论文标题
$ \ hat {\ mathbb {s}}}^{2,4} $中的多项式几乎复杂的曲线
Polynomial Almost-Complex Curves in $\hat{\mathbb{S}}^{2,4}$
论文作者
论文摘要
对于$ \ Mathbb {C} $ offine toda field方程的解决方案，相对于\ emph {polyenmial} holomorphic sextic差异$ q $ \ hat {\ mathbb {s}}}^{2,4} $。 $ \ Mathsf {einsf {einsf {einsf {einsf {einsf {2,3} $，带有$ \ mathsf {deg} Q + 6 $ 6 $ fertices。多边形$δ$满足\ emph {Annihilator属性}，它与$ \ Mathsf {g} _2'$ - 不变式iNDRIANT $ d_3：\ MATHSF {ein}^{ein}^{2,3} \ times \ times \ times \ Mathsf {ein} \ {0,1,2,3 \} $ on $ \ mathsf {ein}^{2,3} $。实际上，我们显示$ \ MATHSF {G} _2'= \ MATHSF {ISOM}（d_3）\ CAP \ MATHSF {diff}（\ Mathsf {ein}^^{2,3}）$。渐近边界定义了图$α：\ Mathsf {MS} _ {K} \ rightArow \ MathSf \ Mathsf {Mp} _ {K+6} $同态形态上的形象。我们还讨论了$ν_q$和相关的最小表面$ f_q：\ mathbb {c} \ rightarrow \ MathSf {g} _2'/k $在对称空间$ \ mathsf {g} _2'/k $中，显示了他们如何实现他们的相互态度fo $ \ nmath $ \ g gele/t}。在开始几何形状之前，我们证明了$ \ Mathbb {C {C} $ offinemial $ q \ in H^0（\ Mathcal {k} _ \ Mathbb {c phodymial $ q \ in $ \ mathbb {c} $ q \ $ \ mathbb {C} _ \ \ m athbbbb {c}^6）的完整（真实）解决方案的存在和独特性。
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 $q \in H^0(\mathcal{K}_\mathbb{C}^6)$.
