Minimal immersions of compact bordered Riemann surfaces with free boundary
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Abstract
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a complete, homogeneously regular Riemannian manifold of dim <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N greater-than-or-equal-to 3"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">N \geq 3</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="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a compact submanifold of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . 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 a compact orientable surface with boundary. We show that for any continuous <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f colon left-parenthesis normal upper Sigma comma partial-differential normal upper Sigma right-parenthesis right-arrow left-parenthesis upper N comma upper M right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi mathvariant="normal"> Σ </mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:mi mathvariant="normal"> Σ </mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo stretchy="false"> → </mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>N</mml:mi> <mml:mo>,</mml:mo> <mml:mi>M</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">f: \left ( \Sigma , \partial \Sigma \right ) \rightarrow \left ( N, M \right )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for which the induced homomorphism <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Subscript asterisk"> <mml:semantics> <mml:msub> <mml:mi>f</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> ∗ </mml:mo> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">f_{*}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on certain fundamental groups is injective, there exists a branched minimal immersion 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> solving the free boundary problem <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis normal upper Sigma comma partial-differential normal upper Sigma right-parenthesis right-arrow left-parenthesis upper N comma upper M right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi mathvariant="normal"> Σ </mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:mi mathvariant="normal"> Σ </mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo stretchy="false"> → </mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>N</mml:mi> <mml:mo>,</mml:mo> <mml:mi>M</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\left ( \Sigma , \partial \Sigma \right ) \rightarrow \left ( N, M \right )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and minimizing area among all maps which induce the same action on the fundamental groups as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Furthermore, under certain nonnegativity assumptions on the curvature of a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -manifold <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation>
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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) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.000 | 0.003 |
| Science and technology studies | 0.000 | 0.003 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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