Hamiltonian stationary submanifolds of compact symplectic manifolds

Hamiltonian stationary submanifolds of compact symplectic manifolds

Tue, 27/01/2009
15:45 		Dominic Joyce (Oxford) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Hamiltonian stationary submanifolds of compact symplectic manifolds
Let $ (M,\omega) $ be a symplectic manifold, and $ g $ a Riemannian metric on $ M $ compatible with $ \omega $. If $ L $ is a compact Lagrangian submanifold of $ (M,\omega) $, we can compute the volume Vol$ (L) $ of $ L $ using $ g $. A Lagrangian $ L $ is called Hamiltonian stationary if it is a stationary point of the volume functional amongst Lagrangians Hamiltonian isotopic to $ L $. Suppose $ L' $ is a compact Lagrangian in $ {\mathbb C}^n $ which is Hamiltonian stationary and rigid, that is, all infinitesimal Hamiltonian deformations of $ L $ as a Hamiltonian stationary Lagrangian come from rigid motions of $ {\mathbb C}^n $. An example of such $ L' $ is the $ n $-torus $  \bigl\{(z_1,\ldots,z_n)\in{\mathbb C}^n:\vert z_1\vert=a_1, \ldots,\vert z_n\vert=a_n\bigr\} $, for small $ a_1,\ldots,a_n>0 $. I will explain a construction of Hamiltonian stationary Lagrangians in any compact symplectic manifold $ (M,\omega) $, which works by `gluing in' $ tL' $ near a point $ p $ in $ M $ for small $ t>0 $.