Packing Lagrangian tori

TL;DR

This paper investigates the packing of symplectic manifolds with integral Lagrangian tori, establishing maximality of the Clifford torus in certain spaces and existence results in symplectic polydisks.

Contribution

It proves the Clifford torus in $S^2 imes S^2$ is a maximal integral Lagrangian packing and shows existence of integral Lagrangian tori in symplectic polydisks beyond standard tori.

Findings

Clifford torus is a maximal integral Lagrangian packing in $S^2 imes S^2$.

Any other integral Lagrangian torus in $S^2 imes S^2$ must intersect the Clifford torus.

In symplectic polydisks with $a,b>2$, there exists an integral Lagrangian torus outside standard product tori.

Abstract

In this paper we consider the problem of packing a symplectic manifold with integral Lagrangian tori, that is Lagrangian tori whose area homomorphsims take only integer values. We prove that the Clifford torus in $S^2 \times S^2$ is a maximal integral packing, in the sense that any other integral Lagranian torus must intersect it. In the other direction, we show that in any symplectic polydisk $P(a,b)$ with $a,b>2$, there is at least one integral Lagrangian torus in the complement of the collection of standard product integral Lagrangian tori.