Spiral Harmonic Products

@misc{zhang2026spiral,
  title = {{Spiral Harmonic Products}},
  author = {Hai Zhang and Yongsheng Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.32191},
  eprint = {2609.32191},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.32191},
}

TY  - UNPB
TI  - Spiral Harmonic Products
AU  - Hai Zhang
AU  - Yongsheng Zhang
PY  - 2026
DA  - 2026/09/26
PB  - arXiv
DO  - 10.48550/arxiv.2609.32191
UR  - https://arxiv.org/abs/2609.32191
AB  - Motivated by the systematic study of spiral minimal products, we consider harmonic maps obtained by coupling two spherical eigenmaps through a profile curve in 𝕊^3, with a doubly warped product metric on the source. The source warps are independent of the profile magnitudes. This gives two complementary problems: reconstructing a compatible source metric from prescribed profile or source data, and finding closed profiles when the source metric is fixed. For arbitrary prescribed warps, homogenization of the reduced Lagrangian gives a strongly convex conic Finsler metric whose oriented graph geodesics are exactly the harmonic profiles. In the unit-speed inverse problem, the phase momenta reduce reconstruction to a scalar magnitude equation. We obtain global reconstructions from a prescribed magnitude, one source warp, or a monotone volume factor, together with periodic families in which profiles closing on finite covers are dense. In the fixed-metric problem, a change of time gives an autonomous Neumann system. One fixed source metric then carries continuous families of closed profiles and infinitely many closed graph geodesics. For fixed even p,q≥ 2, every fixed metric in this Neumann class on 𝕊^1×𝕊^p×𝕊^q supports harmonic maps into 𝕊^p+q+1 of degree 4m for every m≥ 1. In the matched constant-warp case, these maps reduce to explicit eigenmaps.
ER  -