On the injectivity radius in Hofer's geometry
François Lalonde Yasha Savelyev
Electronic Research Announcements 2014, 21(0): 177-185 doi: 10.3934/era.2014.21.177
In this note we consider the following conjecture: given any closed symplectic manifold $M$, there is a sufficiently small real positive number $\rho$ such that the open ball of radius $\rho$ in the Hofer metric centered at the identity on the group of Hamiltonian diffeomorphisms of $M$ is contractible, where the retraction takes place in that ball (this is the strong version of the conjecture) or inside the ambient group of Hamiltonian diffeomorphisms of $M$ (this is the weak version of the conjecture). We prove several results that support the weak form of the conjecture.
keywords: Lagrangian submanifolds Hofer's geometry Quantum characteristic classes.
Proof of the main conjecture on $g$-areas
François Lalonde Egor Shelukhin
Electronic Research Announcements 2015, 22(0): 92-102 doi: 10.3934/era.2015.22.92
In this paper, we prove the main conjecture on $g$-areas arising from [7]. That conjecture was announced by the first author in 2004. It~states that the $g$-area of any Hamiltonian diffeomorphism $\phi$ is equal to the positive Hofer pseudo-distance between $\phi$ and the subspace of Hamiltonian diffeomorphisms that can be expressed as a product of at most $g$ commutators.
keywords: Hofer's geometry Poisson bracket. g-area commutator length

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