Parametric inequalities and Weyl law for the volume spectrum
Bibliographic record
Abstract
We show that the Weyl law for the volume spectrum in a compact Riemannian manifold conjectured by Gromov can be derived from parametric generalizations of two famous inequalities: the isoperimetric inequality and the coarea inequality.We prove two such generalizations in low dimensions and obtain the Weyl law for 1-cycles in 3-manifolds.We also give a new proof of the Almgren isomorphism theorem.35P20, 53A10, 53C23 1. Introduction 863 2. Approximation results for families of cycles 869 3. Homotopy classes of the space of cycles 883 4. Parametric coarea inequality 888 5. Parametric isoperimetric inequality 895 6. Proof of the Weyl law 899 References 901 1.1 Some applications of the Weyl law Irie, Marques and Neves [16] used the Weyl law to prove that for a generic Riemannian metric on an n-dimensional manifold, where 3 Ä n Ä 7, the union of embedded minimal hypersurfaces forms a dense set.Marques, Neves and Song [26] proved a stronger equidistribution property of minimal hypersurfaces; their proof is based on the idea that one can "differentiate" both sides of (1-1) with respect to some cleverly Geometry & Topology, Volume 29 (2025) Ã Ä .1 C C.Á; "//a.n;k/ Vol.M / k=n C c."; Á; M /.p 1=.n 1/ C p k=.n.n 1// /for a constant C."; Á/ with C."; Á/ !0 as Á and " tend to 0. As p ! 1 the second term goes to 0. Since Á and " can be chosen to be arbitrarily small, we conclude that a 2 Ä a.n; k/.This finishes the proof of Theorem 6.1.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.006 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.007 | 0.001 |
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.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".