Harmonic mappings of an annulus, Nitsche conjecture and its generalizations
Bibliographic record
Abstract
\def\xxlong{\mathop{{\rm onto}\atop \longrightarrow}\nolimits} As long ago as 1962 Nitsche conjectured that a harmonic homeomorphism $h \colon \ A(r,R) \xxlong A(r_\ast, R_\ast)$ between planar annuli exists if and only if $ \frac{R_\ast}{r_\ast} \ge \frac{1}{2} \left(\frac{R} {r} + \frac{r}{R}\right)$. We prove this conjecture when the domain annulus is not too wide; explicitly, when $R \le e^{3/2} r$. We also treat the general annuli $A(r,R)$, $ 0 0$r$R\infty$, and obtain the sharp Nitsche bound under additional assumption that either $h$ or its normal derivative have vanishing average along the inner circle of $A(r,R)$. We consider the family of Jordan curves in $A(r_*,R_*)$ obtained as images under $h$ of concentric circles in $A(r,R)$. We refer to such family of Jordan curves as harmonic evolution of the inner boundary of $A(r,R) $. In the borderline case $ \frac{R_\ast}{r_\ast} = \frac{1}{2} \left(\frac{R}{r} + \frac{r}{R}\right)$ the evolution begins with zero speed. It will be shown, as a generalization of the Nitsche Conjecture, that harmonic evolution with positive initial speed results in greater ratio $\frac{R_\ast}{r_\ast}$ in the deformed (target) annulus. To every initial speed there corresponds an underlying differential operator which yields sharp lower bounds of $\frac{R_\ast}{r_\ast}$ in our generalization of the Nitsche Conjecture.
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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 | 0.001 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.001 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
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".