Symplectic Energy and Lagrangian Intersection Under Legendrian Deformations

TL;DR

This paper establishes a lower bound on the intersection number of Lagrangian submanifolds under Legendrian deformations in symplectic manifolds, linking it to topological invariants.

Contribution

It introduces a new estimate for Lagrangian intersections under Legendrian deformations, connecting symplectic topology with Betti number calculations.

Findings

Intersection number bounded below by Betti numbers

Applicable to transversally intersecting Lagrangians

Provides new tools for symplectic topology analysis

Abstract

Let M be a compact symplectic manifold, and L be a closed Lagrangian submanifold which can be lifted to a Legendrian submanifold in the contactization of M. For any Legendrian deformation of L satisfying some given conditions, we get a new Lagrangian submanifold L'. We prove that the number of intersection of L and L' can be estimated from below by the sum of $Z_2$-Betti numbers of L, provided they intersect transversally.