New Invariants for Hamiltonian Isotopy Classes of Monotone Lagrangian Torus Fibers

Based on the exotic Lagrangian tori constructed in CP2

by Vianna and Galkin-Mikhalkin, we construct for each Markov triple three monotone Lagrangian tori in the 4-ball, and for triples with distinct entries show that these tori lie in different Hamiltonian isotopy classes. Defining the outer radius of such a torus as the capacity of the smallest ball containing a representative of the Hamiltonian isotopy class, we compute the outer radius for an infinite sequence of tori and show it distinguishes these tori.

We do a similar study for monotone tori in the cube. If such a torus arises from a degeneration of S2×S2 with triangular moment image, it gives rise to four different Hamiltonian isotopy classes of tori in the cube; on the other hand, if the monotone torus arises from a non-trivial degeneration with quadrilateral moment image, it gives rise to eight different Hamiltonian isotopy classes of tori in the cube. In particular, we find infinitely many pairs of monotone tori in S2×S2

which are symplectomorphic but not Hamiltonian isotopic.

This talk is based on work joint with Richard Hind and Grisha Mikhalkin.