Persistence modules, symplectic Banach–Mazur distance and Riemannian metrics

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.

Infinitesimal Hilbertianity of Locally $$\mathrm{CAT}(\kappa )$$-Spaces

We show that, given a metric space $$(\mathrm{Y},\textsf {d} )$$ ( Y , d ) of curvature bounded from above in the sense of Alexandrov, and a positive Radon measure $$\mu $$ μ on $$\mathrm{Y}$$ Y giving finite mass to bounded sets, the resulting metric measure space $$(\mathrm{Y},\textsf {d} ,\mu )$$ ( Y , d , μ ) is infinitesimally Hilbertian, i.e. the Sobolev space $$W^{1,2}(\mathrm{Y},\textsf {d} ,\mu )$$ W 1 , 2 ( Y , d , μ ) is a Hilbert space. The result is obtained by constructing an isometric embedding of the ‘abstract and analytical’ space of derivations into the ‘concrete and geometrical’ bundle whose fibre at $$x\in \mathrm{Y}$$ x ∈ Y is the tangent cone at x of $$\mathrm{Y}$$ Y . The conclusion then follows from the fact that for every $$x\in \mathrm{Y}$$ x ∈ Y such a cone is a $$\mathrm{CAT}(0)$$ CAT ( 0 ) space and, as such, has a Hilbert-like structure.

An Intrinsic Characterization of Five Points in a CAT(0) Space

Gromov (2001) and Sturm (2003) proved that any four points in a CAT(0) space satisfy a certain family of inequalities. We call those inequalities the ⊠-inequalities, following the notation used by Gromov. In this paper, we prove that a metric space X containing at most five points admits an isometric embedding into a CAT(0) space if and only if any four points in X satisfy the ⊠-inequalities. To prove this, we introduce a new family of necessary conditions for a metric space to admit an isometric embedding into a CAT(0) space by modifying and generalizing Gromov’s cycle conditions. Furthermore, we prove that if a metric space satisfies all those necessary conditions, then it admits an isometric embedding into a CAT(0) space. This work presents a new approach to characterizing those metric spaces that admit an isometric embedding into a CAT(0) space.