Conformal symplectic geometry of cotangent bundles

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.

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Received 20 June 2016

Accepted 25 July 2018

Published 9 September 2019