A special class of $k$-harmonic maps inducing calibrated fibrations
Bibliographic record
Abstract
We consider two special classes of $k$-harmonic maps between Riemannian manifolds which are related to calibrated geometry, satisfying a first order fully nonlinear PDE. The first is a special type of weakly conformal map $u \colon (L^k, g) \to (M^n, h)$ where $k \leq n$ and $α$ is a calibration $k$-form on $M$. Away from the critical set, the image is an $α$-calibrated submanifold of $M$. These were previously studied by Cheng-Karigiannis-Madnick when $α$ was associated to a vector cross product, but we clarify that such a restriction is unnecessary. The second, which is new, is a special type of weakly horizontally conformal map $u \colon (M^n, h) \to (L^k, g)$ where $n \geq k$ and $α$ is a calibration $(n-k)$-form on $M$. Away from the critical set, the fibres $u^{-1} \{ u(x) \}$ are $α$-calibrated submanifolds of $M$. We also review some previously established analytic results for the first class; we exhibit some explicit noncompact examples of the second class, where $(M, h)$ are the Bryant-Salamon manifolds with exceptional holonomy; we remark on the relevance of this new PDE to the Strominger-Yau-Zaslow conjecture for mirror symmetry in terms of special Lagrangian fibrations and to the $\mathrm{G}_2$ version by Gukov-Yau-Zaslow in terms of coassociative fibrations; and we present several open questions for future study.
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.000 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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 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".