Constructing surfaces with first Steklov eigenvalue of arbitrarily large multiplicity
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Bibliographic record
Abstract
Abstract We construct surfaces with arbitrarily large multiplicity for their first nonzero Steklov eigenvalue. The proof is based on a technique by Burger and Colbois originally used to prove a similar result for the Laplacian spectrum. We start by constructing surfaces upper S Subscript p $S_p$ <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> with a specific subgroup of isometry upper G Subscript p Baseline colon equals upper Z Subscript p Baseline right normal factor semidirect product upper Z Subscript p Superscript asterisk $G_p:= \mathbb {Z}_p \rtimes \mathbb {Z}_p^*$ <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mrow> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:mstyle mathvariant="double-struck"> <mml:msub> <mml:mi>Z</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:mstyle> <mml:mo>⋊</mml:mo> <mml:mstyle mathvariant="double-struck"> <mml:msubsup> <mml:mi>Z</mml:mi> <mml:mi>p</mml:mi> <mml:mo>*</mml:mo> </mml:msubsup> </mml:mstyle> </mml:mrow> </mml:math> for each prime p . We do so by gluing surfaces with boundary following the structure of the Cayley graph of upper G Subscript p $G_p$ <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> . We then exploit the properties of upper G Subscript p $G_p$ <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> and upper S Subscript p $S_p$ <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> in order to show that an irreducible representation of high degree (depending on p ) acts on the eigenspace of functions associated with sigma 1 left parenthesis upper S Subscript p Baseline right parenthesis $\sigma _1(S_p)$ <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mrow> <mml:msub> <mml:mi>σ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy="false" form="prefix" fence="true">(</mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo stretchy="false" form="postfix" fence="true">)</mml:mo> </mml:mrow> </mml:math> , leading to the desired result.
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.009 | 0.001 |
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