Spectrally-large scale geometry in cotangent bundles

Jun Zhang; Qi Feng

arxiv: 2401.17590 · v2 · submitted 2024-01-31 · 🧮 math.SG · math.MG

Spectrally-large scale geometry in cotangent bundles

Qi Feng , Jun Zhang This is my paper

Pith reviewed 2026-05-24 03:43 UTC · model grok-4.3

classification 🧮 math.SG math.MG

keywords cotangent bundlesHamiltonian orbitsspectral metricFloer theoryquasi-isometric embeddingLiouville domainslarge-scale geometryboundary depth

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The pith

The Ham-orbit space from a fiber in many cotangent bundles quasi-isometrically embeds an infinite-dimensional normed vector space under the spectral metric.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that the space of Hamiltonian orbits from a fiber in a large family of cotangent bundles contains a quasi-isometric embedding of an infinite-dimensional normed vector space when measured by the Floer-theoretic spectral metric. A similar embedding holds for the group of compactly supported Hamiltonian diffeomorphisms on some of these bundles. The argument proceeds by extending a relation between boundary depth and spectral norm to the setting of Liouville domains and then adapting earlier constructions to produce the embedding. Readers interested in the large-scale geometry of symplectic spaces may care because the result shows that these orbit spaces carry infinite-dimensional features detectable by the spectral metric.

Core claim

The Ham-orbit space from a fiber of a large family of cotangent bundles, as a metric space with respect to the Floer-theoretic spectral metric, contains a quasi-isometric embedding of an infinite-dimensional normed vector space. The same conclusion holds for the group of compactly supported Hamiltonian diffeomorphisms of some cotangent bundles.

What carries the argument

The generalization of a relation between boundary depth and spectral norm to Liouville domains, which permits the adaptation of constructions that produce the desired quasi-isometric embedding.

Load-bearing premise

The relation between boundary depth and spectral norm extends to Liouville domains without extra restrictions that would block the embedding construction.

What would settle it

A concrete Liouville domain in which the boundary depth fails to bound the spectral norm from below in the manner required for the embedding to exist.

read the original abstract

In this paper, we prove that the ${\rm Ham}$-orbit space from a fiber of a large family of cotangent bundles, as a metric space with respect to the Floer-theoretic spectral metric, contains a quasi-isometric embedding of an infinite-dimensional normed vector space. The same conclusion holds for the group of compactly supported Hamiltonian diffeomorphisms of some cotangent bundles. To prove this, we generalize a result, relating boundary depth and spectral norm for closed symplectic manifolds in Kislev-Shelukhin's recent work, to Liouville domains. Then we modify Usher's constructions (which were used to obtain Hofer-large scale geometric properties) to achieve our desired conclusions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The abstract claims an infinite-dimensional quasi-isometric embedding into the spectral metric on Ham orbits of cotangent bundles via a Kislev-Shelukhin generalization, but only the abstract is available.

read the letter

The paper shows that for a large family of cotangent bundles, the space of Ham orbits of a fiber, with the Floer spectral metric, contains a quasi-isometric copy of an infinite-dimensional normed vector space. The same holds for the group of compactly supported Hamiltonian diffeomorphisms on some of these bundles. They get there by first extending a relation from Kislev and Shelukhin that connects boundary depth to the spectral norm, moving it from closed symplectic manifolds to Liouville domains. Then they adjust Usher's earlier constructions, which were originally for Hofer-large scale properties, to fit the spectral metric setting. This is a direct extension of existing techniques rather than a new method. The cotangent bundle examples fit naturally because they are Liouville and have the right structure for Floer theory. The soft spot is obvious: we only have the abstract. There is no derivation or explicit construction visible, so it is not possible to verify whether the generalized relation holds without restrictions that would stop the embedding from working, or whether the modified Usher steps actually produce the infinite-dimensional flat subspace. The citation pattern looks standard, referencing the relevant recent papers on spectral norms and large-scale geometry. This paper is aimed at symplectic geometers who work on metric properties of Hamiltonian groups and orbit spaces. Someone familiar with the Kislev-Shelukhin result and Usher's work would see the value immediately if the details check out. It deserves a serious referee because the statement is precise and the approach is grounded in prior literature, even though the full argument needs examination. I would recommend sending it to peer review, with the note that the generalization to Liouville domains is the part that needs the closest look.

Referee Report

0 major / 1 minor

Summary. The manuscript claims to prove that the space of Ham-orbits of a fiber in a large family of cotangent bundles, equipped with the Floer-theoretic spectral metric, admits a quasi-isometric embedding of an infinite-dimensional normed vector space; the same conclusion is asserted for the group of compactly supported Hamiltonian diffeomorphisms of certain cotangent bundles. The argument proceeds by generalizing the Kislev-Shelukhin relation between boundary depth and spectral norm from closed symplectic manifolds to Liouville domains, followed by a modification of Usher's constructions previously used for Hofer-large-scale geometry.

Significance. If the stated generalization and embedding constructions hold, the result would establish that these spectral metrics on Ham-orbit spaces and compactly supported diffeomorphism groups exhibit infinite-dimensional large-scale geometry, extending known finite-dimensional phenomena and providing a new source of examples with rich quasi-isometric structure in symplectic geometry.

minor comments (1)

The abstract refers to 'a large family' and 'some cotangent bundles' without specifying the precise class of manifolds or the restrictions under which the generalization applies; this vagueness makes it difficult to assess the scope of the claimed result from the provided text alone.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript. The report provides no specific major comments to address point by point. We note the 'uncertain' recommendation but, absent detailed concerns, have no further points to respond to at this time.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The abstract presents a proof strategy that generalizes an external result (Kislev-Shelukhin on closed manifolds) to Liouville domains and modifies constructions from Usher. No equations, fitted parameters, self-citations, or definitional reductions appear in the text. The claimed embedding result is derived from these independent steps rather than reducing to its own inputs by construction. This is the expected non-finding for an abstract-only proof paper without visible internal equivalences.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no explicit free parameters, axioms, or invented entities are stated. The proof strategy relies on background results in Floer theory and Liouville geometry that are not detailed here.

pith-pipeline@v0.9.0 · 5603 in / 1178 out tokens · 19338 ms · 2026-05-24T03:43:35.322973+00:00 · methodology

discussion (0)

Forward citations

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Hofer geometry of $A_3$-configurations
math.SG 2024-02 unverdicted novelty 6.0

A3-configurations of exact Lagrangian spheres imply quasi-isometric embeddings of infinite-dimensional l^∞ spaces into Hofer-metric Lagrangian spaces and into Ham_c(M).