Compact manifolds with fixed boundary and large Steklov eigenvalues
Bibliographic record
Abstract
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper M comma g right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(M,g)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a compact Riemannian manifold with boundary. Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="b greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi>b</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">b>0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the number of connected components of its boundary. For manifolds of dimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="greater-than-or-equal-to 3"> <mml:semantics> <mml:mrow> <mml:mo> ≥ </mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\geq 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , we prove that for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="j equals b plus 1"> <mml:semantics> <mml:mrow> <mml:mi>j</mml:mi> <mml:mo>=</mml:mo> <mml:mi>b</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">j=b+1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> it is possible to obtain an arbitrarily large Steklov eigenvalue <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma Subscript j Baseline left-parenthesis upper M comma e Superscript delta Baseline g right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi> σ </mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mi> δ </mml:mi> </mml:msup> <mml:mi>g</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\sigma _j(M,e^\delta g)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> using a conformal perturbation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="delta element-of upper C Superscript normal infinity Baseline left-parenthesis upper M right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi> δ </mml:mi> <mml:mo> ∈ </mml:mo> <mml:msup> <mml:mi>C</mml:mi> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\delta \in C^\infty (M)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which is supported in a thin neighbourhood of the boundary, with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="delta equals 0"> <mml:semantics> <mml:mrow> <mml:mi> δ </mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\delta =0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on the boundary. For <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="j less-than-or-equal-to b"> <mml:semantics> <mml:mrow> <mml:mi>j</mml:mi> <mml:mo> ≤ </mml:mo> <mml:mi>b</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">j\leq b</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , it is also possible to obtain arbitrarily large eigenvalues, but the conformal factor must spread throughout the interior of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In fact, when working in a fixed conformal class and for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="delta equals 0"> <mml:semantics> <mml:mrow> <mml:mi> δ </mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\delta =0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on the boundary, it is known that the volume of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper M comma e Superscript delta Baseline g right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mi> δ </mml:mi> </mml:msup> <mml:mi>g</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="app
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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 | 0.000 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| 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 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".