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PreprintarXiv licencearXiv

Schwarz-Type Null Curves in ℂ4: Symmetry and Period Reduction

  • Erhan Güler
  • Magdalena Toda
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Abstract

We study a genus-three hyperelliptic holomorphic null-curve family in ℂ4, modelled on the algebraic data of the classical Schwarz P/D family, whose real parts define minimal immersions into ℝ4. For the order-four automorphism ω↦ 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 ℝ4; rather, it provides an explicit symmetry reduction of the period problem associated with this family.

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Cite

BibTeXRIS
BibTeX
@misc{guler2026schwarztype,
  title = {{Schwarz-Type Null Curves in $\mathbbC^4$: Symmetry and Period Reduction}},
  author = {Erhan Güler and Magdalena Toda},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.05509},
  eprint = {2609.05509},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.05509},
}
RIS
TY  - UNPB
TI  - Schwarz-Type Null Curves in ℂ^4: Symmetry and Period Reduction
AU  - Erhan Güler
AU  - Magdalena Toda
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.05509
UR  - https://arxiv.org/abs/2609.05509
AB  - We study a genus-three hyperelliptic holomorphic null-curve family in ℂ^4, modelled on the algebraic data of the classical Schwarz P/D family, whose real parts define minimal immersions into ℝ^4. For the order-four automorphism ω↦ 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 ℝ^4; rather, it provides an explicit symmetry reduction of the period problem associated with this family.
ER  -