Mean curvature in manifolds with Ricci curvature bounded from below
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Bibliographic record
Abstract
Let M be a compact Riemannian manifold of nonnegative Ricci curvature and \Sigma a compact embedded 2-sided minimal hypersurface in M . It is proved that there is a dichotomy: If \Sigma does not separate M then \Sigma is totally geodesic and M\setminus\Sigma is isometric to the Riemannian product \Sigma\times(a,b) , and if \Sigma separates M then the map i_*:\pi_1(\Sigma)\rightarrow \pi_1(M) induced by inclusion is surjective. This surjectivity is also proved for a compact 2-sided hypersurface with mean curvature H\geq(n-1)\sqrt{k} in a manifold of Ricci curvature Ric _M\geq-(n-1)k , k>0 , and for a free boundary minimal hypersurface in an n -dimensional manifold of nonnegative Ricci curvature with nonempty strictly convex boundary. As an application it is shown that a compact (n-1) -dimensional manifold N with the number of generators of \pi_1(N) < n-1 cannot be minimally embedded in the flat torus T^{n} .
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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 | 0.000 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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 it