AI 中文总结
本文研究通过剖面曲线耦合球面特征映射得到的调和映射，提出两种互补问题并利用均匀化与诺伊曼系统获得全局重建及闭合剖面族，证明固定度量下存在任意高阶调和映射。
AI 中文摘要
受螺旋极小积系统研究的启发，我们考虑通过$\nmathbb S^3$中的一条剖面曲线耦合两个球面特征映射而得到的调和映射，源空间上带有双重扭曲乘积度量。源空间的扭曲与剖面幅度无关。这产生了两个互补的问题：从给定的剖面或源数据重建相容的源度量，以及在源度量固定时寻找闭合剖面。对于任意给定的扭曲，约化拉格朗日量的均匀化给出一个强凸锥形芬斯勒度量，其有向图测地线恰好是调和剖面。在单位速度逆问题中，相位动量将重建简化为一个标量幅度方程。我们从给定的幅度、一个源扭曲或单调体积因子获得全局重建，以及周期族，其中在有限覆盖上闭合的剖面是稠密的。在固定度量问题中，时间变换给出一个自治的诺伊曼系统。一个固定的源度量然后承载连续族的闭合剖面和无穷多条闭合图测地线。对于固定的偶数$p,q\geq 2$，$\nmathbb S^1\times\nmathbb S^p\times\nmathbb S^q$上该诺伊曼类中的每个固定度量都支持度为$4m$（对每个$m\geq 1$）的调和映射到$\nmathbb S^{p+q+1}$。在匹配的常数扭曲情形下，这些映射简化为显式特征映射。
英文摘要
Motivated by the systematic study of spiral minimal products, we consider harmonic maps obtained by coupling two spherical eigenmaps through a profile curve in $\mathbb S^3$, 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 $p,q\geq 2$, every fixed metric in this Neumann class on $\mathbb S^1\times\mathbb S^p\times\mathbb S^q$ supports harmonic maps into $\mathbb S^{p+q+1}$ of degree $4m$ for every $m\geq 1$. In the matched constant-warp case, these maps reduce to explicit eigenmaps.
Comments22 pages, based on discussions since last fall and drafts last semester, part of appendices assisted by AI
