AI 中文总结
本文针对 $\mathbb{C}^4$ 中基于 Schwarz P/D 族的亏格三超椭圆全纯零曲线族，显式计算四阶自同构对 Weierstrass 1-形式的作用，导出周期向量恒等式，实现周期问题的对称约化，并确定非退化形变条件。
AI 中文摘要
我们研究了 $\mathbb{C}^4$ 中一个亏格三的超椭圆全纯零曲线族，该曲线族以经典 Schwarz P/D 族的代数数据为模型，其实部定义了到 $\mathbb{R}^4$ 的极小浸入。对于底层 Schwarz 曲线的四阶自同构 $\omega\mapsto i\omega$，我们显式计算了它对四个全纯 Weierstrass $1$-形式的作用，并推导出所有实周期向量所满足的恒等式。因此，对于在诱导目标旋转下不变的每个格，环面周期闭性只需在对称稳定的同调生成集中的每个轨道的一个代表元上检验即可。我们还精确地确定了附加参数何时产生非退化的余维二形变。本文并不声称构造了 $\mathbb{R}^4$ 中新的嵌入周期极小曲面，而是提供了与该族相关的周期问题的显式对称约化。
英文摘要
We study a genus-three hyperelliptic holomorphic null-curve family in $\mathbb{C}^4$, modelled on the algebraic data of the classical Schwarz P/D family, whose real parts define minimal immersions into $\mathbb{R}^4$. For the order-four automorphism $ω\mapsto iω$ of the underlying Schwarz curve, we compute explicitly its action on the four holomorphic Weierstrass $1$-forms and derive the resulting identities for all real period vectors. Consequently, for every lattice invariant under the induced target rotation, torus-period closure can be checked on one representative from each orbit of a symmetry-stable homology generating set. We also identify precisely when the additional parameter produces a nondegenerate codimension-two deformation. The paper does not claim the construction of a new embedded periodic minimal surface in $\mathbb{R}^4$; rather, it provides an explicit symmetry reduction of the period problem associated with this family.
Comments23 pages, 5 figures
