On a class of fully nonlinear equations and their applications in geometry
Bibliographic record
Abstract
In this thesis, we consider two problems in geometry, Weyl's embedding problem and a fully nonlinear flow.In the first part, we study the Weyl's embedding problem in general Riemannian manifolds.We use two different methods to establish the mean curvature estimate.In Chapter 2, we derive a global curvature estimate for immersed hypersurfaces in Riemannian manifolds using maximum principle.The estimate is valid for all dimensions as well as degenerate case.In Chapter 3, we derive an interior curvature estimate for (S 2 , g) embedded in Riemannian manifolds by adopting Heinz-Schulz's interior estimate.Together with the openness results, we obtain an isometric embedding of (S 2 , g) in certain Riemannian manifolds.Moreover, we extend the classical Weyl's embedding theorem in space form to the case that g C 2 with D 2 g Dini continuous.In the second part, we consider an inverse curvature flow in anti-de Sitter-Schwarzschild manifold.The flow is a generalization of Brendle-Hung-Wang's inverse mean curvature flow.We show that the flow exists for all time and the second fundamental form converges exponentially fast. iii ABR G Dans cette thse, nous considrons deux problmes en gomtrie, le problme de plongement de Weyl et un flot totalement nonlinaire.Dans la premiere partie, nous tudions le problme de Weyl dans les varits Riemanniennes en gnral.Nous utilisons deux mthodes pour tablir l'estimation de la courbure moyenne.En chapitre 2, nous drivons une estimation global pour les hypersurfaces immerges en utilisant le principe du maximum.Cette estimation est valide dans toutes les dimensions ainsi que dans le cas dgnr.Dans le chapitre 3, nous drivons une estimation pour la courbure intrieure de (S 2 , g) plong dans les varits Riemanniennes en adoptant une estimation intrieure de Heinz-Schulz.Avec les rsultats douverture, nous obtenons un plongement de (S 2 , g) dans certaines varits Riemanniennes.De plus, nous tendons la thorme classique de Weyl dans le cas o g C 2 avec D 2 g Dini continue.Dans la deuxime partie, nous considrons un flot de courbure inverse dans une varit anti-de-Sitter-Schwarzschild.Ce flot est une gnralisation du flot inverse de courbure moyenne de Brendle-Hung-Wang.Nous dmontrons l'existence du flot pour tout temps et la convergence exponentielle de la forme fondamentale.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.001 | 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".