Symplectic cobordisms and the strong Weinstein conjecture

TL;DR

This paper proves the strong Weinstein conjecture for certain high-dimensional contact manifolds by analyzing holomorphic spheres in symplectic cobordisms, leading to new insights and a symplectic capacity.

Contribution

It introduces a method using holomorphic spheres in symplectic cobordisms to confirm the strong Weinstein conjecture in higher dimensions.

Findings

Confirmed the strong Weinstein conjecture for various high-dimensional contact manifolds.

Established a new symplectic capacity based on the quantitative analysis.

Linked the non-compactness of moduli spaces to the existence of Reeb orbits.

Abstract

We study holomorphic spheres in certain symplectic cobordisms and derive information about periodic Reeb orbits in the concave end of these cobordisms from the non-compactness of the relevant moduli spaces. We use this to confirm the strong Weinstein conjecture (predicting the existence of null-homologous Reeb links) for various higher-dimensional contact manifolds, including contact type hypersurfaces in subcritical Stein manifolds and in some cotangent bundles. The quantitative character of this result leads to the definition of a symplectic capacity.