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Record W1664104325 · doi:10.4310/jsg.2017.v15.n4.a7

The Hofer norm of a contactomorphism

2017· preprint· en· W1664104325 on OpenAlex

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affAt least one author lists a Canadian institution in the pinned OpenAlex snapshot.

Bibliographic record

VenueJournal of Symplectic Geometry · 2017
Typepreprint
Languageen
FieldMathematics
TopicGeometric and Algebraic Topology
Canadian institutionsUniversité de Montréal
Fundersnot available
KeywordsMathematicsInvariant (physics)Pure mathematicsConjectureNorm (philosophy)Degenerate energy levelsMetric (unit)CombinatoricsMathematical physicsPhysicsQuantum mechanics

Abstract

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We show that the $L^{\infty}$-norm of the contact Hamiltonian induces a non-degenerate right-invariant metric on the group of contactomorphisms of any closed contact manifold. This contact Hofer metric is not left-invariant, but rather depends naturally on the choice of a contact form $α,$ whence its restriction to the subgroup of $α$-strict contactomorphisms is bi-invariant. The non-degeneracy of this metric follows from an analogue of the energy-capacity inequality. We show furthermore that this metric has infinite diameter in a number of cases by investigating its relations to previously defined metrics on the group of contact diffeomorphisms. We study its relation to Hofer's metric on the group of Hamiltonian diffeomorphisms, in the case of prequantization spaces. We further consider the distance in this metric to the Reeb one-parameter subgroup, which yields an intrinsic formulation of a small-energy case of Sandon's conjecture on the translated points of a contactomorphism. We prove this Chekanov-type statement for contact manifolds admitting a strong exact filling.

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Full frame distilled prediction

Teacher imitation

Not calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.

metaresearch head score (Codex)0.003
metaresearch head score (Gemma)0.011
Version: codex-gemma-dda1882f352aValidation status: machine_predicted_unvalidated
Candidate categoriesMetaresearch
Consensus categoriesnone
DomainCandidate signal: none · Consensus signal: none
Study designCandidate signal: Theoretical or conceptual · Consensus signal: Theoretical or conceptual
GenreCandidate signal: Empirical · Consensus signal: Empirical
Teacher disagreement score0.449
Threshold uncertainty score0.998

Codex and Gemma teacher scores by category

CategoryCodexGemma
Metaresearch0.0030.011
Meta-epidemiology (narrow)0.0000.000
Meta-epidemiology (broad)0.0010.001
Bibliometrics0.0010.000
Science and technology studies0.0000.000
Scholarly communication0.0000.000
Open science0.0020.001
Research integrity0.0000.002
Insufficient payload (model declined to judge)0.0000.000

Machine scores (provisional)

The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.

Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.

Opus teacher head0.067
GPT teacher head0.350
Teacher spread0.283 · how far apart the two teachers sit on this one work
Validation statusscore_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it