Polynomial Almost-Complex Curves in $\hat{\mathbb{S}}^{2,4}$
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866913326445887488 |
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| author | Evans, Parker |
| author_facet | Evans, Parker |
| contents | For solutions to the $\mathfrak{g}_2$ affine Toda field equations in $\mathbb{C}$ with respect to \emph{polynomial} holomorphic sextic differential $q$, we study the associated almost-complex curves $ν_q: \mathbb{C} \rightarrow \hat{\mathbb{S}}^{2,4}$. The asymptotic boundary $Δ:= \partial_{\infty}(ν_q)$ of $ν_q$ is found to be a polygon in $\mathsf{Ein}^{2,3}$ with $\mathsf{deg} q + 6$ vertices. The polygon $Δ$ satisfies an \emph{annihilator property}, which is related to a $\mathsf{G}_2'$-invariant discrete metric $d_3: \mathsf{Ein}^{2,3} \times \mathsf{Ein}^{2,3} \rightarrow \{0,1,2,3\}$ on $\mathsf{Ein}^{2,3}$. In fact, we show $\mathsf{G}_2' = \mathsf{Isom}(d_3) \cap \mathsf{Diff}(\mathsf{Ein}^{2,3})$. The asymptotic boundary defines a map $α: \mathsf{MS}_{k} \rightarrow \mathsf{MP}_{k+6}$ between the equidimensional moduli spaces of holomorphic polynomial sextic differentials of degree $k$ and of annihilator polygons with $k+6$ vertices and is conjectured to be a homeomorphism onto its image. We also discuss the relationship between $ν_q$ and a related minimal surface $f_q: \mathbb{C} \rightarrow \mathsf{G}_2'/K$ in the symmetric space $\mathsf{G}_2'/K$, showing how to realize their mutual harmonic lift to $\mathsf{G}_2'/T$ geometrically. Before beginning the geometry, we prove the existence and uniqueness of a complete (real) solution to the $\mathfrak{g}_2$ affine Toda field equations in $\mathbb{C}$ associated to polynomial $q \in H^0(\mathcal{K}_\mathbb{C}^6)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_14409 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Polynomial Almost-Complex Curves in $\hat{\mathbb{S}}^{2,4}$ Evans, Parker Differential Geometry Analysis of PDEs Primary: 53C43. Secondary: 53C26, 53C38 For solutions to the $\mathfrak{g}_2$ affine Toda field equations in $\mathbb{C}$ with respect to \emph{polynomial} holomorphic sextic differential $q$, we study the associated almost-complex curves $ν_q: \mathbb{C} \rightarrow \hat{\mathbb{S}}^{2,4}$. The asymptotic boundary $Δ:= \partial_{\infty}(ν_q)$ of $ν_q$ is found to be a polygon in $\mathsf{Ein}^{2,3}$ with $\mathsf{deg} q + 6$ vertices. The polygon $Δ$ satisfies an \emph{annihilator property}, which is related to a $\mathsf{G}_2'$-invariant discrete metric $d_3: \mathsf{Ein}^{2,3} \times \mathsf{Ein}^{2,3} \rightarrow \{0,1,2,3\}$ on $\mathsf{Ein}^{2,3}$. In fact, we show $\mathsf{G}_2' = \mathsf{Isom}(d_3) \cap \mathsf{Diff}(\mathsf{Ein}^{2,3})$. The asymptotic boundary defines a map $α: \mathsf{MS}_{k} \rightarrow \mathsf{MP}_{k+6}$ between the equidimensional moduli spaces of holomorphic polynomial sextic differentials of degree $k$ and of annihilator polygons with $k+6$ vertices and is conjectured to be a homeomorphism onto its image. We also discuss the relationship between $ν_q$ and a related minimal surface $f_q: \mathbb{C} \rightarrow \mathsf{G}_2'/K$ in the symmetric space $\mathsf{G}_2'/K$, showing how to realize their mutual harmonic lift to $\mathsf{G}_2'/T$ geometrically. Before beginning the geometry, we prove the existence and uniqueness of a complete (real) solution to the $\mathfrak{g}_2$ affine Toda field equations in $\mathbb{C}$ associated to polynomial $q \in H^0(\mathcal{K}_\mathbb{C}^6)$. |
| title | Polynomial Almost-Complex Curves in $\hat{\mathbb{S}}^{2,4}$ |
| topic | Differential Geometry Analysis of PDEs Primary: 53C43. Secondary: 53C26, 53C38 |
| url | https://arxiv.org/abs/2208.14409 |