Hallo,


I am looking for a preprint "Analytic Lagrangian Submanifolds" by Guillemin, Sternberg. I googled it but without any success. Does any one know how I could get this preprint. Or are there similar ones? I am actually interested in understanding better the construction of a defining phase function for a Lagrangian submanifold and to understand the uniqueness. Is there any other literature on that? Actually I am interested in the proof of the following: Let $M$ be a connected Lagrangian submanifold of a Kähler manifold $\Omega$, then there is a neighbourhood $U$ of $M$ in $\Omega$ and a unique defining phase function $\phi$ on $U$ for $M$. 


hapchiu 



Analytic Lagrangian Submanifolds
Dear Hapchiu,


To understand the generating function construction of Lagrangian submanifolds I recommend the following:


1. Take the few pages in "Geometric Asymptotics" by Guillemin and Sternberg, where they very neatly describe the construction of a Lagrangian submanifold in $T^\* M$ as the reduction of the graph of the differential of a function on $M \times R^n$ (actually you can replace other manifolds for $R^n$ and this *sometimes* turns out to be useful).
2. There is a little Comptes Rendus note by Giroux that explains what are the topological obstructions for the existence of a generating function. As far as I know this note has not made its way into other literature.
3. The most important remark on generating functions is that if $M$ is compact and $L$ is a Lagrangian submanifold in $T^\*M$ that is hamiltonian isotopic to the zero section, then $L$ admits a very special type of generating function (quadratic at infinity). This allows you to prove via Morse-Conley theory that $L$ intersects the zero section as much as Morse or Luisternik-Schnirelman theory requires. This you can find in papers by Sikorav and Laudenbach and Chaperon.
4. The second most important remark is that there is a sort of uniqueness (modulo certain basic operations) in the choice of generating function quadratic at infinity. Thisus due to Viterbo (Symplectic topology as the geometry of generating functions, a remarkable paper well worth reading), but unfortunately there is a gap in the proof of the uniqueness result. This gap was filled by David Theret in his thesis. The interest of the uniqueness result is that it allows you to define symplectic capacities and other symplectic topological invariants.


I'm sorry about not being more precise with the references, but google and MathScinet will guide you better than my memory. 


