Journal of Symplectic Geometry

Volume 17 (2019)

Number 3

Conformal symplectic geometry of cotangent bundles

Pages: 639 – 661



Baptiste Chantraine (Département de Mathématiques, Laboratoire de Mathématiques Jean Leray, Université de Nantes, France)

Emmy Murphy (Department of Mathematics, Northwestern University, Evanston, Illinois, U.S.A.)


We prove a version of the Arnol’d conjecture for Lagrangian submanifolds of conformal symplectic manifolds: a Lagrangian $L$ which has non-zero Morse–Novikov homology for the restriction of the Lee form $\beta$ cannot be disjoined from itself by a $C^0$-small Hamiltonian isotopy. Furthermore for generic such isotopies the number of intersection points equals at least the sum of the free Betti numbers of the Morse–Novikov homology of $\beta$. We also give a short exposition of conformal symplectic geometry, aimed at readers who are familiar with (standard) symplectic or contact geometry.

Received 20 June 2016

Accepted 22 February 2018

Published 9 September 2019