Faber and Grunsky operators corresponding to bordered Riemann surfaces
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Bibliographic record
Abstract
Let<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper R"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="fraktur">R</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathfrak {R}</mml:annotation></mml:semantics></mml:math></inline-formula>be a compact Riemann surface of finite genus<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German g greater-than 0"><mml:semantics><mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="fraktur">g</mml:mi></mml:mrow><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:annotation encoding="application/x-tex">\mathfrak {g}>0</mml:annotation></mml:semantics></mml:math></inline-formula>and let<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma"><mml:semantics><mml:mi mathvariant="normal">Σ</mml:mi><mml:annotation encoding="application/x-tex">\Sigma</mml:annotation></mml:semantics></mml:math></inline-formula>be the subsurface obtained by removing<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n greater-than-or-equal-to 1"><mml:semantics><mml:mrow><mml:mi>n</mml:mi><mml:mo>≥</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:annotation encoding="application/x-tex">n\geq 1</mml:annotation></mml:semantics></mml:math></inline-formula>simply connected regions<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega 1 Superscript plus Baseline comma ellipsis comma normal upper Omega Subscript n Superscript plus"><mml:semantics><mml:mrow><mml:msubsup><mml:mi mathvariant="normal">Ω</mml:mi><mml:mn>1</mml:mn><mml:mo>+</mml:mo></mml:msubsup><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msubsup><mml:mi mathvariant="normal">Ω</mml:mi><mml:mi>n</mml:mi><mml:mo>+</mml:mo></mml:msubsup></mml:mrow><mml:annotation encoding="application/x-tex">\Omega _1^+, \dots , \Omega _n^+</mml:annotation></mml:semantics></mml:math></inline-formula>from<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper R"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="fraktur">R</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathfrak {R}</mml:annotation></mml:semantics></mml:math></inline-formula>with non-overlapping closures. Fix a biholomorphism<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Subscript k"><mml:semantics><mml:msub><mml:mi>f</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:annotation encoding="application/x-tex">f_k</mml:annotation></mml:semantics></mml:math></inline-formula>from the unit disc onto<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega Subscript k Superscript plus"><mml:semantics><mml:msubsup><mml:mi mathvariant="normal">Ω</mml:mi><mml:mi>k</mml:mi><mml:mo>+</mml:mo></mml:msubsup><mml:annotation encoding="application/x-tex">\Omega _k^+</mml:annotation></mml:semantics></mml:math></inline-formula>for each<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"><mml:semantics><mml:mi>k</mml:mi><mml:annotation encoding="application/x-tex">k</mml:annotation></mml:semantics></mml:math></inline-formula>and let<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold f equals left-parenthesis f 1 comma ellipsis comma f Subscript n Baseline right-parenthesis"><mml:semantics><mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="bold">f</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">\mathbf {f}=(f_1, \dots , f_n)</mml:annotation></mml:semantics></mml:math></inline-formula>. We assign a Faber and a Grunsky operator to<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper R"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="fraktur">R</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathfrak {R}</mml:annotation></mml:semantics></mml:math></inline-formula>and<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold f"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="bold">f</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathbf {f}</mml:annotation></mml:semantics></mml:math></inline-formula>when all the boundary curves of<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma"><mml:semantics><mml:mi mathvariant="normal">Σ</mml:mi><mml:annotation encoding="application/x-tex">\Sigma</mml:annotation></mml:semantics></mml:math></inline-formula>are quasicircles in<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper R"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="fraktur">R</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathfrak {R}</mml:annotation></mml:semantics></mml:math></inline-formula>. We show that the Faber operator is a bounded isomorphism and the norm of the Grunsky operator is strictly less than one for this choice of boundary curves. A characterization of the pull-back of the holomorphic Dirichlet space of<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma"><mml:semantics><mml:mi mathvariant="normal">Σ</mml:mi><mml:annotation encoding="application/x-tex">\Sigma</mml:annotation></mml:semantics></mml:math></inline-formula>in terms of the graph of the Grunsky operator is provided.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| 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