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.