# Celebratio Mathematica

## Dusa McDuff

This Honoree is in Preparation. Elements may be Missing or Nonfunctional.

### 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 $$\label{e:Bai} \bigsqcup_{i=1}^k \operatorname{B}^4(a_i) \stackrel{s}{\hookrightarrow} \operatorname{B}^4(A)$$ 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 $$\label{eq:ee} \sum_i m_i = 3d-1, \qquad \sum_i m_i^2 = d^2+1$$ 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}: $$\label{e:equiv} \operatorname{E}(1,a) \stackrel{s}{\hookrightarrow} \operatorname{B}^4(A) \quad\Longleftrightarrow\quad \bigsqcup_{i=1}^k \operatorname{B}^4(a_i) \stackrel{s}{\hookrightarrow} \operatorname{B}^4(A)$$ 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.

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 $$\label{e:Hofer} \operatorname{E}(a,b) \stackrel{s}{\hookrightarrow} \operatorname{E}(c,d) \quad\Longleftrightarrow\quad N_k(a,b) \leqslant N_k(c,d) \quad\mbox{for all } k .$$ 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