Isoperimetric inequalities for Laplace and Steklov eigenvalues
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Bibliographic record
Abstract
Isoperimetric inequalities for eigenvalues of geometric operators have received a lot of attention in recent years. Such inequalities are of particular interest for the eigenvalues of Laplace and Dirichlet-to-Neumann operators on Riemannian manifolds, where they exhibit a surprising connection to the theory of minimal submanifolds. This bridge between analysis and geometry provides a path to solving problems from both fields. In the present manuscript we apply geometric methods to prove several new isoperimetric inequalities. In particular, we obtain an isoperimetric inequality for the first Laplace eigenvalue on non-orientable surfaces, improving upon results of P. Li and S.-T. Yau. We prove an isoperimetric inequality for all Steklov eigenvalues on orientable surfaces, improving upon results of A. Girouard and I. Polterovich. We also provide a high-dimensional analog of the latter inequality by considering Dirichlet-to-Neumann operators on differential forms. Finally, jointly with N. Nadirashvili, A. Penskoi and I. Polterovich we establish a sharp inequality for all Laplace eigenvalues on a two-dimensional sphere settling the conjecture of Nadirashvili proposed in 2002.
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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 | 0.003 | 0.010 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 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 it