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Pré-Publication, Document De Travail Année : 2022

Algebraic intersection in regular polygons

Résumé

We study the function $$\mbox{KVol} : (X,\omega)\mapsto \mbox{Vol} (X,\omega) \sup_{\alpha,\beta} \frac{\mbox{Int} (\alpha,\beta)}{l_g (\alpha) l_g (\beta)}$$ defined on the moduli spaces of translation surfaces. More precisely, let $\mathcal T_n$ be the Teichm\"uller discs of the original Veech surface $(X_n,\omega_n)$ arising from right-angled triangle with angles $(\pi/2,\pi/n,(n-2)\pi/2n)$ by the unfolding construction for $n\geq 5$. For $n \equiv 1 \mod 2$ and any $(X,\omega)\in \mathcal T_n$, we establish the (sharp) bounds $$ \frac{n}{2} \cot \frac{\pi}{n} \leq \mbox{KVol}(X,\omega) \leq \frac{n}{2} \cot \frac{\pi}{n} \cdot \frac1{\sin \frac{2\pi}{n}}.$$ The lower bound is uniquely realized at $(X_n,\omega_n)$.

Dates et versions

hal-03909001 , version 1 (21-12-2022)

Identifiants

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Julien Boulanger, Erwan Lanneau, Daniel Massart. Algebraic intersection in regular polygons. 2022. ⟨hal-03909001⟩
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