Relative growth rate and contact Banach–Mazur distance

Abstract

In this paper, we define non-linear versions of Banach-Mazur distance in the contact geometry set-up, called contact Banach-Mazur distances and denoted by d CBM . Explicitly, we consider the following two set-ups, either on a contact manifold W × S 1 where W is a Liouville manifold, or a closed Liouville-fillable contact manifold M. The inputs of d CBM are different in these two cases. In the former case the inputs are (contact) star-shaped domains of W × S 1 which correspond to the homotopy classes of positive contact isotopies, and in the latter case the inputs are contact 1-forms of M inducing the same contact structure. In particular, the contact Banach-Mazur distance d CBM defined in the former case is motivated by the concept, relative growth rate, which was originally defined and studied by Eliashberg-Polterovich. The main results are the large-scale geometric properties in terms of d CBM . In addition, we propose a quantitative comparison between elements in a certain subcategory of the derived categories of sheaves of modules (over certain topological spaces). This is based on several important properties of the singular support of sheaves and Guillermou-Kashiwara-Schapira's sheaf quantization.

Symplectic embeddings from concave toric domains into convex ones

“…The space of contact forms giving the standard contact structure on S 3 has infinite quasi-flat rank with respect to d CBM . Indeed, by [18], Lemma 8, the metric d CBM on the boundary forms is bounded below by the metric d SM applied to our starshaped domains. However our upper bounds on d SM are derived from explicit symplectic embeddings, namely inclusions, which imply the same upper bounds for d CBM .…”

“…There is an analogue of the symplectic Banach-Mazur distance in contact geometry, more precisely for the space of contact forms on a fixed contact structure, called the contact Banach-Mazur distance, denoted d CBM ; we will not state the definition here for brevity, referring the reader instead to [18,Defn. 4].…”

“…In other words our quasi-flats mapping R N into the space of starshaped domains can be composed with the map taking starshaped domains to contact forms on S 3 , then the resulting composition is a quasi-flat in the space of contact forms. For previous work on the contact Banach-Mazur distance, we refer the reader to [18,16].…”

Hamiltonian knottedness and lifting paths from the shape invariant

“…It is Sikorav's work [Sik89] and Eliashberg's work [Eli91] that first observed the application of the shape invariant to the study of the rigidity of symplectic or contact embeddings. For further development in this direction, see [Mül19, RZ21, HZ21].…”