International Congress of Mathematicians

2.2. Balls intersect ballsGeometry of Symplectic Intersections 243Denote by B 2n (R) the closed Euclidean ball of radius F, endowed with thestandard symplectic structure induced from R 2 ". Denote by CF" the complexprojective space, endowed with its standard Kahler form a, normalized so that/cpi f = 7T. The following obstruction for symplectic packing was discovered byGromov [22]:Theorem C. Let M be either B 2n (l) or 1then B Vl n B V2 ^ 0.Since symplectic embeddings are also volume preserving there is an obviousvolume obstruction for having B lfil fl B V2 = 0. However, volume considerationspredict an intersection only if R 2n + Ff" > 1 (moreover for volume preservingembeddings the latter inequality is sharp).When one considers embeddings of several balls things become more complicatedand interesting. Here results are currently available only in dimension 4.Theorem D. Let M be either F 4 (l) or CP 2 , and let B Vï ,...,B VN c M be theimages of symplectic embeddings tp k : B 4 (R) —t M, k = 1,...,N, of N balls ofthe same radius R. Then there exist i ^ j such that B ipi n B lfii ^ $ in each of thefollowing cases:1. N = 2 or 3 and R 2 > 1/2.2. N = 5 or 6 and R 2 > 2/5.3. N = 7 and R 2 > 3/8.4. N = 8 and R 2 > 6/17.Moreover all the above inequalities are sharp in the sense that in each case if theinequality on R is not satisfied then there exist symplectic embeddings tpi,..., tpMas above with disjoint images B iPl ,..., F^ c M.Statement 2 for N = 5 was proved by Gromov [22]. The rest was establishedby McDuff and Polterovich [27]. Let us mention that for N = 4 and any N > 9this intersection phenomenon completely disappears in the sense that an arbitrarilylargeportion of the volume of M can be filled by a disjoint union of N equal balls(see [27] for N = 4 and N = k 2 , and [5, 6] for the remaining cases).2.3. Balls intersect LagrangiansIt turns out that there exist (symplectically) irremovable intersections alsobetween contractible domains (e.g. balls) and Lagrangian submanifolds.Denote by RF" C CF" the Lagrangian n-dimensional real projective space(embedded as the fixed point set of the standard conjugation of CP n ). The followingwas proved in [7]:Theorem E. Let B v c CF" be the image of a symplectic embedding tp : B 2n (R) —tCP n . If R 2 > 1/2 then B v n RF" ^ 0. Moreover the inequality is sharp, namelyfor every R 2 < 1/2 there exists a symplectic embedding tp : B 2n (R) —t CP n whoseimage avoids RF".In fact this pattern of intersections occurs in a wide class of examples (see [7]):