International Congress of Mathematicians

246 P. Biransymplectic packings in dimension higher than 4. The only packing obstructionsknown in these dimensions are described in Theorem C. Note that CF" admits fullpacking by N = k n equal balls [27], but it is unclear what happens for other valuesof N. In view of this and Theorem C, the first unknown case (for n > 3) is ofN = 2 n + 1 equal balls.The situation in dimension 4 is only slightly better. Except of CF 2 and a fewother rational surfaces no packing obstructions are known. It is known that for everysymplectic 4-manifold (M,oj) with [OJ] £ H 2 (M;Q) packing obstruction (for equalballs) disappear once the number of balls is large enough (see [6]), but nothing isknown when the number of balls is small. In fact even the case of one ball is poorlyunderstood(namely, what is the maximal radius of a ball that can be symplecticallyembedded in M). The reason here is that the methods yielding packing obstructionsstrongly rely on the geometry of algebraic and pseudo-holomorphic curves in themanifold. The problem is that most symplectic manifolds have very few (or noneat all) J-holomorphic curves for a generic choice of the almost complex structure.Thus, even in dimension 4 it is unknown whether or not packing obstructions is aphenomenon particular to a sporadic class of manifolds such as CF 2 .Is everything Lagrangian? Weinstein's famous saying could be relevant for theintersection described in Theorems C,D and E. In other words, it could be thatthese intersections are in fact Lagrangian intersections under disguise. To be moreconcrete, let | < R 2 < ^r and consider a Lagrangian LR lying on the boundarydB 2n (R). Is it possible to Hamiltonianly separate LR from itself inside F 2 "(l) ?If we can find a Lagrangian LR for which the answer is negative then thiswould strongly indicate that Theorem C is in fact a Lagrangian intersections result.Namely it would imply Theorem C for Fi = F2 under the additional assumptionthat tpi,tp2 are symplectically isotopie. A good candidate for LR seems to be thesplit torus dB 2 (x/R/n) x • • • x dB 2 (x/R/n) C dB 2n (R), but one could try otherLagrangians as well.Attempts to approach this question with traditional Floer homology fail. Thereason is that Floer homology is blind to sizes: both to the "size" of the LagrangianLR as well as to the "size" of the domain in which we work F 2 "(l). Indeed itis easy to see that HF(LR,LR) whether computed inside F 2 "(l) or in R 2 " is thesame, hence vanishes. The meaning of "sizes" can be made precise: the size of LRis encoded in its Liouville class, and the size of F 2 "(l) could be encoded here bythe action spectrum of its boundary.It would be interesting to try a mixture of symplectic field theory [19] withFloer homology. This would require a sophisticated counting of holomorphic discswith k punctures (for all k > 0), where the boundary of the discs goe to LR andthe punctures to periodic orbits on 9F 2 "(1).It is interesting to note that when the radii of the balls are not equal thingsbecome more complicated. Indeed suppose that R\ + F| > 1 and consider twoLagrangian submanifolds LR X , LR 2 lying on the boundaries of the balls B iPl , B V2 .Then clearly LR X and LR 2 can be disjoint even though the balls B iPl , B V2 do intersect(e.g. two concentric balls B iPl c B iP2 , where Fi < F2). It would be interesting