Screw motion surfaces in $\Bbb H\sp 2\times\Bbb R$ and $\Bbb S\sp 2\times\Bbb R$
Bibliographic record
Abstract
In this paper we study the geometry of constant mean curvature H-screw motion surfaces in H 2 × R and S 2 × R. We compute the Abresch-Rosenberg holomorphic quadratic differential Q.For instance, if the ambient space is H 2 × R, we derive that, given ℓ, if 0 < 4H 2 < 1, then there exists a complete H-screw motion surface with pitch ℓ immersed in H 2 × R, such that Q = 0.An analogous result holds if the ambient space is S 2 × R. We deduce a general non-parametric formula for the mean curvature H(ρ).When H is constant, we find a first integral and use it to get an explicit two parameter family of complete, embedded, simply connected, minimal screw motion surfaces in H 2 × R with pitch ℓ.If ℓ = 1, each such surface has Gaussian curvature K ≡ -1.We deduce that any two isometric screw motion minimal immersions in H 2 × R or S 2 × R are associate, i.e., the absolute values of their Hopf functions are the same.
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 0.000 |
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.
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".