Compactly supported Hamiltonian loops with a non-zero Calabi invariant

Abstract

We give examples of compactly supported Hamiltonian loops with a non-zero Calabi invariant on certain open symplectic manifolds. 1 Introduction Let (M, ω) be an open symplectic 2n-dimensional manifold. Denote by Ham(M) the group of Hamiltonian diffeomorphisms generated by compactly supported Hamiltonian functions. Denote by Ham(M) the universal cover of Ham(M). We write elements of Ham(M) as [{f t } t∈[0,1] ], where {f t } t∈[0,1] is a smooth path of Hamiltonian diffeomorphisms with f 0 = Id, and [{f t } t∈[0,1] ] stands for the homotopy class of {f t } t∈[0,1] with fixed end points. In what follows we use the notation Vol(M) := M ω n . Introduce the Calabi homomorphism Cal : Ham(M) → R, as Cal([{f t } t∈[0,1] ]) = * Partially supported by European Research Council advanced grant 338809

On the rank of $π_1(\text{Ham})$

“…As a byproduct of the arguments used to prove the main result, we are also able to show that Calabi's morphism on the one-point blow up of (R 4 , ω 0 ) is non trivial. The first examples of open manifolds whose Calabi's morphism is non trivial are due to A. Kislev [5]. Alongside we prove that the rank of the fundamental group of Ham( M , ω κ ) is positive where ( M , ω κ ) is the one-point blow of weight κ of (M, ω).…”