Singular Lagrangian intersections and the non-linear Cauchy-Riemann equation
A classical result of Lojasiewicz says that a bounded gradient flow trajectory of a real analytic function converges to a unique limit. I will discuss an analogous result for maps from a Riemann surface into a symplectic manifold that satisfy the non-linear Cauchy-Riemann equation with real analytic Lagrangian boundary conditions. The proof relies on an isoperimetric inequality that controls the singularities of real analytic Lagrangian intersections.
The Floer cohomology of a pair of Lagrangian submanifolds is defined using solutions of the non-linear Cauchy-Riemann equation, and depends in general on the global geometry of the ambient symplectic manifold. However, as a consequence of our result and Gromov's compactness theorem, we see that in certain situations, the Floer cohomology of a pair of Lagrangian submanifolds is a local invariant. This fits nicely with conjectures relating Floer cohomology and algebraic invariants of singular Lagrangian intersections arising from deformation quantization and perverse sheaves.
No background in symplectic geometry will be assumed. This talk is based on joint work with M. Verbitsky.