \] for any \( c > 0 \). It­er­at­ing this and us­ing meth­ods from Mc­Duff–Schlenk, [8] they found that \( p(\operatorname{B}^{2n}) \) is fi­nite for all \( n \geqslant 3 \). Us­ing also \eqref{e:Hofer} these bounds be­come reas­on­ably low, [e11]: \[ p(\operatorname{B}^{2n}) \in [2^n, 3^n] \quad\text{for all }n , \] and \( p(\operatorname{B}^{6}) \in \) \( [8, 21] \). Is it true that \( p(\operatorname{B}^6) =8 \)? If not, a really new ob­struc­tion must be found.

The el­lips­oid em­bed­ding prob­lem in di­men­sions \( \geqslant 6 \) is wide open: We in gen­er­al have no clue when \[ \operatorname{E}(a_1,\dots,a_n) \stackrel{s}{\hookrightarrow} \operatorname{E} (b_1,\dots,b_n) \] for \( n \geqslant 3 \). By work of Larry Guth [e6] there is no com­bin­at­or­i­al ob­struc­tion as in \eqref{e:Hofer}; in fact, \[ \operatorname{E} (1,\infty,\infty) \stackrel{s}{\hookrightarrow}\operatorname{E} (3,3,\infty) .\] Fur­ther, the case where \( a_i = b_i = \infty \) for \( i \geqslant 3 \) is partly un­der­stood: For \( a \in [1,\tau^4] \) one has \[ \operatorname{E}(1,a) \times \mathbb{C}^{n-2} \,\stackrel{s}{\hookrightarrow}\, \operatorname{B}^4(A) \times \mathbb{C}^{n-2} \quad\Longleftrightarrow\quad \operatorname{E}(1,a) \stackrel{s}{\hookrightarrow} \operatorname{B}^4(A) , \] that is, the Fibon­acci stairs sur­vives sta­bil­iz­a­tion [e12].

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

  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 \( S^1 \)-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 \( \mathbb{R}^4/\mathbb{Z}^4 \), 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

[5] D. Mc­Duff: “From sym­plect­ic de­form­a­tion to iso­topy,” pp. 85–​99 in Top­ics in sym­plect­ic 4-man­i­folds (Irvine, CA, 28–30 March 1996). Edi­ted by R. J. Stern. First In­ter­na­tion­al Press Lec­ture Series 1. In­ter­na­tion­al Press (Cam­bridge, MA), 1998. MR 1635697 Zbl 0928.​57018 ArXiv dg-​ga/​9606004 incollection

[6] D. Mc­Duff: “Sym­plect­ic em­bed­dings of 4-di­men­sion­al el­lips­oids,” J. To­pol. 2 : 1 (2009), pp. 1–​22. MR 2499436 Zbl 1166.​53051 article

[7] D. Mc­Duff: “The Hofer con­jec­ture on em­bed­ding sym­plect­ic el­lips­oids,” J. Diff. Geom. 88 : 3 (2011), pp. 519–​532. MR 2844441 Zbl 1239.​53109 article

[8] D. Mc­Duff and F. Schlenk: “The em­bed­ding ca­pa­city of 4-di­men­sion­al sym­plect­ic el­lips­oids,” Ann. Math. (2) 175 : 3 (2012), pp. 1191–​1282. MR 2912705 Zbl 1254.​53111 article

[9] J. Latschev, D. Mc­Duff, and F. Schlenk: “The Gro­mov width of 4-di­men­sion­al tori,” Geom. To­pol. 17 : 5 (2013), pp. 2813–​2853. MR 3190299 Zbl 1277.​57024 article