Compactness and noncompactness of Yamabe-type problems on manifolds with boundary
Bibliographic record
Abstract
The study of semilinear partial differential equations has proven to be of great importance in the fields of Physics and Geometry.Solutions to such equations correspond in particular to standing waves in Schrdinger equations and metrics with constant curvature on Riemannian manifolds.In this manuscript, we obtain existence results of blowing-up solutions as well as compactness results to Yamabe-type equations on manifolds.In particular, in a joint work with Jrme Vtois, we construct blowing-up solutions to a Yamabe-type equation on the standard sphere with dimension n = 4.Then, we construct blowing-up solutions on the standard half-sphere in dimensions n 3. Finally, joint with Srgio Almaraz and Olivaine Queiroz, we prove a compactness theorem to the boundary Yamabe problem in the scalar flat case in dimension n = 3. iii ABR G L'tude des quations aux drives partielles semi-linaires s'est rvle d'une grande importance dans les domaines de la physique et de la gomtrie.Des solutions de telles quations correspondent en particulier des ondes stationnaires dans les quations de Schrdinger et des mtriques de courbure constante sur des varits riemanniennes.Dans ce manuscrit, nous obtenons des rsultats d'existence de solutions explosives ainsi que des rsultats de compacit pour des quations de type Yamabe sur les varits.En particulier, dans un travail en collaboration avec Jrme Vtois, nous construisons des solutions explosives pour une quation de type Yamabe sur la sphre standard en dimension n = 4. Ensuite, nous construisons des solutions explosives sur la demi-sphre standard en dimension n 3. Finalement, en collaboration avec Srgio Almaraz et Olivaine Queiroz, nous prouvons un thorme de compacit pour le problme de Yamabe bord dans le cas courbure scalaire nulle en dimension n = 3. iv
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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 | 0.000 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 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".