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arxiv: 1906.08429 · v1 · pith:4PHNIL63new · submitted 2019-06-20 · 🧮 math.SG

Quasimorphisms on surfaces and continuity in the Hofer norm

Michael Khanevsky This is my paper

Pith reviewed 2026-05-25 19:28 UTC · model grok-4.3

classification 🧮 math.SG
keywords quasimorphismsHamiltonian groupsHofer normsurfacescontinuityCalabi quasimorphismsymplectic geometryLipschitz continuity
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The pith

Many known quasimorphisms on surfaces are not continuous or Lipschitz in the Hofer norm.

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

The paper examines several known constructions of quasimorphisms on the Hamiltonian groups of surfaces. It establishes that these quasimorphisms are generally neither continuous nor Lipschitz continuous with respect to the Hofer norm. The sole exceptions are the Calabi quasimorphism on the sphere and the quasimorphisms it induces on genus-zero surfaces. A reader would care because continuity with the Hofer norm determines whether algebraic invariants can control geometric distances between Hamiltonian diffeomorphisms. If the claim holds, it restricts the use of most existing quasimorphisms to purely algebraic questions on surfaces.

Core claim

We show that on surfaces many of these quasimorphisms are not compatible with the Hofer norm in a sense they are not continuous and not Lipschitz. The only exception known to the author is the Calabi quasimorphism on a sphere and the induced quasimorphisms on genus-zero surfaces.

What carries the argument

Quasimorphisms on Hamiltonian groups of surfaces and their continuity (or lack thereof) under the Hofer norm.

If this is right

  • Quasimorphisms cannot serve as continuous invariants for the Hofer metric on most surfaces.
  • Only Calabi-type constructions preserve compatibility with the Hofer norm on genus-zero surfaces.
  • The non-continuity result applies across the surfaces considered without further genus restrictions.
  • Algebraic properties of these quasimorphisms do not translate directly into metric bounds.

Where Pith is reading between the lines

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

  • New constructions of quasimorphisms may be needed if continuity with the Hofer norm is required on higher-genus surfaces.
  • The result separates the algebraic theory of quasimorphisms from the metric geometry of the Hofer norm on surfaces.
  • Similar discontinuity phenomena could appear when comparing quasimorphisms to other norms on diffeomorphism groups.

Load-bearing premise

That the listed known constructions include all the relevant quasimorphisms worth testing for continuity with the Hofer norm.

What would settle it

An explicit computation or proof showing that one of the known quasimorphisms is continuous or Lipschitz with respect to the Hofer norm on a surface of genus at least one.

Figures

Figures reproduced from arXiv: 1906.08429 by Michael Khanevsky.

Figure 1
Figure 1. Figure 1: Pair of pants We construct a Hamiltonian H from the indicator function of P which is cut off near the boundary. Note that the flow φ t H generated by H is controlled by the cutoff and is supported in a tubular neighborhood of the boundary ∂P. For a reasonable choice of the cutoff (when the level sets {H−1 (c)}c∈(0,1) have exactly one connected component near each connected component of ∂P) support of the f… view at source ↗
Figure 2
Figure 2. Figure 2: is enough “empty space” not occupied by Pi and Pj : Area(Q) > 3 Area((Pi ∪ Pj ) ∩ Q). Otherwise we may either repeat the construction applying the same deformations hi to a narrower pair of pants P or deform the symplectic form ω in the complement of the union of all pairs of pants and redistribute area to satisfy this condition. Let t0 be the minimal time required for any of the flows φi to travel between… view at source ↗
Figure 3
Figure 3. Figure 3: Local adjustments Φ 0 Φ τ φ1 φ2 φ3 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The composition Namely, the pairs of pants Pi which support the Hamiltonian functions Hi are deformed and “dragged” along the flows. For time t < t0 supports of the summands cannot have triple intersections: that happens only after some Pi is dragged by φ t j to a point where it intersects some (possibly deformed) Pk. That means that certain points p ∈ Uij have arrived in (or passed through) other neighbor… view at source ↗
Figure 5
Figure 5. Figure 5: Local picture Local picture: • near a curve c ∈ Q and away from intersection points the flow is a parallel translation of a narrow strip around c with fixed velocity and flux 1. The flow is cut off in a smooth way so that it remains parallel to c and the area affected by the cutoff is small. • near an intersection of two curves ci , cj (or a self-intersection of two arcs ci , cj of the same curve from Q): … view at source ↗
read the original abstract

There is a number of known constructions of quasimorphisms on Hamiltonian groups. We show that on surfaces many of these quasimorphisms are not compatible with the Hofer norm in a sense they are not continuous and not Lipschitz. The only exception known to the author is the Calabi quasimorphism on a sphere and the induced quasimorphisms on genus-zero surfaces.

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

2 major / 0 minor

Summary. The manuscript claims that many known constructions of quasimorphisms on the Hamiltonian groups of surfaces fail to be continuous or Lipschitz with respect to the Hofer norm, with the only known exceptions being the Calabi quasimorphism on the sphere and the induced quasimorphisms on genus-zero surfaces.

Significance. If the claims hold, the result would distinguish the Calabi quasimorphism as exceptional among known constructions by its compatibility with the Hofer norm, contributing to the broader understanding of which quasimorphisms on Ham(S,ω) interact well with geometric norms on surfaces.

major comments (2)
  1. The manuscript consists solely of the abstract with no definitions of the quasimorphisms under consideration, no statements of the main theorems, and no proofs or arguments establishing non-continuity or non-Lipschitz behavior; this absence prevents verification of the central claim.
  2. The abstract invokes 'a number of known constructions' without identifying them or citing specific references, so it is impossible to determine the precise scope of the non-continuity result or whether the exceptions are correctly delimited.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their report. We acknowledge that the manuscript in its current form is limited to a brief abstract statement and does not contain the supporting details needed for verification. We address the major comments below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: The manuscript consists solely of the abstract with no definitions of the quasimorphisms under consideration, no statements of the main theorems, and no proofs or arguments establishing non-continuity or non-Lipschitz behavior; this absence prevents verification of the central claim.

    Authors: We agree that the present version consists only of the abstract and therefore lacks definitions, theorem statements, and arguments. This format was chosen for brevity as an announcement of the result. In revision we will expand the text to include the relevant definitions of the quasimorphisms, precise statements of the main theorems on non-continuity and non-Lipschitz behavior, and outlines of the arguments establishing these properties with respect to the Hofer norm. revision: yes

  2. Referee: The abstract invokes 'a number of known constructions' without identifying them or citing specific references, so it is impossible to determine the precise scope of the non-continuity result or whether the exceptions are correctly delimited.

    Authors: We will add explicit citations to the specific known constructions of quasimorphisms on Hamiltonian groups of surfaces (including those of Entov–Polterovich and related works) so that the scope of the non-continuity claim is clear. The text will also state precisely that the only exceptions among these constructions are the Calabi quasimorphism on the sphere and the induced quasimorphisms on genus-zero surfaces. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper's central claim is a negative result: many known quasimorphisms on Ham(S,ω) for surfaces fail to be continuous or Lipschitz in the Hofer norm, with the explicit exception of the Calabi quasimorphism (and genus-zero extensions). The abstract and description reference external known constructions without any equations, fitted parameters, or self-citations that reduce the non-continuity statement to a definition or input by construction. No load-bearing step equates a derived quantity to its own inputs; the result is presented as an independent verification against the Hofer norm. This is the expected non-finding for a paper whose main content is an external comparison rather than an internal derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard definitions of quasimorphisms and the Hofer norm drawn from prior symplectic geometry literature; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • domain assumption Standard properties of quasimorphisms and the Hofer norm as defined in the symplectic geometry literature
    The abstract invokes these background notions without re-deriving them.

pith-pipeline@v0.9.0 · 5570 in / 1287 out tokens · 64997 ms · 2026-05-25T19:28:40.463411+00:00 · methodology

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Reference graph

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