[Paper Review] Symplectic Banach-Mazur distances between subsets of C^n

Michael Usher|arXiv (Cornell University)|Nov 2, 2018

Topological and Geometric Data Analysis12 references4 citations

TL;DR

This paper introduces coarse and fine symplectic Banach-Mazur distances on open Liouville domains in $ℂ^n$, distinguishing them via filtered equivariant symplectic homology. It shows that while sequences may converge coarsely to an ellipsoid, they can diverge infinitely under the fine distance, and constructs quasi-isometric embeddings of $ℝ^D$ into the space of star-shaped domains for any finite $D$, demonstrating rich geometric complexity beyond symplectic capacities.

ABSTRACT

Following proposals of Ostrover and Polterovich, we introduce and study "coarse" and "fine" versions of a symplectic Banach-Mazur distance on certain open subsets of $\mathbb{C}^n$ and other open Liouville domains. The coarse version declares two such domains to be close to each other if each domain admits a Liouville embedding into a slight dilate of the other; the fine version, which is similar to the distance on subsets of cotangent bundles of surfaces recently studied by Stojisavljević and Zhang, imposes an additional requirement on the images of these embeddings that is motivated by the definition of the classical Banach-Mazur distance on convex bodies. Our first main result is that the coarse and fine distances are quite different from each other, in that there are sequences that converge coarsely to an ellipsoid but diverge to infinity with respect to the fine distance. Our other main result is that, with respect to the fine distance, the space of star-shaped domains in $\mathbb{C}^n$ admits quasi-isometric embeddings of $\mathbb{R}^D$ for every finite dimension $D$. Our constructions are obtained from a general method of constructing $(2n+2)$-dimensional Liouville domains whose boundaries have Reeb dynamics determined by certain autonomous Hamiltonian flows on a given $2n$-dimensional Liouville domain. The bounds underlying our main results are proven using filtered equivariant symplectic homology via methods from prior joint work with Gutt.

Motivation & Objective

• To define and study coarse and fine symplectic Banach-Mazur distances on open Liouville domains in $ℂ^n$, generalizing classical Banach-Mazur concepts to symplectic geometry.
• To investigate the topological and geometric distinctions between the coarse and fine versions of the distance, particularly in relation to convergence and divergence of sequences.
• To demonstrate that the fine distance captures significantly more structure than the coarse distance, using advanced tools from symplectic topology.
• To construct quasi-isometric embeddings of Euclidean spaces $ℝ^D$ into the space of star-shaped domains under the fine distance, revealing high-dimensional complexity.
• To extend the framework of symplectic capacities by incorporating filtered equivariant symplectic homology to derive stronger obstructions than classical capacities allow.

Proposed method

• Introduces open Liouville domains as exact symplectic manifolds with a Liouville vector field whose negative-time flow shrinks the domain into compact subsets.
• Defines the coarse symplectic Banach-Mazur distance via existence of Liouville embeddings between dilates of domains.
• Implements the fine distance by adding a compatibility condition on the compositions of embeddings, inspired by classical Banach-Mazur geometry.
• Constructs $(2n+2)$-dimensional Liouville domains from $2n$-dimensional ones using autonomous Hamiltonian flows on the boundary, controlling Reeb dynamics.
• Applies filtered equivariant symplectic homology to detect non-trivial persistence modules and derive lower bounds on the fine distance.
• Uses Conley–Zehnder indices and spectral invariants to analyze closed Reeb orbits and track changes in homology ranks across scales.

Experimental results

Research questions

• RQ1 How do the coarse and fine symplectic Banach-Mazur distances differ in their topological behavior, particularly in terms of convergence and divergence of sequences?
• RQ2 Can the fine distance detect geometric complexity beyond what is captured by symplectic capacities or the coarse distance?
• RQ3 To what extent can the space of star-shaped domains in $ℂ^n$ contain quasi-isometric copies of high-dimensional Euclidean spaces under the fine distance?
• RQ4 What role does filtered equivariant symplectic homology play in distinguishing symplectic embeddings where classical capacities fail?
• RQ5 How do irrational parameters in Hamiltonian flows affect the structure of Reeb orbits and the resulting homological invariants?

Key findings

• The coarse and fine symplectic Banach-Mazur distances are topologically distinct: sequences can converge coarsely to an ellipsoid while diverging infinitely under the fine distance.
• For any finite dimension $D$, the space of star-shaped domains in $ℂ^n$ admits a quasi-isometric embedding of $ℝ^D$ under the fine distance, indicating high intrinsic geometric complexity.
• The fine distance satisfies the lower bound $\delta_f((W_{H_{\vec{\zeta}}}^\circ,\hat{\lambda}),(W_{H_{\vec{\epsilon}}}^\circ,\hat{\lambda})) \geq \max_{1\leq m\leq D}\left(\min\left\{\frac{1}{\epsilon_{m}},\left(\frac{\zeta_{m}}{\epsilon_{m}}\right)^2\right\}\right)$, derived from spectral invariants in filtered equivariant symplectic homology.
• Filtered equivariant symplectic homology detects non-degenerate Reeb orbits with Conley–Zehnder indices congruent to $n$ modulo 2, and their persistence across scales provides strong obstructions to symplectic embeddings.
• The canonical map $CH^{s}_{2-3n}(W_{H_{\vec{\epsilon}}}^\circ,\hat{\lambda})\to CH^{t}_{2-3n}(W_{H_{\vec{\epsilon}}}^\circ,\hat{\lambda})$ is injective for $s<t<1$, which is used to derive the main lower bounds.
• When $\epsilon_m$ are irrational, the number of simple Reeb orbits of Conley–Zehnder index $2-3n$ with period at most $b$ is exactly $m$, and this count increases with $m$, enabling precise control over homological ranks.