Symplectic energy and Lagrangian intersection under Legendrian deformations

Hai-Long Her

Vol. 231 (2007), No. 2, 417–435

DOI: 10.2140/pjm.2007.231.417

Abstract

Let $M$ be a compact symplectic manifold, and $L\subset M$ 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 $L^{\prime }$. We prove that the number of intersection $L\cap L^{\prime }$ can be estimated from below by the sum of ${\mathbb{Z}}_2$-Betti numbers of $L$, provided they intersect transversally.

Keywords

Arnold conjecture, Floer homology, Lagrangian intersection, Symplectic energy

Mathematical Subject Classification 2000

Primary: 57R22, 53D40, 53D12, 53D10

Milestones

Received: 11 January 2006

Revised: 10 April 2007

Accepted: 12 April 2007

Published: 1 June 2007