ON CERTAIN QUANTIFICATIONS OF GROMOV’S NON-SQUEEZING THEOREM

. Let R > 1 and let B be the Euclidean 4-ball of radius R with a closed subset E removed. Suppose that B embeds symplectically into the unit cylinder D 2 × R 2 . By Gromov’s non-squeezing theorem, E must be non-empty. We prove that the Minkowski dimension of E is at least 2, and we exhibit an explicit example showing that this result is optimal at least for R ≤ √ 2. In an appendix by Jo´e Brendel, it is shown that the lower bound is optimal for R < √ 3. We also discuss the minimum volume of E in the case that the symplectic embedding extends, with bounded Lipschitz constant, to the entire ball.

Geometry & Topology msp Volume 28 (2024) On certain quantifications of Gromov's nonsqueezing theorem 1 Introduction Consider R 2n with its standard symplectic structure !D P dx i ^dy i .A prototypical question in symplectic geometry is to ask whether one domain of R 2n symplectically embeds into another (ie via an embedding ˆwith ˆ !D !).At the very least, a symplectic embedding preserves the standard volume form .1=n!/! n .However, there is more rigidity in symplectic geometry than just volume.We recall the most famous result certifying this bold claim.Let B 2n .R 2 / R 2n be the open ball of radius R, and let Z 2n .r 2 / D B 2 .r 2 / R 2n 2 R 2n be the open cylinder of radius r .Theorem 1.1 (Gromov's nonsqueezing theorem [11]) A symplectic embedding of B 2n .R 2 / into Z 2n .r 2 / exists if and only if R Ä r .
Our goal in this paper is to try to quantify the failure of B 2n .R 2 / to symplectically embed into Z 2n .r 2 /, when R > r , via the following motivating question: Motivating question How much do we need to remove from B 2n .R 2 / so that it embeds symplectically into Z 2n .r 2 /?
Let us start by discussing the constructive side.Observe that if we remove a union of codimension-one affine hyperplanes along a sufficiently fine grid, we end up with many connected components, each of which embeds into Z./ by translations.Interestingly, at least in a certain range of R, one can do better and find a two-dimensional submanifold E whose complement embeds into Z./.
To explain this result we need to introduce some notation.Let us consider the Lagrangian disk Let us also define Ᏹ. ; 4 / R 4 to be the open ellipsoid ˚.x 1 ; y 1 ; x 2 ; y 2 / j x 2 1 C y In particular, removing a Lagrangian plane from B.2 / halves its Gromov capacity.Our proof of Theorem 1.3 has a great deal in common with Section 3 of Oakley and Usher's beautiful paper [25], where the same geometries are used for a different purpose.In fact, we show in Section 4 how the projective space CP 2 is symplectomorphic to the boundary reduction of the unit cotangent bundle D RP 2 by using the explicit map of [25,Lemma 3.1].Theorem 1.3 can also be derived from the proof of Biran's general decomposition theorem [2, Theorem 1.A; Example 3.1.2],and Opshtein [26,Lemma 3.1].We use the latter in our argument as well.
On the obstructive side, we show that removing a two-dimensional subset as in Theorem 1.3 is the best one can do in general: Theorem 1.4 Let E be a closed subset of R 4 and let R > 1. Suppose that B. R 2 / n E symplectically embeds into the cylinder Z. /.Then the lower Minkowski dimension of E is at least 2.
For the proof of Theorem 1.4, we build on Gromov's original nonsqueezing argument by adding a key new ingredient: the waist inequality, which was also introduced by Gromov [12]; see also Memarian [23].Crucially, we require the sharp version due to Akopyan and Karasev [1], as well as the Heintze-Karcher [15] bound on the volumes of tubes around minimal surfaces, in place of the monotonicity inequality for minimal surfaces.
Remark 1.5 Let R sup 2 .1; 1 be the supremum of the radii R such that there is a codimension-two subset of B. R 2 / whose complement can symplectically embed into Z./.In the first version of this article we had conjectured that R sup should be equal to p 2. However shortly after its appearance, Joé Brendel informed us that using a construction inspired by Hacking and Prokhorov [13], he can prove R sup p 3. As a consequence, we changed our conjecture to a question; see Section 6.3.His construction appears in the appendix.
We further remark that in Theorem 1.3, we remove a Lagrangian plane.In the construction of Joé Brendel in the appendix realizing R sup p 3, he removes a union of Lagrangians together with a symplectic divisor.In higher dimensions (see eg Section 6.1), this distinction could be interesting.

The Lipschitz problem
Recall that for fixed L > 1, we are asking for the smallest volume of the region E.ˆ/ WD B. R 2 / n ˆ 1 .Z. // D ˆ 1 .R 4  n Z. // over all symplectic embeddings ˆW B. R 2 / ,! R 4 of Lipschitz constant bounded above by L. (We note that although we use the letter L for both the Lipschitz constant as well as for the Lagrangian disk of Theorem 1.3, there will be no confusion given the context.) On the obstructive side, we obtain the following as a corollary of the proof of the obstructive bound for the Minkowski question (Theorem 1.4): Then there exists a constant c D c.R/ > 0 such that for all constants L and all symplectic embeddings ˆW B. R 2 / ,! R 4 with Lipschitz constant at most L, we have It is worth noting that one may use the standard nonsqueezing theorem alone to find a weaker quantitative obstructive bound of c=L 3 as follows.Suppose we had an L-Lipschitz symplectic embedding Then by Gromov's nonsqueezing theorem and the Lipschitz condition, one can check that there is a ball of radius of order 1=L embedded inside E. /.Hence, Vol.E.// & 1 L 4 : With a little more effort, one may find order L many disjoint such balls inside E. /, yielding the obstructive bound c=L 3 .However, jumping from c=L 3 to our obstructive bound of c=L 2 appears to require a new tool, which in our case is Gromov's waist inequality.
On the constructive side, we adapt Katok's ideas in [16] to prove the following: One would obviously like to push the obstructive and constructive bounds together.

Organization of the paper
We start by recalling some definitions and known theorems in Section 2, which are then applied in Section 3 to prove an obstructive bound implying Theorem 1.4.In Section 4 we construct the symplectomorphism of Theorem 1.3.The Lipschitz problem, including Theorems 1.6 and 1.7, are discussed in Section 5. We also list several related questions in the final Section 6.In the appendix, written by Joé Brendel, the construction mentioned in Remark 1.5 appears.
Let S R n be any bounded subset.Let N t .S/ denote the open t-neighborhood of S with respect to the standard metric.If † is a compact submanifold (possibly with boundary), let V t .†/ be the exponential t-tube of †, ie the image by the normal exponential map of the open t-neighborhood of the zero-section in the normal bundle of † (which is endowed with the natural metric).
We denote by Vol n the Euclidean n-volume of a set and set Note that when n is a natural number, ˛n D Vol n .B n / is precisely the Euclidean volume of the unit n-dimensional ball B n R n .For s 0, the s-dimensional lower Minkowski content of S is defined as Vol n .N t .S// ˛n s t n s : Note that the normalization is chosen so that if The lower Minkowski dimension of S is defined as There are similar notions of upper Minkowski dimension and upper Minkowski content, which we will not need in this paper since a lower bound of the lower Minkowski content implies by definition the same lower bound for the upper Minkowski content.There are also equivalent definitions using ball packings.Replacing S by its closure x S does not change the Minkowski upper/lower dimensions.

Waist inequalities
The waist inequality for round spheres proved by Gromov [12] and with more details by Memarian [23] was extended to the case of maps from Euclidean balls by Akopyan and Karasev [ It will be useful for our application that the above estimate is uniform in f .This uniform estimate is indeed implied by the proof of [1, Theorem 1] as they compare the t-neighborhood of a fiber to the t-neighborhood of an equatorial unit sphere S nC1 k S nC1 (cf the second-last equation of their proof, with correct normalization).The latter is independent of f , and by explicit calculation one may verify that actually h n;k 2 O.t kC2 /.
We remark that the waist inequality for spheres [12; 23] describes a stronger property than the above statement, since it gives optimal bounds on all (not just small) neighborhoods of the big fiber.

Tubes around minimal submanifolds
The Heintze-Karcher inequality [15] estimates the volume of tubes around compact submanifolds.We need the case of minimal submanifolds in Euclidean space (covered by [15,Theorem 2.3] with ı D 0 and Remark 2 on page 453 in [15]), which may be stated as follows: Theorem 2.2 (Heintze-Karcher inequality) For any positive integers n and k, and any smooth compact k-dimensional minimal submanifold † k R n with boundary, for t > 0 we have Theorem 2.2 is again a uniform estimate on the exponential t-neighborhood; one may compare it to the statement that the (upper) Minkowski k-content of a closed submanifold † k R n is Vol k .†/.The point of Theorem 2.2 is that the constant in this estimate does not depend on the minimal k-submanifold † or t.The main ingredient for its proof is an estimate for the Jacobian determinant of the normal exponential, which in Euclidean space is 1 to lowest order.The following term is controlled by the mean curvature H , which one may expect from the interpretation of mean curvature as the first variation of (k-)area.Consequently, for a general compact submanifold †, (2-2) will have an error term of order c.max † jH j/t n kC1 .Proof The following argument is standard in the symplectic community.As mentioned before the statement of the proposition, the ideas are due to Gromov [11], though more thorough analytic details may be found elsewhere; see eg [22].

Gromov foliation and maps into the cylinder
We define A WD r 2 for brevity.Since B. R 2 / n U has compact closure in B. R 2 / n E, the image of B. R 2 / n U under lands in B 2 .r 2 0 / OE K; K 2 for some large constant K and 0 < r 0 < r (possibly depending upon U ).Let S 2 .A/ denote the 2-sphere with standard symplectic form scaled to have total area A, and let T 2 K D .R=4KZ/ 2 be the 2-torus with symplectic form induced by the standard form on R 2 .Then we have a symplectic embedding B 2 .r 2 0 / OE K; K 2 S 2 .A/ T 2 K , and upon composing with , we arrive at a symplectic embedding, also denoted by (by abuse of notation), K : Let J 0 denote the standard complex structure on B. R 2 /, and let J 1 denote the standard (split) complex structure on S 2 .A/ T 2 K .We pick a special almost complex structure J on S 2 .A/ T 2 K which incorporates by requiring that it satisfies the following three properties: On the image of , J D .J 0 /.J D J 1 in a neighborhood of f1g T 2 K .Everywhere, J is compatible with the symplectic form.
By [22,Proposition 9.4.4], which encompasses standard 4-dimensional techniques, the evaluation map evW ᏹ 0;1 .ˇ;J / !S 2 .A/ T 2 K is a diffeomorphism, where ᏹ 0;1 .ˇ;J / is the moduli space of J -holomorphic spheres with one marked point and in the class ˇ.Meanwhile, the map which forgets the marked point W ᏹ 0;1 .ˇ;J / !ᏹ 0;0 .ˇ;J / is a smooth fibration, with fibers diffeomorphic to S 2 .By positivity of intersections (see [22,Theorem 2.6.3]), each fiber of , which is a J -holomorphic sphere, intersects f1g T 2 K once and transversely, so by the implicit function theorem we have a canonical diffeomorphism g of T 2 K with ᏹ 0;0 .ˇ;J /.We therefore obtain a map hW S . This is the function f we desired in the statement of the proposition, and we must now check it satisfies both of the desired properties.
The fact that f 1 .y/ is a complex submanifold is simply because it is by definition a subset of a Jholomorphic sphere, and J is chosen to equal .J 0 / on the image of .Finally, the area bound comes from the fact that the area of f 1 .y/ is at most the symplectic area of the corresponding sphere (the one passing through .1;OEy/), which is just A since symplectic area is purely homological.The main idea of our proof is that the waist inequality guarantees a fiber with large-volume neighborhoods, but by the area bound (and the structure of tubes) this can only happen if the fiber accumulates near the exceptional set.Accordingly, a key component is the following covering claim.

A quantitative obstruction to partial symplectic embeddings
Claim 1 Let f ı;t be the restriction of f ı to the ball B. .R 2t / 2 /.Then Indeed, by definition of † ı and the supposition ı < t we have that Now given any submanifold †, its t-neighborhood N t .†/ is always contained in the union of the tube V t .†/ and the t-neighborhood N t .@†/ of its boundary.So since @ † ı @U ı , we have that By the triangle inequality it follows that But by definition of U ı , we have N t .@Uı / N t .@B. .R ı/ for instance, concludes the proof.
An immediate corollary is a lower bound for the lower Minkowski dimension.We will see that this bound is sharp in the next section, at least for radii R which are not too large.
In particular, the lower Minkowski dimension dim ᏹ .E/ is at least 2.

Squeezing the complement of a Lagrangian plane
In this section it will be more convenient to use complex coordinates for the standard symplectic R 4 .Therefore we consider C 2 with its standard Kähler structure, ie if x and y are the complex coordinates, then the symplectic form is i 2 .dx^d x x C dy ^d x y/: Let us recall the main objects in the statement of Theorem 1.3 in complex notation for convenience.Let B.2 / C 2 be the open ball of radius p 2 centered at the origin.Let R 2 C 2 be the real part and define We also define Ᏹ. ; 4 / C 2 to be the open ellipsoid and let Let us introduce the main actors in the proof.Let CP 2 .2/ be the symplectic manifold obtained from coisotropic reduction of the sphere S 5 of radius p 2 in C 3 .Denoting the complex coordinates on C 3 by z 1 , z 2 and z 3 , we are using the symplectic structure i 2 .dz 1 ^dx z 1 C dz 2 ^dx z 2 C dz 3 ^dx z 3 / on C 3 .There is a canonical identification of CP 2 .2/ with the complex manifold CP 2 WD Gr C .1; 3/, whose homogeneous coordinates we will denote by OEz 1 W z 2 W z 3 .Of course, CP 2 .2/ is nothing but CP 2 equipped with the Fubini-Study symplectic form scaled so that a complex line (eg fz 1 D 0g CP 2 ) has area 2 .
Let us also specify some submanifolds of CP 2 .2/ using its canonical identification with CP 2 .
L RP is the real part Consider RP as a smooth manifold in the standard way.Let be the tautological one-form on T RP and V be the Liouville vector field, which is a vertical vector field equal to the Euler vector field in each fiber (which is defined on any vector space independently of a basis).We have where !D d .We denote the zero-section submanifold on T RP by Z RP .
The Riemannian metric on S 2 obtained from its embedding as the round sphere of radius 1 in R 3 induces a metric on its quotient by the antipodal map, which is canonically diffeomorphic to RP .We call this the round metric on RP and denote it by g RP .We have the diffeomorphism g ] RP W T RP !T RP , which is in particular linear on the fibers.We can transport the function K W T RP !R, given by lengths of tangent vectors to On T RP we have the geodesic flow; under the identification by g ] RP this becomes the Hamiltonian flow of the function 1  2 .K ] / 2 .The normalized geodesic flow on T RP n RP becomes the Hamiltonian flow of K ] .Let us call these the geodesic flow and normalized geodesic flow on T RP .Note that the normalized geodesic flow on T RP nZ RP is a -periodic action of R.
Any unparametrized oriented geodesic circle in RP with its round metric defines a symplectic submanifold C in T RP with boundary on Z RP by taking points .q;p/ such that q 2 and p D g q .v;/, where v is nonnegatively tangent to .Let us denote by the geodesic circle with opposite orientation.Clearly, C and C intersect along Z RP and form a symplectic submanifold of T RP , which is diffeomorphic to a cylinder.
Let D RP T RP be the closed unit-disk bundle, which is given by the subset K ] Ä 1.Also let U RP D .K ] / 1 .1/be the unit-sphere bundle, ie the boundary of D RP , with its induced contact structure Â WD Ã .
The symplectic reduction U RP =S 1 is a two-sphere equipped with a canonical symplectic form.Let us call this symplectic manifold .Q; !Q /.The points of Q are canonically identified with unparametrized oriented geodesic circles in round RP .
Let us denote the boundary reduction symplectic manifold of D RP by D RP 2 (see Definition 3.9 of [30]; also note the interpretation as one half of a symplectic cut [18]).Let 0 be the oriented unparametrized geodesic on RP which corresponds to the quotient of the horizontal great circle in S 2 R 3 oriented as the boundary of the lower hemisphere.We define S WD S 0 .Proposition 4.1 There is a symplectomorphism D RP 2 !CP 2 .2/ with the following properties: The proof of this proposition is postponed to Section 4.1.Let us continue with an immediate corollary.
Note that V K ] D K ] on T RP nZ RP , which means that K ] is an exponentiated Liouville coordinate for V on T RP nZ RP .Hence, we obtain a Liouville isomorphism where K ] is matched with the function r .The Hamiltonian vector field X r gives the r -translation invariant Reeb vector field (using r D 1) on the contact levels.Finally, observe that C \ .T RP nZ RP /'s are obtained as the traces of the Reeb orbits on U RP under the Liouville flow.
In particular, we have a foliation of D RP n Z RP is symplectomorphic to an area-standard symplectic disk bundle of .Q; !Q / in the sense of Biran [2, Section 2.1] in such a way that D is sent to the fiber over for every 4 The cohomology class of !Q = is integral and it admits a unique lift to H 2 .Q; Z/.What we mean by an area-standard symplectic disk bundle of .Q; !Q / is an area-1 standard symplectic disk bundle of .Q; !Q = / with its symplectic form multiplied by .
Proof The symplectomorphism D RP nZ RP ' U RP .0; 1 r induces a symplectomorphism of the boundary reductions of both sides.We note that on U RP .0;1/, where we define z Â D pr Â for clarity.
Here we use the maps in the following commutative diagram, where Pr and pr are the obvious projections, and U RP 2 !Q is the symplectic reduction map: Notice that the map U RP !Q has the structure of a principal U.1/ D R=Z bundle structure using the Reeb flow of the contact form Â= , and the associated complex line bundle ᏸ is precisely the fiberwise blow-down of U RP OE0; 1/ with respect to its canonical projection Pr to Q.The integral Chern class of this complex line bundle is !Q = (using that H .Q; Z/ has no torsion) and a transgression 1-form is given by the pull-back of Â= by Pr.By definition, the open unit disk bundle < 1 inside ᏸ, endowed with the symplectic form is an area-1 standard symplectic disk bundle of .Q; !Q = /.Therefore, if we multiply this form by , we obtain an area-standard symplectic disk bundle of .Q; !Q /.
Now consider the following commutative diagram: Here the left map is the restriction of the fiberwise blowdown map and the right map is the boundary reduction map.Here we use a symplectomorphism between Q n f 0 g and the two-dimensional open ellipsoid of area 4 which sends 0 to the origin, so that D 0 is sent to Ꮿ by this symplectomorphism.

Proof of Proposition 4.1
We will freely use the canonical identification of CP n .2/ with .C nC1 n f0g/=C .Note that the homogenous coordinates OEz 1 W W z nC1 denote the class OEz 1 e 1 C Cz nC1 e nC1 , where e i is the standard basis of R nC1 and its complexification C nC1 .We also realize T RP n as f.q; p/ 2 S n R nC1 j hq; pi D 0g=f˙1g; where S n R nC1 is the unit sphere.It is a straightforward computation that the standard symplectic form on T RP n descends from the restriction of P nC1 iD1 dp i ^dq i on R 2nC2 under this identification.Note also that K ] .OEq; p/ D jpj away from the zero-section.
In [25,  To do so, it suffices to show that x ˆis smooth.Indeed, if x ˆis smooth, then by continuity it follows that x ˆpreserves the symplectic form.This in particular shows that x ˆis an immersion, and hence a diffeomorphism that preserves the symplectic forms.
The following point is crucial.The canonical (linear) action of the group G D SO.nC1/ on R nC1 induces an action on D RP n (and in turn on D RP n ) and C nC1 (and in turn on CP n .2/).It is clear from the definition (4-4) that x ˆis G-equivariant with respect to these actions.
We first prove smoothness in the case n D 1. Equip CP 1 .2/ with the induced Riemannian metric (the so called Fubini-Study metric), which makes it isometric to a round S 2 .We will use the fact that the image of a linear Lagrangian subspace in C 2 n f0g (with standard Kähler structure) under the canonical projection to CP 1 .2/ is a geodesic circle.We will denote CP 1 .2/ by CP 1 for brevity.
Note that Z RP 1 is sent to the real part L RP 1 CP 1 under x ˆ.Moreover, D RP 1 nint.D RP 1 / consists of two points which map to OE1 W ˙i .Finally, notice that the images of cotangent fibers (that is, line segments of constant q) are sent to geodesic segments connecting OE1 W i and OE1 W i .It is easy to see that these geodesics are orthogonal to the geodesic circle L RP 1 .Also recall that x ˆis SO.2/-equivariant as explained above.
Let D R 2 be the unit disk jxj 2 C jyj 2 Ä 1 with the symplectic form dx ^dy.It is well-known that there is a symplectic embedding D ,! CP1 which sends the origin to OE1 W i , the unit circle to L RP 1 , radial rays to the geodesic segments that connect L RP 1 and OE1 W i , in a way that preserves angles at the intersections, and finally, the disks centered at the origin to balls around OE1 W i (in the Fubini-Study metric).
We now consider the induced continuous map …W .D RP 1 / C ! D defined by the commutative diagram Let .; / denote polar coordinates on Dnf0g.We deduce, from our discussion thus far, the following facts: There exists a constant c such that … D 2Â C c.
There exists a function hW OE0; 1/ !.0; 1 such that These facts imply that h satisfies the differential equation h 0 .x/h.x/ D 1 2 , which, with the initial condition h.0/ D 1, has the unique solution h.x/ D p 1 x. 1 We thus observe that the map … W .D RP 1 / C ! D is a model for the boundary reduction of .D RP 1 / C at U RP 1 ; see [30, equation (3.1)].This proves that the map x ˆis smooth at both points of D RP 1 n int.D RP 1 / as we can repeat the same argument on the other half of D RP 1 .
We now move on to the case n > 1.We start with a preliminary lemma.Lemma 4.7 Let G be a Lie group acting smoothly on M and N. Suppose that S is a smooth submanifold of M and that the multiplication map G S !M is a surjective submersion.If W M !N is a G-equivariant map and j S is smooth , then is also smooth.
Proof We have the following commutative diagram in which each map is known to be smooth except the bottom map: Since the left map is smooth surjective submersion, it has local smooth sections M U ! G S. The commutativity then implies that is smooth.
Note that the orbits of G D SO.nC1/ on D RP n are the submanifolds of constant jpj.Fix any unoriented geodesic circle in RP n with its round metric g.We obtain an embedding of T RP 1 in T RP n by taking points .q;p/ such that q 2 and p D g q .v;/, where v is tangent to .This restricts to a smooth embedding of D RP x and that the focal set of RP n is precisely FQ n .This yields an alternative construction of ˆas well as its extension x ˆ.

Lipschitz symplectic embeddings of balls
In this section, we aim to prove Theorems 1.6 and 1.7 from the introduction.To begin, we introduce some slightly more general notation, in which we also vary the radius of the cylinder.Suppose that ˆW B. R 2 / !R 4 is a symplectic embedding with Lipschitz constant L > 0. Then we may set Hence, we lose no information by restricting to the case r D 1.

Remark 5.2
The proof demonstrates that we may take c.R/ R 4 as R grows large.
On the other hand, we wish to prove the constructive bound, in which we must find an embedding Proof of Theorem 1.7 The fact that Vol 4 .E.ˆ// can be made arbitrarily small if there is no restriction on the Lipschitz constant is exemplified by ideas of Katok [16].Our proof is a quantitative refinement obtained using constructions which appear in symplectic folding.The basic idea is to break the ball B. R 2 / into a number of cubes and Hamiltonian isotope each of these cubes into Z./, where the cubes are separated by walls of width 1=L.To begin, we make three simplifications.First, we replace the domain B. R 2 / with the cube K.R/ D OE R; R 4 , as the volume defect can only increase in size.Second, we replace K.R/ with the rectangular prism K 0 .R/ D .OE0; 4R 2 OE0; 1/ .OE0; 4R 2 OE0; 1/; where the parentheses indicate a symplectic splitting.Explicitly, the factors refer to the coordinates x 1 , y 1 , x 2 , y 2 , with symplectic form ! D dx 1 ^dy 1 C dx 2 ^dy 2 .Notice that the natural symplectomorphism between K.R/ and K 0 .R/ has Lipschitz constant 2R, and in particular, the effect of replacing K.R/ with K 0 .R/ only affects our proposition by a factor of R which gets absorbed into the constant C .And finally, we allow the Lipschitz constant to be O R .L/, by which we mean it is bounded by AL, where A is a constant which again depends upon R; we arrive at the proposition as stated by absorbing this constant in C .Consider now each symplectic factor OE0; 4R 2 OE0; 1 R 2 of K 0 .R/.For i 2 Z with 0 Ä i Ä 4R 2 , let X i be the region in this rectangle with Our goal is to fit each X i X j K 0 .R/ into Z./ -indeed, the complement of the union of these regions has volume ‚.R 4 =L/; the R 4 factor gets absorbed by the constant C .
To begin, there is an area-preserving map of the rectangle OE0; 4R 2 OE0; 1 into R 2 of Lipschitz constant at most O.L/ which translates X i in the x-direction by i .That is, X 0 stays fixed, X 1 gets shifted to the right Figure 1: We stretch the region between X i and X iC1 in an area-preserving way so that the images Y i and Y iC1 are separated by distance just over 1.
by 1, and the region between them gets stretched like taffy in an area-preserving way to accommodate for this shift.See Figure 1.Let Y i be the image of X i under this map.Then Y i and Y iC1 are separated by distance 1 C 1=L, with Explicitly, a model for the taffy-stretching map is given as W OE0; 1=L OE0; The first two conditions imply that the constructed stretching map glues to the rigid translations of the X i in a C 1 manner.The latter two conditions imply the desired Lipschitz constant bound of O.L/, since with each entry in this matrix of order O.L/.
An example of such a desired function f may be given in the form 2 f .x/D Z x 0 1 1 Cg.y/ dy; 2 There are of course many such choices for function f , and we do not claim to make an optimal choice.In an earlier draft, we claimed that the Lipschitz constant of the stretching map could be made less than 2L.We only need that the Lipschitz constant is O.L/ for our purposes, and it is unclear whether 2L can be achieved.We thank the reviewer for pointing out this lack of clarity.
where C is a constant dependent upon L and g (as we will explain soon), and where g W OE0; 1=L !OE0; 1=4L is a smooth function which is C 0 -close to the continuous function We may take g so that g.x/ Á 0 identically near the endpoints x D 0 and x D 1=L, and such that jg 0 .x/jÄ 1 C for any chosen > 0. The constant C is chosen so that f .1=L/D 1 C 1=L, ie such that We claim that such a constant C exists.Notice that the value of the integral I.C /, as a function of C , is continuous and monotonically nondecreasing on the interval OE0; 1= supfg.x/g/,with where the limit on the right follows because g attains its maximum value on the interior of the interval OE0; 1=L, and because g is smooth, we have g 0 D 0 at this maximum value.The existence of C now follows from the intermediate value theorem, and monotonicity implies uniqueness.We notice that because supfg.x/g1=4L and C < 1= supfg.x/g,we have that C 2 O.L/.
With these choices, all of the conditions on f are now met, so long as we take a close enough approximation g of g 0 (where the closeness depends upon L).The first bullet point follows because g.x/ Á 0 near the endpoints.The second follows because we chose C accordingly.The fourth is guaranteed since and we have constructed g so that jg 0 .x/jÄ 1 C and C 2 O.L/.
That leaves the third bullet point, which we now verify.The fact that f 0 .x/ 1 is clear, so it suffices to check f 0 .x/ 2 O.L/.Solving for C 0 using g 0 , we find Notice that the value of C depends continuously on the function g (in the C 0 -topology), so for any > 0 there is a choice of approximation g so that supff 0 .x/gD sup Hence, f 0 .x/ 2 O.L/, as required.
Applying the stretching map to each symplectic factor, each region X i X j is sent to Y i Y j .It suffices now to find a symplectomorphism of R 4 of Lipschitz constant O R .1/so that each Y i Y j has image in the cylinder Z. /, since in such a case, we compose with our stretching map, and each X i X j under this composition lands in Z. /, where the composition map has total Lipschitz constant O R .L/.We construct this by sliding each Y i Y j in two steps.We begin by separating the y 2 -coordinates of the various blocks based on their x 1 -coordinates.That is, we translate Y i Y j in the y 2 -direction by 2i units, in other words so that it gets translated to Explicitly, we use a Hamiltonian of the form H D .x 1 /x 2 , where is a step function with the properties The corresponding Hamiltonian vector field X H is of the form 2i @ y 2 when 2i Ä x 1 Ä 2i C 1, and hence translates Y i Y j as desired.Explicitly, the time-1 Hamiltonian flow is with derivative All terms are O.1/ except for 00 .x 1 /x 2 , because x 2 can grow large.But on the image of our cube after the taffy-stretching step, x 2 is at most 4R 2 , and so the relevant Lipschitz constant of this sliding step is O.R 2 /.
A similar construction, using a Hamiltonian of the form .y 2 /y 1 for the same function , allows us to then take each of these new blocks and translate them in the x 1 -direction.After we complete both of these steps, the image of A final translation simultaneously in the x 1 and y 1 coordinates by 1 2 lands each of these blocks in the cylinder Z. /, concluding the proof.
Remark 5.3 As in Remark 5.1, we may also vary the radius of the cylinder r , but where the new constant is r 4 C.R=r /.
Remark 5. 4 The construction as presented has C.R/ R 9 .Indeed, our volume defect came with a factor of R 4 , but the Lipschitz constant has an extra factor of R 5 .The first factor of R appearing in the Lipschitz constant came from replacing K.R/ with K 0 .R/.An extra two factors of R 2 came from our slide moves.One can optimize a little by performing a single diagonal slide move instead of two separate orthogonal slide moves.Hence, in the end, we may take C.R/ R 7 .We suspect this is far from optimal, though decreasing the exponent appears to require a new idea.
6 Further questions

Minkowski dimension problem in higher dimensions
In this section we pose the simplest Minkowski dimension question that one could ask in dimensions higher than four, and make a couple of remarks about it.Let us assume n > 2 throughout this section.
Question 1 What is the smallest d 2 R such that for some A > , there exists a closed subset E B 2n .A/ of Minkowski dimension d such that B 2n .A/ n E symplectically embeds into Z 2n ./? Assume that for some A > and a closed subset E B 2n .A/ of Minkowski dimension d , B 2n .A/ n E symplectically embeds into Z 2n ./.We find it plausible that a version of our obstructive argument would still give d 2, even though we do not have a proof of this.
The argument in Proposition 2.3 suggests that the problem is related to the question of how the 2-width of a round ball changes after the removal of a closed subset.One can explicitly see that for E being the intersection of B 2n .A/ with a linear Lagrangian subspace L of R 2n , an n-dimensional submanifold, there is a (holomorphic!)sweepout of B 2n .A/ n E with width 1  2 A. Namely, we take the foliation by half-disks that are the connected components of the intersections with B 2n .A/ n E of affine complex planes that intersect L nontransversely.

Capacity after removing a linear plane
Throughout this section let B WD B 4 .2/ C 2 , where C 2 is equipped with its standard Kähler structure.Let us denote the complex coordinates by x and y.
Let us denote by c Gr the Gromov width, which is a capacity defined on any symplectic manifold Y 2n as the supremum of r 2 , where B 2n .r 2 / symplectically embeds into Y .Definition 6.1 Let V C 2 be a real subspace of dimension 2. We define the symplecticity of V as j! st .e 1 ; e 2 /j, where e 1 ; e 2 is any orthonormal basis of T 0 V .
Notice that V is a Lagrangian plane if and only if its symplecticity is 0. On the other extreme, V is a complex plane if and only if its symplecticity is 1.
The symplecticity defines a surjective continuous function Gr R .

Minkowski dimension problem for large R
Let R sup 2 .1; 1 be the supremum of the radii R such that there is a closed subset E of Minkowski dimension 2 inside B. R 2 / whose complement symplectically embeds into Z./.In Section 4, we showed that R sup Ä p 2. This inequality will be improved by Joé Brendel to R sup Ä p 3 using a different construction; see Remark 1.5.An intriguing aspect of both of the squeezing constructions is that they fail for large radii R.This motivates the following: Question 3 Is R sup a finite number?

Minkowski dimension problem for extendable embeddings
Here is a variant of our Minkowski dimension question, which is also more directly related to the Lipschitz number question.Assume R > r .What is the smallest Minkowski dimension of a subset E B 4 .R 2 / with the property that for any neighborhood U of E, there is a symplectic embedding Corollary 6.5 Let U be a neighborhood of L that is disjoint from Á.Then, we cannot extend the restriction to B n U of our symplectic embedding B n L into Z./ to a symplectic embedding B ! C 2 .
Proof If there were such an embedding, the action of the image of Á would have to be simultaneously zero and nonzero, which is absurd.

Bounds on Minkowski content of the defect region
We have shown in Corollary 3.2 that the lower Minkowski dimension bound dim ᏹ .E/ 2 is optimal in the range R 2 .1;p 2. Is the estimate on the 2-content ᏹ 2 .E/ .R 2 1/ also sharp?That is, does there exist R 2 .1;p 2 and E with ᏹ 2 .E/ D .R 2 1/ such that B. R 2 / n E symplectically embeds into Z./?

Speculations on the Lipschitz question
Consider the volume loss function VL.L; R/ WD inf fVol.E.ˆ//g ; where the infimum is taken over all symplectic embeddings ˆW B 4 .R 2 / !R 4 with Lipschitz constant at most L, and In Section 5, we proved Theorems 1.6 and 1.7, which may be summarized by the statement that where we tacitly assume R > 1.Even more, we noted in Remarks 5.2 and 5.4 that our methods show that we may take c.R/ R 4 and C.R/ R 7 .
Our notation corresponds to that of the main body of the text by setting ˛D R 2 .The idea of the proof is to view B 4 .˛/as CP 2 .˛/n CP 1 and to use almost toric fibrations of CP 2 .As observed in [32], for every Markov triple .a;b; c/, there is a triangle a;b;c .˛/R 2 and an almost toric fibration on CP 2 .˛/with a base diagram whose underlying polytope is a;b;c .˛/.Now note that the toric moment map image of Z 4 .1/ is the half-strip D R >0 OE0; 1/.We shall show that if the triangle a;b;c .˛/fits into (after applying an integral affine transformation), then there is a symplectic embedding of CP 2 .˛/into Z 4 .1/at the cost of removing a certain subset † 0 from CP 2 .˛/.The point here is that one can get a good understanding of the subset one needs to remove.Indeed, we show that † 0 is a union of three Lagrangian pinwheels (defined as in [7]) and a symplectic torus.In particular, this set has Minkowski dimension 2. A combinatorial argument shows that for every ˛< 3, there is a Markov triple .a;b; c/ and an inclusion a;b;c .˛/; see Lemma A.5.
Remark A.2 As was pointed out to us by Leonid Polterovich, our results can be combined with Gromov's nonsqueezing to show that any symplectic ball B 4 .1C"/CP 2 .˛/intersects the set † 0 CP 2 discussed above.See Corollary A.9 for more details.
Remark A. 3 The same strategy may work to produce symplectic embeddings of the polydisk of capacity ˛< 2 minus a union of some two-dimensional manifolds into the cylinder.Indeed, one can view the polydisk as the affine part of S 2 S 2 and use almost toric fibrations of the latter space to carry out the same argument.
The relationship between Markov triples and the complex and symplectic geometry of CP 2 has generated a lot of interest in recent years.It first appeared in the work of Galkin and Usnich [8], who conjectured that for every Markov triple there is an exotic Lagrangian torus in CP 2 .This conjecture was proved and generalized by Vianna [32; 33] by the use of almost toric fibrations; see also Symington [30].
On the algebrogeometric side, Hacking and Prokhorov [13] showed that a complex surface X with quotient singularities admits a Q-Gorenstein smoothing to CP 2 if and only if X is a weighted projective space CP .a 2 ; b 2 ; c 2 / and .a;b; c/ forms a Markov triple.In [7], Evans and Smith studied embeddings of Lagrangian pinwheels into CP 2 .This is directly related to [13], since Lagrangian pinwheels appear naturally as vanishing cycles of the smoothings of CP .a 2 ; b 2 ; c 2 / to CP 2 .See also the recent work by Casals and Vianna [5] and the forthcoming paper joint with Mikhalkin and Schlenk [4] for other applications of almost toric fibrations to symplectic embedding problems.
As was pointed out to us by Leonid Polterovich, one can combine Theorem A.1 with Gromov's nonsqueezing theorem to get certain rigidity results, reminiscent of [2, Theorem 1.B].
Note that for a fixed 1 < ˛< 3 we get infinitely many sets † 0 CP 2 to which Corollary A.9 applies and all of these consist of a union of a symplectic torus and Lagrangian pinwheels.

A.4 Proof of Proposition A.7
Following the exposition in [7, Example 2.5] we consider smoothings of certain orbifold quotients of C 2 .This yields the local version from Lemma A.10 of the symplectomorphism in Proposition A.7.
Let a and q be coprime integers with 1 6 q < a and take the quotient of C 2 by the action of .a 2 / th roots of unity (A-14) :.z 1 ; z 2 / D .z 1 ; aq 1 z 2 /; where a 2 D 1: We denote this quotient by C 2 = a;q .It can be embedded as fw 1 w 2 D w a 3 g into the quotient C 3 =Z a by the action (A-15) Á:.w 1 ; w 2 ; w 3 / D .Áw 1 ; Á 1 w 2 ; Á q w 3 /; Á a D 1: The smoothing is given by which we view as a degeneration by projecting to the t-component, W ᐄ !C t .We denote the fibers by X t D 1 .t/.The smooth fiber X 1 is a rational homology ball and the vanishing cycle of the degeneration is a Lagrangian pinwheel L a;q .This follows from the description of ᐄ as Z a -quotient of an A a 1 -Milnor fiber.Let s 2 X 0 be the unique isolated singularity of X 0 , and ᐄ reg D ᐄ n fsg its complement.The restriction of the standard symplectic form ! 0 on C 4 D C 3 C t yields a symplectic manifold .ᐄreg ; /.Note that the smooth loci of the fibers X t ¤0 and X 0 n fsg are symplectic submanifolds.Let us now construct a symplectomorphism (A-17) W X 1 n L a;q !X 0 n fsg D C 2 = a;q n f0g: For this, we take the connection on ᐄ reg defined as the symplectic complement to the vertical distribution, (A-18) x D .ker./ x / D fv 2 T x ᐄ reg j .v;w/ D 0 for all w 2 T x 1 ..x//g: symplectic form.This shows that there is a symplectomorphism which intertwines the (almost) toric structures on the preimages of a complement of neighborhoods of the vertices.For example, one can choose W a;b;c as in Figure 2. We use Lemma A.10 to extend this symplectomorphism.
Let us make a few preparations.In particular, we discuss how to use the quotient C 2 = a;q as a local toric model for CP .a 2 ; b 2 ; c 2 /.This is the orbifold version of the toric ball embedding into CP 2 one obtains by the inclusion of the simplex with one edge removed into the standard simplex 1;1;1 .The toric structure on C 2 = a;q is induced by the standard toric structure .z 1 ; z 2 / 7 ! .jz 1 j 2 ; jz 2 j 2 / on C 2 .Indeed, note that the a;q -action is obtained by restricting the standard T 2 D .R=Z/ 2 -action to a discrete subgroup Z a 2 .This implies that we obtain an induced action by T 2 = a;q on C 2 = a;q .This action is Hamiltonian and its moment map image, under a suitable identification T 2 Š T 2 = a;q , is given by See for example [7,Remark 2.7] or [30,Section 9].Note that a ball B 4 .d/ D f .jz 1 j 2 Cjz 2 j 2 / < dg C 2 quotients to an orbifold ball B 4 .d/=a;q C 2 = a;q , which is fibered by the induced toric structure on the quotient.Furthermore, the boundary sphere S 3 .d/C 2 quotients to a lens space † a;q .d/D S 3 .d/=a;q of type .aq1; a 2 / equipped with its canonical contact structure and which fibers over a segment in † a;q .We will use this fact in the proof of Proposition A.7.
Let us now show that, for a suitable choice of q, the toric system C 2 = a;q !† a;q can be used as a local model around one of the orbifold points of the toric system on CP .a 2 ; b 2 ; c 2 /.In order to get a concrete description of this toric system on the weighted projective space, recall that the symplectic orbifold CP .a 2 ; b 2 ; c 2 / can be defined as a symplectic quotient of C 3 , (A-21) CP .a 2 ; b This description (A-21) has the advantage that it is naturally equipped with a Hamiltonian T 2 -action inherited from the standard T 3 -action on C 3 .This induced action is toric and its moment map image is given by the intersection of the plane defined by H and the positive orthant, Note that this is a polytope in R By the definition of integral affine embedding, the definition (A-23) makes sense and the polytope it defines is independent of the choice of ˆup to applying an integral affine transformation.We now show that there is a natural number q and an integral affine embedding ˆa;q such that the triangle ˆ 1 a;q .z a;b;c / R 2 is obtained by intersecting † a;q with a half-plane.Indeed, set where q satisfies bq D 3c mod a; see also [7,Example 2.6].The map ˆa;q has image and it is an integral affine embedding, as can be checked by a computation.Furthermore ˆmaps 0 to the vertex .b 2 c 2 ; 0; 0/ (corresponding to a 2 ) of z a;b;c , and v 1 and v 2 to the outgoing edges at .b 2 c 2 ; 0; 0/.This means that there is an integral vector . 1 ; 2 / defining a half-plane K D f 1 x 1 C 2 x 2 6 kg such that (A-25) ˆ 1 a;q .z a;b;c / D K \ † a;q : From this we deduce the desired toric model.Let z E a z a;b;c be the edge opposite the vertex .b 2 c 2 ; 0; 0/.
Proposition A.12 The subset in CP .a 2 ; b 2 ; c 2 / fibering over z a;b;c n z E a is fibered (orbifold ) symplectomorphic to the subset in C 2 = a;q fibering over Int K \ † a;q .
Proof We have shown above that z a;b;c n z E a and Int K \ † a;q are integral affine equivalent.This implies the claim by the classification of compact toric orbifolds by their moment map images; see [19].Compactness is not a problem here, since we can compactify the subset fibering over K \ † a;q by performing a symplectic cut at f 1 x 1 C 2 x 2 D kg.Proof of Proposition A.7 The main part of the proof will be concerned with proving the existence of the symplectomorphism (A-10) and for readability, we postpone the proof of the existence of the global map z and the computation of z 1 .Ᏸ/ to Step 5.

Let us now fix a moment map
Step 1 We start by setting up some notation on the side of the weighted projective space.The orbifold points s a ; s b ; s c 2 CP .a 2 ; b 2 ; c 2 / are mapped to the vertices v a ; v b ; v c 2 a;b;c under the moment map W CP .a 2 ; b 2 ; c 2 / 7 !a;b;c .Let us first focus on the orbifold point s a .Denote the edge opposite to v a by E a .By Proposition A.12, there is an orbifold symplectomorphism .Int K \ † a;q /; which intertwines the toric structures.Now let B 4 .d/=a;q C 2 = a;q be a closed orbifold ball for B 4 .d/D f .jz 1 j 2 Cjz 2 j 2 / 6 dg and d > 0. Its boundary S 3 .d/=a;q C 2 = a;q is a lens space equipped with the standard contact structure.Note that both the orbifold ball and its boundary are fibered by the moment map C 2 = a;q .Since a intertwines the toric structures, the image sets (A-28) B orb a D 1 a .B 4 .d/=a;q / and † a D 1 a .S 3 .d/=a;q / are fibered by .Then the image of the pair .B a ; † a / under is a pair .V a ; `a/ consisting of a segment contained in a triangle around the vertex v a .Note also that the lens space † a is naturally equipped with its standard contact structure.We do the same procedure around the remaining vertices v for all j 2 fa; b; cg.Again, see Figure 2.
Step 2 Now consider the almost toric fibration of CP 2 associated to the triangle a;b;c .In the conventions of [30; 32] the triangle a;b;c is decorated with three dashed line segments of prescribed slope between the vertices and the nodal points.The latter are usually marked by a cross.There is a map W CP 2 !a;b;c which is a standard toric fibration away from the dashed lines, but which is only continuous on the preimages of the dashed lines (which encode monodromy of the integral affine structure).By applying nodal slides if necessary, we may assume that the dashed lines lie outside of the subset W a;b;c .Since the projection is standard toric away from the dashed lines, this implies that there is a symplectomorphism (A-29) 0 W CP 2 1 .W / ! 1 .W / CP .a 2 ; b 2 ; c 2 /; which intertwines and .Define the preimages By [30, Section 9], the set B 0 a is a closed rational homology ball and † 0 a is a lens space of type .aq1; a 2 / equipped with its standard contact structure.In fact, 0 maps the copy † 0 a of the lens space to the copy † a .However, contrary to B orb a , the rational homology ball B 0 a is smooth.The same discussion holds for b and c.
Step 3 The key part of the proof is finding extensions (A-31) j W B 0 j n L j ;q j !B orb j n fs j g of the map 0 j † 0 j , where L j ;q j are Lagrangian pinwheels for j 2 fa; b; cg.For this we use Lemma A.10.Again, restricting our attention to a, let a be the symplectomorphism from Lemma A.10.Note that we have already established the correspondence between B orb a and C 2 = a;q by a and that this correspondence is compatible with the toric picture.We now establish a correspondence between the rational homology sphere B 0 a and the space S a;q coming from the smoothing in Lemma A. 10.Define yet another copy † 00 a of the lens space by setting † 00 a D 1 a .a .† a;q //.This lens space is also equipped with the standard contact structure and it bounds a rational homology ball B 00 a by [7,Example 2.5].We now have two pairs .B 0 a ; † 0 a / and .B 00 a ; † 00 a / consisting of a rational homology ball bounded by a lens space carrying its standard contact structure.By [9, Proposition A.2], which relies on [20], this implies that .B 0 a ; † 0 a / and .B 00 a ; † 00 a / are equivalent up to symplectic deformation.Let a W .B 0 a ; † 0 a / !.B 00 a ; † 00 a / be the diffeomorphism we obtain from this.Note that we cannot directly use the symplectic deformation to conclude, since the symplectomorphism obtained from a Moser-type argument may not restrict to the desired map on the boundary.More precisely, we obtain a diagram of diffeomorphisms of lens spaces (A-32) a † 00 a S 3 .d/=a;q B 4 .d/=a;q 0 j † 0 a a j † 0 a a j †a a j † 00 a and this diagram does not commute.We may, however, correct the diffeomorphism a so that (A-32) commutes.Recall that † a;q is a lens space of type .aq1; a 2 /.Since .aq1/ 2 ¤ ˙1 mod a 2 , it follows from [3, Théorème 3(a)] that the space of diffeomorphisms of † 00 a has two components, namely the one of the identity and the one of the involution induced by the involution .z 1 ; z 2 / 7 !.xz 1 ; x z 2 / of S 3 .The diffeomorphism extends to a diffeomorphism z of B 00 p;q .Up to postcomposing a with z , we may thus assume that . 1 a a 0 1 a /j † 00 a is isotopic to the identity by an isotopy ' t .Using this isotopy, we can correct the diffeomorphism a such that the diagram (A-32) commutes.Indeed, the set 1 .V a \ W / is a collar neighborhood † 0 a OE0; 2/ and thus we can use the collar coordinate together with the isotopy ' t to define a corrected diffeomorphism z a , which coincides with the original diffeomorphism a on 1 .V a n W / and with . 1 a a 0 /j † 0 a on † 0 a .Recall that two collar neighborhoods which agree on the boundary coincide up to applying a smooth isotopy; see Munkres [24,Lemma 6.1].This means that, after applying an isotopy in .B 00 a ; † 00 a /, we can assume that the corrected version of a and 1 a a 0 agree on a smaller collar † a OE0; 1/.Denoting the diffeomorphism we obtain in this way by z a , this allows us to define a diffeomorphism (A- 33) a D 1 a ı a ı z a j B 0 a nL a;qa W B 0 a n L a;q a !B orb a n fs a g; which extends 0 in the sense that it agrees with 0 on a collar of † 0 a .Since a is defined outside of a Lagrangian pinwheel L a;q a B 00 a , the diffeomorphism a is defined outside of a pinwheel (which we again denote by L a;q a ) in B 0 a .We repeat this procedure for b and c to obtain diffeomorphisms b and c .
Step 4 By construction, the diffeomorphisms a , b and c extend the initial symplectomorphism and hence we obtain a diffeomorphism (A-34) y W CP 2 n.L a;q a t L b;q b t L c;q c / !CP .a 2 ; b 2 ; c 2 / n fs a ; s b ; s c g: We now turn to the symplectic forms.On CP 2 , we define a symplectic form y ! which turns y into a symplectomorphism as follows.On 1 .W n .V a [ V b [ V c // we define y ! to be the usual Fubini-Study form !. On V j we define y ! as the pullback form z j !B 00 j , where z j is the corrected diffeomorphism constructed at the end of Step 3.This yields a well-defined symplectic form which turns y into a symplectomorphism.Indeed, this follows from the fact that the maps 0 , j and j are symplectomorphisms and j is defined as their composition (A-33).This also implies that the symplectic form y ! has the same total volume as the Fubini-Study form.By the Gromov-Taubes theorem [22,Remark 9.4.3(ii)], the form z ! is symplectomorphic to the Fubini-Study form and hence postcomposing y from (A-34) with this symplectomorphism yields the desired symplectomorphism (A-10).
Step 5 The definition of the global map z W CP 2 !CP .a 2 ; b 2 ; c 2 / is obtained by replacing a from (A-33) by z a D 1 a ı z a ı z a and carrying out the rest of the construction as above.The map z a is given by Lemma A.10.Let us now identify z 1 .Ᏸ/, where Ᏸ D 1 .@a;b;c /.Let z W W be the subset of W where z coincides with 0 .Since 0 intertwines the toric structures on 1 .W / CP 2 and 1 .W / CP .a 2 ; b 2 ; c 2 /, the set z 1 .Ᏸ/ \ 1 .z W / fibers over the three pieces of the boundary given by z W \ @ a;b;c and hence consists of three disjoint symplectic cylinders.
We use Lemma A.10 to prove that the missing pieces z 1 .Ᏸ/ \ B 0 j for j 2 fa; b; cg are also given by symplectic cylinders and that the union of the six cylinders is given by a torus.We again discuss the case of j D a since the other two are completely analogous.Recall that a W B orb a !B 4 .d/=a;q is compatible with the toric structure and thus a .Ᏸ \ B orb a / D Ᏸ a;q a \ a .B orb a /, where Ᏸ a;q a D fz 1 z 2 D 0g= a;q a as in Lemma A.10.Indeed, this follows from the fact that Ᏸ a;q a fibers over the boundary of the moment map image † a;q a .By Lemma A.10, we deduce that z 1 a .z 1 a .a .Ᏸ \ B orb a /// is given by the union of a pinwheel with a piece of a cylinder.Furthermore, recall that near the lens spaces at the respective boundaries, the map z 1 a z a a coincides with 0 .This implies that the cylinder contained in B 0 a has two boundary components at @B 0 a D † 0 a , which are smoothly identified in a collar neighborhood with boundary components of the set z 1 .Ᏸ/ \ 1 .z W / discussed in the previous paragraph.This proves the claim.
Remark A. 13 We suspect that there are shorter and more natural proofs of Proposition A. 7.In particular, one should be able to avoid Gromov-Taubes.One possibility we have hinted at above is working with a global degeneration and trying to analyze its vanishing cycle.This would completely avoid the use almost toric fibrations.Another possibility, in the spirit of [27; 14], is to equip the explicit local degeneration from [17] with a family of integrable systems avoiding the pinwheel and extending the given toric structure on the boundary.The symplectomorphism from Proposition A.7 then follows from the usual toric arguments and it is automatically equivariant.Although this construction is elementary, it is somewhat outside the scope of this appendix and we hope to carry out the details elsewhere.This is also reminiscent of [10, Section 7], and it is plausible one can apply results from this paper to prove the same result.

Theorem 1 . 7 Remark 1 . 8
Let R > 1.Then there exists a constant C D C.R/ > 0 such that for all constants L, there exists a symplectic embedding ˆW B 4 .R/ ,! R 4 with Lipschitz constant at most L such that Vol 4 E.ˆ/ Ä C L : As was pointed out to us by Felix Schlenk, our construction is a simplified version of multiple symplectic folding; see Schlenk [28, Sections 3 and 4].
Note that Q and Z RP sit naturally inside D RP 2 .The Poincaré dual of the homology class of Q is 1= times the symplectic class.The cylinders .C [ C / \ D RP become symplectic 2-spheres in D RP 2 .They intersect Q positively in two points and Z RP along the circle .Let us call these spheres S , now indexed by unoriented unparametrized geodesic circles on RP .Each S has self-intersection number 1.

2 nZ
RP by open disks which are the reductions of the C \ .D RP nZ RP /'s.Let us denote this Q-family of submanifolds by D , where 2 Q.Proposition 4.3 D RP 2 is automatically area-preserving; see D computed below) with f W OE0; 1=L !OE0; 1C1=L a family of functions, depending upon L, such that f 0 .x/D 1 for x in an open neighborhood of the endpoints 0 and 1=L, f .0/D 0 and f .1=L/D 1 C 1=L, 1 Ä f 0 .x/ 2 O.L/, jf 00 .x/=.f 0 .x// 2 j 2 O.L/.

Proposition 6 . 4
r 2 /?Our obstructive Theorem 1.4 still gives a bound, but our construction in Section 4 does not apply.Recall B WD B 4 .2/.Consider the embedding B n L ,! Z. / that we constructed in Theorem 1.3.There exists an embedded circle Á in iL \ B that is disjoint from L, and which maps into fx 1 D y 1 D 0g.Proof Recall that our symplectomorphism first sendsB n L to CP 2 .2/ n .L RP [ S OE0W0W1 / via the restriction of a symplectomorphism b W B ! CP 2 .2/ n S OE0W0W1 .The image of L \ B under b is L RP n S OE0W0W1 , whereas the image of iL \ B is iL RP n S OE0W0W1, where we defineiL RP WD fOEz 1 W z 2 W z 3 j Re.z 1 / D Re.z 2 / D Im.z 3 / D 0g: FQ and iL RP intersect along the circle y C D fOEi cos Â W i sin Â W 1 j Â 2 OE0; 2 g: Note that y C is disjoint from L RP [ SOE0W0W1 .Now recall that to complete our symplectomorphism from B n L to Ᏹ. ; 4 / n Ꮿ we use the Oakley-Usher symplectomorphism between CP 2 .2/ n .L RP [ S OE0W0W1 / and D RP 2 n .Z RP [ S/, and then the Opshtein symplectomorphism.From Opshtein's formula we see that all the points on Qn.Z RP [S/ and therefore on FQn.L RP [S OE0W0W1 / are sent to points in fx 1 D y 1 D 0g in Ᏹ. ; 4 /.This means that b 1 .y C / satisfies the condition in the statement.

(A- 26 )Figure 2 :
Figure 2: The triangle a;b;c as union W [ V a [ V b [ V c , on the left as the toric moment polytope of CP .a 2 ; b 2 ; c 2 / and on the right as almost toric base diagram of CP 2 .In both cases the fibration is toric over W and lens spaces fiber over the segments `a; `b; `c.
b and v c , and denote the corresponding sets by B orb b ; B orb c ; † b ; † c CP .a 2 ; b 2 ; c 2 / and by V b ; V c ; `b; `c a;b;c .We choose the sizes so that B orb a , B orb b and B orb c are mutually disjoint.Furthermore, we choose a set W a;b;c such that a;b

Issue 3 (
pages 1005-1499) 2024 1005 Homological mirror symmetry for hypertoric varieties, I: Conic equivariant sheaves MICHAEL MCBREEN and BEN WEBSTER 1065 Moduli spaces of Ricci positive metrics in dimension five MCFEELY JACKSON GOODMAN 1099 Riemannian manifolds with entire Grauert tube are rationally elliptic XIAOYANG CHEN 1113 On certain quantifications of Gromov's nonsqueezing theorem KEVIN SACKEL, ANTOINE SONG, UMUT VAROLGUNES and JONATHAN J ZHU 1153 Zariski dense surface groups in SL.2k C 1; Z/ D DARREN LONG and MORWEN B THISTLETHWAITE 1167 Scalar and mean curvature comparison via the Dirac operator SIMONE CECCHINI and RUDOLF ZEIDLER 1213 Symplectic capacities, unperturbed curves and convex toric domains DUSA MCDUFF and KYLER SIEGEL 1287 Quadric bundles and hyperbolic equivalence ALEXANDER KUZNETSOV 1341 Categorical wall-crossing formula for Donaldson-Thomas theory on the resolved conifold YUKINOBU TODA 1409 Nonnegative Ricci curvature, metric cones and virtual abelianness JIAYIN PAN 1437 The homology of the Temperley-Lieb algebras RACHAEL BOYD and RICHARD HEPWORTH For r < R, recall that B. R 2 / and Z. r 2 / denote the open ball and open cylinder of radius R and r , respectively, in R [11]n this subsection, we give a slight modification of the holomorphic foliation argument of Gromov[11]in dimension n D 4.Proposition 2.3 Let R; r > 0. Let E be a compact subset of R 4 and let W B. R 2 / n E !R 4 be a smooth symplectic embedding into the cylinder Z. r 2 /.Let U be the closure of an open neighborhood of @B.R 2 / [ E in R 4 .Then there exists a smooth map f W B. R 2 / n U ! R 2 such thatf has no critical points on B. R 2 / n U , and for all y 2 R 2 , if f 1 .y/\ B. R 2 / n U is nonempty, then it is a two-dimensional complex submanifold of Euclidean area less than r 2 .
As usual, for r < R, B. R 2 / and Z. r 2 / refer to the open ball and open cylinder of radius R and r , respectively, in R 4 .The main estimate of this section is the following obstructive bound: Theorem 3.1 Let E be a compact subset of R 4 and suppose that B. R 2 / n E symplectically embeds into the cylinder Z. r 2 / R 4 .Then there is a function k R 2 o.t 2 / such that for any t > 0,Vol 4 .N t .E// 2 .R 2 r 2 /t 2 k Let U ı be the closure of N ı .@B.R 2 / [ E/ in R 4 and let z f ı W B. R 2 / n U ı !R 2 bethe map given by Proposition 2.3.Note that B. R 2 / n U ı D B. .R ı/ 2 / n N ı .E/.We then take f ı W B. .R ı/ 2 / !R 2 to be any continuous extension of z f ı .Since f ı agrees with z f ı on B. .R ı/ 2 / n U ı , by the conclusions of Proposition 2.3 we have for any y 2 R 2 that † ı WD f 1 ı .y/\ B. .R ı/ 2 / n U ı is a minimal submanifold with area less than A WD r 2 .
t .† ı // Ä Vol 2 .† ı / t 2 Ä A t 2 : The upper map sends .x;/ 2 U RP OE0; 1/ to .x; 1 2 /, and it is a homeomorphism overall as well as a diffeomorphism of the interiors.By construction of symplectic boundary reduction we deduce that there is a canonical diffeomorphism F making this diagram commutative.
Combined with Corollary 4.2, the following finishes the proof of Theorem 1.3.Proposition 4.5 D RP 2 n .S [ Z RP / is symplectomorphic to Ᏹ. ; 4 / n Ꮿ. Proof We use our Proposition 4.3 and [26, Lemma 2.1] to find an explicit symplectomorphism from the complement of D 0 in D RP 2 n Z RP to Ᏹ. ; 4 /.
Lemma 3.1], Oakley and Usher considered the map /=x 2 on .0; 1 and f .0/D 1 2 .They proved that ˆjint.D RP n / is a symplectomorphism onto its image CP n .2/ n FQ n , where FQ n D n OEz 0 W W z n 2 CP n .2/ ˇX z 2 k D 0 o is the Fermat quadric.As before, we will denote by D RP n the boundary reduction of D RP n , and by Z RP n the zero-section.We have the following, which implies Proposition 4.1: Proposition 4.6 The Oakley-Usher map ˆW D RP n !CP n .2/ descends to a symplectomorphism x ˆW D RP n !CP n .2/: Proof Note that D RP n is canonically homeomorphic to the quotient D RP n = , where x y if x; y 2 U RP n and they are in the same orbit of the geodesic flow.Therefore, it is easy to see from the computations of Oakley-Usher that ˆdescends to a bijective continuous map x ˆW D RP n !CP n .2/.We will show that this map is a symplectomorphism.
1into D RP n , and of D RP 1 into D RP n (the last point is particularly clear in the description of the boundary reductions at hand as in the proof of Proposition 4.3, which we keep in mind for the next point as well.)It is easy to see that the multiplication map G D RP1! D RP n is indeed a surjective submersion using that G T RP1! T RP n is one.Applying the lemma with S D D RP 1 , smoothness of x follows from the smoothness of the n D 1 case x ˆ1 W D RP 1 !CP 1 .Remark 4.8 In [29, Chapter V], Seade gives a description of CP n as a double mapping cylinder via the natural SO.n C 1/ action.One may follow this discussion to obtain the corresponding description of D RP n , and that the map ˆfactors as the normal exponential map of RP n ,! CP n (with respect to the Fubini-Study metric) composed with the map D RP n !T RP n ' N RP n induced by jpj 7 ! 1 2 sin 1 jpj; note that xf .x/D tan 1 2 sin 1 We may more generally ask about the volume of the region E.ˆ; r / for R > r .By a scaling argument, we find that Vol 4 .E.ˆ; r // r 4 c.R=r / L 2 : r 2 //: Recall now that Theorem 1.6 is the statement that Vol 4 .E.ˆ; 1// c=L 2 for some constant c D c.R/ > 0.Proof of Theorem 1.6 Let ı be any number strictly between 0 and R 1. Observe that by the Lipschitz bound, we haveN ı=L .E.ˆ; 1 C ı// E.ˆ; 1/: R 2 /nE.ˆ;1Cı/WB.R 2 / n E.ˆ; 1 C ı/ !Z.1 C ı/; we obtain Vol 4 .E.ˆ; 1// 2 .R 2 .1 C ı/ 2 / ı 2 L 2 o Â ı 2 L 2 Ã ;which implies the desired bound after fixing some value for ı, for instance ı D 1 2 .R 1/.Remark 5.1 Gr .B n V s /, where V s 2 Gr R .2;4/ with symplecticity s. 0g [ f.e i t x; y/ [ .x; e i t y/ j Im.x/ D Im.y/ D 0; Re.x/ 0; Re.y/ 0; t 2 OE0; 2 g : Hence, we have that stc.1/D 2 .On the other hand, it follows immediately from our Theorem 1.3 that stc.0/D .We finish with the obvious question.
2; 4/ !OE0; 1: Lemma 6.2 Two elements of Gr R .2;4/ have the same symplecticity if and only if there is an element of U.2/ sending one to the other.Question 2 What is the function stc?Is it continuous?
[32, not R 2 .We get a Markov triangle a;b;c .abc/R 2 as in same triangles (up to integral affine equivalence) as those obtained from almost toric fibrations of CP 2 as discussed in Section A.2; see[32, Section 2].Hence it makes sense to denote them by a;b;c .abc/.The normalization ˛D abc of the triangle comes from the choice of level at which we have reduced in (A-21).Definition A.11 An affine map ˆW R 2 !R 3 given by x 7 ! A C b for A 2 Z 3 2 and b 2 R 3 is called an integral affine embedding if it is injective and if A.Z 2 / D A.R 2 / \ Z 3 .