Preservation of Area Under Hamiltonian Isotopy

This is a write up of the proof that area is perserved under Hamiltonian isotopy. It is very similar to the proof we came up with in the case of a Hamiltonian flow for a fixed hamiltonian vector field \(X_H\), but slightly different as I tried to push the proof though in the general case where our vector field is time dependent, i.e. \(X_{H_t}\), epsecially since we were at this point considering general isotopies of our Lagrangians.

Hamiltonian_Isotopy

Ongoing Questions

Questions that came up during discussions that we want to keep in mind for later. Questions may be put aside to give time to consult references, because we wanted someone else’s input, or simply because the question was becoming a time suck. We may take questions off the list as they are resolved to our satisfaction.

Current questions:

  1. What is a coherent sheaf? Is there a good intuition around them or deciding when a given sheaf is coherent? What is a non-example? (Consider looking at Polterovich’s sheafs in symplectic topology notes)
  2. What is Denis Arroux referencing when he references a rank \( p\) degree \(q\) vector bundle? In particular is there an intuition we should be using when he mentions the degree aspect?

Resolved questions (see other posts in “Notes on Ongoing Questions”)

  1. When creating the operator on our psuedoholomorphic disks we create a time dependent family of almost complex structures \(J_t\) in order to ensure the transversality of the operator. This is the same \(t\) as the \(t\)-coordinate in our holomorphic strip \(\mathbb{R}\times [0,1]\). We are confused why the \(t\) dependence is there.