Monotone homotopies and contracting discs on Riemannian surfaces
Bibliographic record
Abstract
A monotone homotopy is a homotopy composed of simple closed curves which are also pairwise disjoint. In this paper, we prove a “gluing” theorem for monotone homotopies; we show that two monotone homotopies which have appropriate overlap can be replaced by a single monotone homotopy. The ideas used to prove this theorem are used in [G. R. Chambers and Y. Liokumovich, Existence of minimal hypersurfaces in complete manifolds of finite volume, arXiv:1609.04058] to prove an analogous result for cycles, which forms a critical step in their proof of the existence of minimal surfaces in complete non-compact manifolds of finite volume. We also show that, if monotone homotopies exist, then fixed point contractions through short curves exist. In particular, suppose that [Formula: see text] is a simple closed curve of a Riemannian surface, and that there exists a monotone contraction which covers a disc which [Formula: see text] bounds consisting of curves of length [Formula: see text]. If [Formula: see text] and [Formula: see text], then there exists a homotopy that contracts [Formula: see text] to [Formula: see text] over loops that are based at [Formula: see text] and have length bounded by [Formula: see text], where [Formula: see text] is the diameter of the surface. If the surface is a disc, and if [Formula: see text] is the boundary of this disc, then this bound can be improved to [Formula: see text].
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot 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.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 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 teacher head, 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".