The Geometry and Dynamics of Spiral Minimal Products
Abstract
We study the spiral product Gγ(t,x,y)=(z1(t)f1(x),z2(t)f2(y)), which couples two spherical 𝒞-totally real immersed factors through a profile curve γ=(z1,z2)⊂𝕊3. Its minimality is governed by a weighted-geodesic system: Gγ is minimal precisely when both factors are minimal and γ is an unparametrized geodesic of |z1|2k1|z2|2k2g𝕊3. This flow is Liouville integrable and, when k1+k2>0, its phase map has an open dense full-rank locus. Consequently, closed profiles of arbitrarily large primitive order occur densely. For compact minimal factors, Routh reduction and Sturm oscillation give Ind(Gγ)≥ Ind(f1)+Ind(f2)+2mγ−3 for a closed oscillatory profile of primitive closing order mγ. On the contact level, a factor-adapted choice of profiles produces, from any prescribed pair of compact connected embedded special Legendrians, embedded special Legendrian products of every sufficiently large prime closing order. Finally, canonical finite horizontal lifts and Hopf projection give Delaunay-type minimal Lagrangians in complex projective spaces. For compact embedded inputs and ordinarily closed profiles, the primitive spherical quotient is embedded, while its projective quotient is embedded exactly when the reduced relative winding is one.
Related
BibTeX
@misc{li2026geometry,
title = {{The Geometry and Dynamics of Spiral Minimal Products}},
author = {Haizhong Li and Yongsheng Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.02370},
eprint = {2608.02370},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.02370},
}
RIS
TY - UNPB
TI - The Geometry and Dynamics of Spiral Minimal Products
AU - Haizhong Li
AU - Yongsheng Zhang
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2608.02370
UR - https://arxiv.org/abs/2608.02370
AB - We study the spiral product G_γ(t,x,y)=(z_1(t)f_1(x),z_2(t)f_2(y)), which couples two spherical 𝒞-totally real immersed factors through a profile curve γ=(z_1,z_2)⊂𝕊^3. Its minimality is governed by a weighted-geodesic system: G_γ is minimal precisely when both factors are minimal and γ is an unparametrized geodesic of |z_1|^2k_1|z_2|^2k_2g_𝕊^3. This flow is Liouville integrable and, when k_1+k_2>0, its phase map has an open dense full-rank locus. Consequently, closed profiles of arbitrarily large primitive order occur densely. For compact minimal factors, Routh reduction and Sturm oscillation give Ind(G_γ)≥ Ind(f_1)+Ind(f_2)+2m_γ−3 for a closed oscillatory profile of primitive closing order m_γ. On the contact level, a factor-adapted choice of profiles produces, from any prescribed pair of compact connected embedded special Legendrians, embedded special Legendrian products of every sufficiently large prime closing order. Finally, canonical finite horizontal lifts and Hopf projection give Delaunay-type minimal Lagrangians in complex projective spaces. For compact embedded inputs and ordinarily closed profiles, the primitive spherical quotient is embedded, while its projective quotient is embedded exactly when the reduced relative winding is one.
ER -