The space of compact self-shrinking solutions to the Lagrangian mean curvature flow in ℂ2{\mathbb{C}^{2}}
Bibliographic record
Abstract
Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>F</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> F_{n} : (Σ, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>h</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> h_{n} ) <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo>→</m:mo> </m:math> \to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℂ</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> \mathbb{C}^{2} be a sequence of conformally immersed Lagrangian self-shrinkers with a uniform area upper bound to the mean curvature flow, and suppose that the sequence of metrics <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo>{</m:mo> </m:math> \{ <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>h</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> h_{n} <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo>}</m:mo> </m:math> \} converges smoothly to a Riemannian metric h . We show that a subsequence of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo>{</m:mo> </m:math> \{ <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>F</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> F_{n} <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo>}</m:mo> </m:math> \} converges smoothly to a branched conformally immersed Lagrangian self-shrinker <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>F</m:mi> <m:mi>∞</m:mi> </m:msub> </m:math> F_{\infty} : (Σ, h ) <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo>→</m:mo> </m:math> \to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℂ</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> \mathbb{C}^{2} . When the area bound is less than 16π, the limit <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>F</m:mi> <m:mi>∞</m:mi> </m:msub> </m:math> {F_{\infty}} is an embedded torus. When the genus of Σ is one, we can drop the assumption on convergence <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>h</m:mi> <m:mi>n</m:mi> </m:msub>
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.006 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.002 | 0.000 |
| Scholarly communication | 0.001 | 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".