On the large-scale geometry of domains in an exact symplectic 4-manifold
Dan Cristofaro-Gardiner, Richard Hind

TL;DR
This paper investigates the large-scale geometry of open subsets in exact symplectic 4-manifolds, revealing infinite-dimensional structures and contrasting behaviors between convex and dynamically convex domains.
Contribution
It demonstrates that the space of open subsets in any complete exact symplectic 4-manifold has infinite dimension under the symplectic Banach-Mazur distance, with new constructions for dynamically convex domains.
Findings
The space of convex domains in ^4 is quasi-isometric to a plane.
The space of dynamically convex domains in ^4 has infinite dimension.
The space of star-shaped domains in T^*S^2 has infinite dimension.
Abstract
We show that the space of open subsets of any complete and exact symplectic -manifold has infinite dimension with respect to the symplectic Banach-Mazur distance; the quasi-flats we construct take values in the set of dynamically convex domains. In the case of , we therefore obtain the following contrast: the space of convex domains is quasi-isometric to a plane, while the space of dynamically convex ones has infinite dimension. In the case of , a variant of our construction resolves a conjecture of Stojisavljevi\'{c} and Zhang, asserting that the space of star-shaped domains in has infinite dimension. Another corollary is that the space of contact forms giving the standard contact structure on has infinite dimension with respect to the contact Banach-Mazur distance.
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Taxonomy
TopicsGeometric and Algebraic Topology · Geometry and complex manifolds · Topological and Geometric Data Analysis
