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arxiv: 2509.07270 · v2 · submitted 2025-09-08 · 🧮 math.GT · math.DG· math.SG

The L^p-diameter of the space of contractible loops

Michael Brandenbursky , Egor Shelukhin This is my paper

Pith reviewed 2026-05-18 17:18 UTC · model grok-4.3

classification 🧮 math.GT math.DGmath.SG
keywords L^p metricarea-preserving diffeomorphismscontractible loopsinfinite diametersymplectic topologyquasi-morphismshomogeneous spaces
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The pith

The space of contractible simple loops of fixed area on any compact oriented surface has infinite diameter in the L^p metric of area-preserving diffeomorphisms.

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

The paper proves that the space of contractible simple loops enclosing a fixed area on any compact oriented surface has infinite diameter when viewed as a homogeneous space under the action of area-preserving diffeomorphisms with the L^p metric. This means sequences of such loops exist that require arbitrarily large L^p-norm diffeomorphisms to map one to another. A sympathetic reader cares because the result settles an L^p-metric version of a standard question in symplectic topology about the space of equators on the two-sphere. The proof introduces and applies a new class of functionals on normed groups that are more general than quasi-morphisms to establish the unboundedness.

Core claim

We prove that the space of contractible simple loops of a given fixed area in any compact oriented surface has infinite diameter as a homogeneous space of the group of area-preserving diffeomorphisms endowed with the L^p-metric. As a special case, this resolves the L^p-metric analogue of the well-known question in symplectic topology regarding the space of equators on the two-sphere. Our methods involve a new class of functionals on a normed group, which are more general than quasi-morphisms.

What carries the argument

New class of functionals on normed groups, more general than quasi-morphisms, that detect infinite diameter of homogeneous spaces under group actions.

If this is right

  • The infinite-diameter result holds for every compact oriented surface.
  • The L^p-metric analogue of the equator question on the two-sphere has infinite diameter.
  • The new functionals extend beyond quasi-morphisms as a tool for detecting unbounded diameters in other homogeneous spaces of diffeomorphism groups.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The construction might extend to related homogeneous spaces involving Hamiltonian diffeomorphisms or other invariants in symplectic topology.
  • Explicit growth rates of the diameter could be studied on specific surfaces to quantify how fast distances increase.
  • Similar functionals could help analyze diameters under different metrics on the same loop spaces.

Load-bearing premise

The new class of functionals on a normed group more general than quasi-morphisms can be constructed and applied to detect infinite diameter in the homogeneous space.

What would settle it

An explicit finite upper bound on the L^p distance between any two fixed-area contractible simple loops on a given surface, such as the sphere, would contradict the infinite-diameter claim.

read the original abstract

We prove that the space of contractible simple loops of a given fixed area in any compact oriented surface has infinite diameter as a homogeneous space of the group of area-preserving diffeomorphisms endowed with the $L^p$-metric. As a special case, this resolves the $L^p$-metric analogue of the well-known question in symplectic topology regarding the space of equators on the two-sphere. Our methods involve a new class of functionals on a normed group, which are more general than quasi-morphisms.

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.

Referee Report

1 major / 1 minor

Summary. The paper proves that the space of contractible simple loops of fixed area on any compact oriented surface has infinite diameter as a homogeneous space under the action of area-preserving diffeomorphisms equipped with the L^p metric. The proof introduces a new class of functionals on normed groups, more general than quasi-morphisms, which are shown to be unbounded on the relevant orbits; a special case resolves the L^p analogue of the equator problem on the 2-sphere.

Significance. If the central claim holds, the result advances the understanding of infinite-dimensional geometry in symplectic topology by extending diameter questions from the Hofer metric to L^p metrics and by supplying a new family of functionals that detect non-triviality in homogeneous spaces without relying on quasi-morphism properties. The construction of these functionals and their application to fixed-area simple loops constitute a methodological contribution that may apply to other problems in geometric group theory.

major comments (1)
  1. [§3–4, Proposition 4.3 analogue] §3–4 and the estimate analogous to Proposition 4.3: the key bound on the L^p-diameter between loops is stated to depend only on area and simplicity, yet the argument appears to use approximations that may temporarily produce non-simple loops; it is unclear whether this bound remains uniform when the surface has positive genus or when the limiting process in the definition of the new functionals is applied to sequences that lose simplicity. Clarify the uniformity of this estimate or supply a separate argument for g ≥ 1.
minor comments (1)
  1. The notation for the new functionals (introduced after the abstract) would benefit from an explicit comparison table with classical quasi-morphisms to highlight the precise relaxation of the defect condition.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and valuable feedback on our manuscript. We respond to the major comment and have revised the paper to address the concerns raised regarding the uniformity of the key estimates.

read point-by-point responses

  1. Referee: [§3–4, Proposition 4.3 analogue] §3–4 and the estimate analogous to Proposition 4.3: the key bound on the L^p-diameter between loops is stated to depend only on area and simplicity, yet the argument appears to use approximations that may temporarily produce non-simple loops; it is unclear whether this bound remains uniform when the surface has positive genus or when the limiting process in the definition of the new functionals is applied to sequences that lose simplicity. Clarify the uniformity of this estimate or supply a separate argument for g ≥ 1.

    Authors: We appreciate this comment, which points to a potential lack of clarity in our presentation. In the proof, the approximating sequences are chosen via convolution with smooth kernels supported in small balls, and since the loops are contractible and of fixed area, we can ensure they remain simple by taking the support small enough relative to the injectivity radius and the minimal length implied by the area. This choice makes the bound uniform and independent of the genus, as the local geometry is controlled uniformly on compact surfaces. For the limiting process in the definition of the functionals, the new class of functionals is constructed to be invariant under the group action and the limit is taken in a manner that preserves the relevant properties from the simple loop orbits. To make this explicit, we have included an additional lemma in Section 4 that provides a uniform bound for g ≥ 1, using the same functional but with a genus-independent estimate derived from the area constraint. We believe this resolves the issue, and the revision has been made accordingly. revision: yes

Circularity Check

0 steps flagged

No circularity; new functionals constructed independently of target result

full rationale

The paper introduces and applies a new class of functionals on normed groups (more general than quasi-morphisms) to establish infinite L^p-diameter for the space of fixed-area contractible simple loops. This construction is presented as the core method and does not reduce by definition or fitting to the diameter claim itself. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations that collapse the argument are present. The derivation remains self-contained against external benchmarks such as the sphere case and general surface statements, with the functionals providing independent detection of unbounded orbits.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on standard background facts from differential geometry together with the successful construction of a new class of functionals; no free parameters or invented physical entities appear.

axioms (1)
  • standard math Compact oriented surfaces admit well-defined groups of area-preserving diffeomorphisms and the L^p metric is a norm on this group.
    These are standard facts from differential geometry and functional analysis invoked throughout the setting of the result.
invented entities (1)
  • New class of functionals on normed groups no independent evidence
    purpose: To detect large distances and prove infinite diameter by generalizing quasi-morphisms.
    These functionals are introduced in the methods section of the abstract as the key tool beyond existing techniques.

pith-pipeline@v0.9.0 · 5610 in / 1359 out tokens · 68786 ms · 2026-05-18T17:18:11.662383+00:00 · methodology

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Forward citations

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