Hofer geometry of $A_3$-configurations
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866908952131796992 |
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| author | Dawid, Adrian |
| author_facet | Dawid, Adrian |
| contents | Let $L_0,L_1,L_2 \subset M$ be exact Lagrangian spheres in a Liouville domain $M$ with $2c_1(M)=0$. If $L_0,L_1,L_2$ form an $A_3$-configuration, we show that $\mathscr{L}(L_0)$ and $\mathscr{L}(L_2)$ endowed with the Hofer metric contain quasi-isometric embeddings of $(\mathbb{R}^\infty, \|\cdot\|_\infty)$, i.e. infinite-dimensional quasi-flats. A corollary of the proof presented here establishes that $\text{Ham}_c(M)$ itself contains an infinite-dimensional quasi-flat. We also show that for a Dehn twist $τ: M \to M$ along $L_1$ the boundary depth of $CF(τ^{2\ell}(L_0), L')$ is unbounded in $L' \in \mathscr{L}(L_2)$ for any $\ell \in \mathbb{N}_0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_16773 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hofer geometry of $A_3$-configurations Dawid, Adrian Symplectic Geometry Let $L_0,L_1,L_2 \subset M$ be exact Lagrangian spheres in a Liouville domain $M$ with $2c_1(M)=0$. If $L_0,L_1,L_2$ form an $A_3$-configuration, we show that $\mathscr{L}(L_0)$ and $\mathscr{L}(L_2)$ endowed with the Hofer metric contain quasi-isometric embeddings of $(\mathbb{R}^\infty, \|\cdot\|_\infty)$, i.e. infinite-dimensional quasi-flats. A corollary of the proof presented here establishes that $\text{Ham}_c(M)$ itself contains an infinite-dimensional quasi-flat. We also show that for a Dehn twist $τ: M \to M$ along $L_1$ the boundary depth of $CF(τ^{2\ell}(L_0), L')$ is unbounded in $L' \in \mathscr{L}(L_2)$ for any $\ell \in \mathbb{N}_0$. |
| title | Hofer geometry of $A_3$-configurations |
| topic | Symplectic Geometry |
| url | https://arxiv.org/abs/2402.16773 |