Surgery on Lagrangian and Legendrian Singularities

Abstract

Let $π: E\to M$ be a smooth fiber bundle whose total space is a symplectic manifold and whose fibers are Lagrangian. Let $L$ be an embedded Lagrangian submanifold of $E$. In the paper we address the following question: how can one simplify the singularities of the projection $π: L \to M$ by a Hamiltonian isotopy of $L$ inside $E$? We give an answer in the case when $dim L = 2$ and both $L$ and $M$ are orientable. A weaker version of the result is proved in the higher-dimensional case. Similar results hold in the contact category. As a corollary one gets an answer to one of the questions of V.Arnold about the four cusps on the caustic in the case of the Lagrangian collapse. As another corollary we disprove Y.Chekanov's conjecture about singularities of the Lagrangian projection of certain Lagrangian tori in $R^4$.