Packing Lagrangian tori (Q6620571)

The paper studies maximal collections (packings) of disjoint Lagrangian tori in the simplest precompact symplectic manifolds. Maximality (packing) means that any other Lagrangian torus must intersect one of those in the collection. To make the question non-trivial, attention is restricted to integral tori, those whose area homomorphism takes only integer values.\N\NThe first main result is that in \(S^2\times S^2\), with the \(S^2\) symplectic form scaled to make its area equal \(2\), the monotone Clifford torus \(L_{1,1}\) (the product of equators) is already a packing. When \(1<a,b<2\), the same result then follows for the standard Clifford torus in the symplectic polydisk \N\[\NP(a,b):=\{(z_1,z_2)\in\mathbb{C}^2: \pi|z_1|^2<a,\ \pi|z_2|^2<b\}\subset\mathbb{R}^4.\N\]\NFor \(a,b>2\), a natural candidate packing may seem to be the collection of Lagrangian tori \(L_{k,l}\), the products of circles around the origin bounding disks of areas \(k\) and \(l\) in the \(z_1\) and \(z_2\) planes, respectively. However, the second main result of the paper is that this collection is \textit{never} a packing. As a side remark, a recent result of \textit{J. Hicks} and \textit{C. Y. Mak} [``Some cute applications of Lagrangian cobordisms towards examples in quantitative symplectic geometry'', Preprint, \url{arXiv:2208.14498}] states that there are at least six disjoint tori in a packing when \(2<a,b<3,\) and there are at least three when \(2<b<3\).\N\NThe proof of the packing property is based on the work of \textit{R. Hind} and \textit{E. Opshtein} [Comment. Math. Helv. 95, No. 3, 535--567 (2020; Zbl 1453.53075)], and first shows that any integral Lagrangian torus in \(S^2\times S^2\) is monotone. A hypothetical integral torus \(\mathbb{L}\) disjoint from \(L_{1,1}\) is then considered, and a foliation with the normal form near \(\mathbb{L}\) and \(L_{1,1}\) due to \textit{G. Dimitroglou Rizell} et al. [Geom. Funct. Anal. 26, No. 5, 1297--1358 (2016; Zbl 1358.53089)] is utilized to construct two symplectic spheres that have special intersection properties with the leaves and each other. Then \(S^2\times S^2\) is modified into a manifold \(X\), where \(\mathbb{L}\) and \(L_{1,1}\) are not only disjoint and monotone, but also Hamiltonian isotopic to a Clifford torus in \(X\). The constructed spheres are used to prove this homotopy based on the work of \textit{K. Cieliebak} and \textit{M. Schwingenheuer} [Pac. J. Math. 299, No. 2, 427--468 (2019; Zbl 1435.53057)]. It then follows from the Lagrangian Floer theory that \(\mathbb{L}\) and \(L_{1,1}\) must intersect, contrary to the assumption. The negative result is proved by explicit embedding of the closure of \(P(1,1)\) into the complement of the union of \(L_{k,l}\) using a time-dependent Hamiltonian flow. The disjoint Lagrangian is then on the boundary of the image.