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Article Dans Une Revue International Journal of Mathematics Année : 2021

Persistence modules, symplectic Banach–Mazur distance and Riemannian metrics

Résumé

We use persistence modules and their corresponding barcodes to quantitatively distinguish between different fiberwise star-shaped domains in the cotangent bundle of a fixed manifold. The distance between two fiberwise star-shaped domains is measured by a nonlinear version of the classical Banach–Mazur distance, called symplectic Banach–Mazur distance and denoted by [Formula: see text]. The relevant persistence modules come from filtered symplectic homology and are stable with respect to [Formula: see text]. Our main focus is on the space of unit codisc bundles of orientable surfaces of positive genus, equipped with Riemannian metrics. We consider some questions about large-scale geometry of this space and in particular we give a construction of a quasi-isometric embedding of [Formula: see text] into this space for all [Formula: see text]. On the other hand, in the case of domains in [Formula: see text], we can show that the corresponding metric space has infinite diameter. Finally, we discuss the existence of closed geodesics whose energies can be controlled.

Dates et versions

hal-03914081 , version 1 (27-12-2022)

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Vukašin Stojisavljević, Jun Zhang. Persistence modules, symplectic Banach–Mazur distance and Riemannian metrics. International Journal of Mathematics, 2021, 32 (07), pp.2150040. ⟨10.1142/S0129167X21500403⟩. ⟨hal-03914081⟩
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