The Weinstein conjecture in the presence of submanifolds having a Legendrian foliation

TL;DR

This paper extends Hofer's holomorphic curve techniques to higher-dimensional contact manifolds, proving the Weinstein conjecture in new cases involving submanifolds with Legendrian foliations, including connected sums with real projective spaces.

Contribution

It adapts Hofer's method to higher dimensions and specific submanifold configurations, establishing the existence of Reeb orbits in these cases.

Findings

Connected sum with real projective space has a closed contractible Reeb orbit.

Extension of holomorphic curve techniques to higher-dimensional contact manifolds.

Validation of the Weinstein conjecture in new geometric contexts.

Abstract

Helmut Hofer introduced in '93 a novel technique based on holomorphic curves to prove the Weinstein conjecture. Among the cases where these methods apply are all contact 3--manifolds $(M,\xi)$ with $\pi_2(M) \ne 0$. We modify Hofer's argument to prove the Weinstein conjecture for some examples of higher dimensional contact manifolds. In particular, we are able to show that the connected sum with a real projective space always has a closed contractible Reeb orbit.