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

TL;DR

This paper establishes new energy-capacity inequalities in symplectic topology, leading to results on the uniqueness of continuous Hamiltonians and improvements in C^0-rigidity of the Poisson bracket.

Contribution

It introduces a novel energy-capacity inequality and applies it to prove the uniqueness of continuous Hamiltonians, advancing understanding in C^0-symplectic topology.

Findings

Proved a new energy-capacity inequality for rational symplectic manifolds.

Showed that continuous functions approximated by Hamiltonians with converging flows must vanish.

Enhanced results on the C^0-rigidity of the Poisson bracket.

Abstract

We prove a new variant of the energy-capacity inequality for closed rational symplectic manifolds (as well as certain open manifolds such as 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 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.