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Commentarii Mathematici Helvetici

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Volume 90, Issue 1, 2015, pp. 1–21
DOI: 10.4171/CMH/343

Published online: 2015-02-23

New energy-capacity-type inequalities and uniqueness of continuous Hamiltonians

Vincent Humilière[1], Rémi Leclercq[2] and Sobhan Seyfaddini[3]

(1) Université Pierre et Marie Curie, Paris, France
(2) Université Paris-Sud, Orsay, France
(3) University of California, Berkeley, USA

We prove a new variant of the energy-capacity inequality for closed rational symplectic manifolds (as well as certain open manifolds such as $\mathbb R^{2n}$, cotangent bundle of closed manifolds…) and we derive some consequences to $C^0$–symplectic topology. Namely, we prove that a continuous function which is a uniform limit of smooth normalized Hamiltonians whose flows converge to the identity for the spectral (or Hofer’s) distance must vanish. This gives a new proof of uniqueness of continuous generating Hamiltonian for hameomorphisms. This also allows us to improve a result by Cardin and Viterbo on the $C^0$–rigidity of the Poisson bracket.

Keywords: Symplectic manifolds, Hamiltonian diffeomorphism group, $C^0$–symplectic topology, Hofer’s distance, spectral invariants

Humilière Vincent, Leclercq Rémi, Seyfaddini Sobhan: New energy-capacity-type inequalities and uniqueness of continuous Hamiltonians. Comment. Math. Helv. 90 (2015), 1-21. doi: 10.4171/CMH/343