Conformal Symplectic structures, Foliations and Contact Structures

TL;DR

This paper establishes two h-principles for conformal symplectic and leafwise conformal symplectic structures, enabling deformations of foliations into contact structures using advanced topological and geometric tools.

Contribution

It introduces new existence h-principles for conformal symplectic structures on closed and foliated manifolds, extending contact topology techniques.

Findings

Proves existence of conformal symplectic structures on closed manifolds.

Shows how to deform certain foliations into contact structures.

Utilizes the Borman-Eliashberg-Murphy h-principle and foliated Morse theory.

Abstract

This paper presents two existence h-principles, the first for conformal symplectic structures on closed manifolds, and the second for leafwise conformal symplectic structures on foliated manifolds with non empty boundary. The latter h-principle allows to linearly deform certain codimension-$1$ foliations to contact structures. These results are essentially applications of the Borman-Eliashberg-Murphy h-principle for overtwisted contact structures and of the Eliashberg-Murphy symplectization of cobordisms, together with tools pertaining to foliated Morse theory, which are elaborated here.