Surgery on Lagrangian and Legendrian Singularities

TL;DR

This paper investigates how to simplify singularities of Lagrangian projections via Hamiltonian isotopies, providing new results in low dimensions, contact geometry, and addressing longstanding conjectures in symplectic topology.

Contribution

It offers a method to reduce singularities of Lagrangian projections through Hamiltonian isotopies, with specific results for 2-dimensional cases and extensions to higher dimensions and contact geometry.

Findings

Simplification of singularities in 2D Lagrangian projections

Disproof of Chekanov's conjecture on Lagrangian tori

Resolution of Arnold's question on caustic cusps

Abstract

Let $\pi : E\to M$ 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 $\pi: L \to M$ 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 $R^4$.