#### 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
Guillemin–Sternberg 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
Taubes–Seiberg–Witten
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
Nakai–Moishezon 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,
Li–Li and
Li–Liu: An embedding __\eqref{e:Bai}__ exists if and only if

(Volume constraint)

__\( \sum_i a_i^2 \leqslant A^2 \)__;(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.

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]):

On the interval

__\( [1,\tau^4] \)__the function__\( c(a) \)__is given by the Fibonacci stairs.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.__\( c(a)=\sqrt a \)__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 __\( \geqslant 6 \)__:

positivity of intersections of

__\( J \)__-holomorphic curves is an important tool in dimension four;__\( J \)__-holomorphic curves in dimension four are much better understood, since often their existence comes from Taubes–Seiberg–Witten theory;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.