Algebraic intersection in regular polygons
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866917844465221632 |
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| author | Boulanger, Julien Lanneau, Erwan Massart, Daniel |
| author_facet | Boulanger, Julien Lanneau, Erwan Massart, Daniel |
| contents | We study the function $$\mbox{KVol} : (X,ω)\mapsto \mbox{Vol} (X,ω) \sup_{α,β} \frac{\mbox{Int} (α,β)}{l_g (α) l_g (β)}$$
defined on the moduli spaces of translation surfaces. More precisely, let $\mathcal T_n$ be the Teichmüller discs of the original Veech surface $(X_n,ω_n)$ arising from right-angled triangle with angles $(π/2,π/n,(n-2)π/2n)$ by the unfolding construction for $n\geq 5$. For $n \equiv 1 \mod 2$ and any $(X,ω)\in \mathcal T_n$, we establish the (sharp) bounds $$ \frac{n}{2} \cot \fracπ{n} \leq \mbox{KVol}(X,ω) \leq \frac{n}{2} \cot \fracπ{n} \cdot \frac1{\sin \frac{2π}{n}}.$$
The lower bound is uniquely realized at $(X_n,ω_n)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_14235 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Algebraic intersection in regular polygons Boulanger, Julien Lanneau, Erwan Massart, Daniel Dynamical Systems Differential Geometry 37D40, 32G15, 53C22 We study the function $$\mbox{KVol} : (X,ω)\mapsto \mbox{Vol} (X,ω) \sup_{α,β} \frac{\mbox{Int} (α,β)}{l_g (α) l_g (β)}$$ defined on the moduli spaces of translation surfaces. More precisely, let $\mathcal T_n$ be the Teichmüller discs of the original Veech surface $(X_n,ω_n)$ arising from right-angled triangle with angles $(π/2,π/n,(n-2)π/2n)$ by the unfolding construction for $n\geq 5$. For $n \equiv 1 \mod 2$ and any $(X,ω)\in \mathcal T_n$, we establish the (sharp) bounds $$ \frac{n}{2} \cot \fracπ{n} \leq \mbox{KVol}(X,ω) \leq \frac{n}{2} \cot \fracπ{n} \cdot \frac1{\sin \frac{2π}{n}}.$$ The lower bound is uniquely realized at $(X_n,ω_n)$. |
| title | Algebraic intersection in regular polygons |
| topic | Dynamical Systems Differential Geometry 37D40, 32G15, 53C22 |
| url | https://arxiv.org/abs/2110.14235 |