Schwarz-Type Null Curves in $\mathbb{C}^4$: Symmetry and Period Reduction

Abstract

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 $\omega\mapsto i\omega$ 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.