Constructing surfaces with first Steklov eigenvalue of arbitrarily large multiplicity
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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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 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.007 | 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 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".