The simplification of singularities of Lagrangian and Legendrian fronts

TL;DR

This paper proves a comprehensive h-principle allowing the simplification of singularities in Lagrangian and Legendrian fronts via Hamiltonian isotopies, provided no homotopy obstructions exist.

Contribution

It establishes a full, parametric h-principle for simplifying singularities of Lagrangian and Legendrian fronts, extending the understanding of their geometric flexibility.

Findings

Full h-principle for singularity simplification

Hamiltonian isotopies achieve the simplification

No homotopy obstructions are necessary

Abstract

We establish a full $h-$principle ($C^0-$close, relative, parametric) for the simplification of singularities of Lagrangian and Legendrian fronts. More precisely, we prove that if there is no homotopy theoretic obstruction to simplifying the singularities of tangency of a Lagrangian or Legendrian submanifold with respect to an ambient foliation by Lagrangian or Legendrian leaves, then the simplification can be achieved by means of a Hamiltonian isotopy.