Exact lagrange submanifolds, periodic orbits and the cohomology of free loop spaces

Abstract

We prove that there are obstructions to the existence of an exact Lagrange embedding from a closed manifold L to T* N. This may be seen as an extension of Gromov’s theorem as formulated by Lalonde and Sikorav, showing that no such embedding exists for N open. For example we answer positively a question by Lalonde and Sikorav on the non-existence of exact Lagrange embeddings from T2 into T*S2E . Our obstruction is in terms of the cohomology of the loop space of L and N and the map induced by the embedding in the cohomologies of these loop spaces. In particular, we give obstructions to the existence of an exact Lagrangian embedding inducing a degree-zero map from L to N. As another application of our method, we prove the Weinstein conjecture in cotangent bundles of simply connected manifolds (removing an assumption in a previous joint paper with H. Hofer). A number of these results had been announced in [48] and [49] © 1997 Journal of Differential Geometry. © 1997 Applied Probability Trust.