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 ÉG ÉDans cette thèse, nous considérons deux problèmes en géométrie, le problème de plongement de Weyl et un flot totalement nonlinéaire.Dans la premiere partie, nous étudions le problème de Weyl dans les variétés Riemanniennes en général.Nous utilisons deux méthodes pour établir l'estimation de la courbure moyenne.En chapitre 2, nous dérivons une estimation global pour les hypersurfaces immergées en utilisant le principe du maximum.Cette estimation est valide dans toutes les dimensions ainsi que dans le cas dégénéré.Dans le chapitre 3, nous dérivons une estimation pour la courbure intérieure de (S 2 , g) plongé dans les variétés Riemanniennes en adoptant une estimation intérieure de Heinz-Schulz.Avec les résultats douverture, nous obtenons un plongement de (S 2 , g) dans certaines variétés Riemanniennes.De plus, nous étendons la théorème classique de Weyl dans le cas où g ∈ C 2 avec D 2 g Dini continue.Dans la deuxième partie, nous considérons un flot de courbure inverse dans une variété anti-de-Sitter-Schwarzschild.Ce flot est une généralisation du flot inverse de courbure moyenne de Brendle-Hung-Wang.Nous démontrons l'existence du flot pour tout temps et la convergence exponentielle de la forme fondamentale.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 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".