Variations at infinity in contact form geometry

Abstract

Let v be a nonsingular Morse–Smale vector field in the kernel of a contact form α, with Reeb vector field \(\xi\), defined on M3. We establish that the associated variational problem at infinity defined by the action functional on the stratified space \(\bigcup \Gamma_{2k}\) of curves made of \(\xi\)-pieces of orbits alternating with \(\pm v\)-pieces of orbits satisfies the Palais–Smale condition. This result takes a more special form for the standard contact structure of S3.